NIKOLAI Е. KOROTKOV
Edited by ALEXANDER N. KOROTKOV
TABLE OF INTEGRALS
RELATED TO ERROR FUNCTION
2019
i Table of integrals related to error function by Nikolai E. Korotkov Retired Leading Researcher, Voronezh Institute of Communications, Russia edited by Alexander N. Korotkov Professor, University of California, Riverside, USA
2019 Nikolai E. Korotkov and Alexander N. Korotkov
ii Preface
This book presents a table of integrals related to the error function, including indefinite and improper definite integrals. Since many tables of integrals have been published previously and, moreover, computers are widely used nowadays to find integrals numerically and analytically, a natural question is why such a new table would be useful. There are at least three reasons for that. First, to the best of our knowledge, this is the first book (except Russian versions of essentially the same book), which presents a comprehensive collection of integrals related to the error function. Most of the formulas in this book have not been presented in other tables of integrals or have been presented only for some special cases of parameters or for integration only along the real axis of the complex plane. Second, many of the integrals presented here cannot be obtained using a computer (except via an approximate numerical integration). Third, for improper integrals, this book emphasizes the necessary and sufficient conditions for the validity of the presented formulas, including the trajectory for going to infinity on the complex plane; such conditions are usually not given in computer-assisted analytical integration and often not presented in the previously published tables of integrals. We hope that this book will be useful to researchers whose work involves the error function (e.g., via probability integrals in communication theory). It can also be useful to a broader audience.
Alexander N. Korotkov and Nikolai E. Korotkov
iii CONTENTS
Notations and definitions ...... 1
Introduction ...... 2
Part 1. Indefinite integrals ...... 5
1.1. Integrals of the form ∫ z n exp [ (α z + β)2 ] dz ...... 5 1.2. Integrals of the form ∫ zn exp ( α22 z +β z +γ) dz ...... 7
1.3. Integrals of the form ∫ erf n (α z + β) exp[ −(α z + β)2 ] dz ...... 8
1.4. Integrals of the form ∫ z n erf (α z + β) exp[ −(α z + β)2 ] dz ...... 8
1.5. Integrals of the form z n erf (α z + β) exp (β z + γ ) dz ...... ∫ 1 10 + 1.6. Integrals of the form ∫ z 2m 1 erf (α z + β) exp( α 2 z 2 + γ ) dz ...... 11
2m+1 2 1.7. Integrals of the form ∫ z erf (αz + β)exp(α1 z + γ ) dz ...... 12
1.8. Integrals of the form ∫ z n erf 2 (α z )exp ( α 2 z 2 )dz ...... 14 1.9. Integrals of the form ∫ z n erf (α z + β) d z ...... 15 1.10. Integrals of the form ∫ z nerf 2 (α z + β) dz ...... 15
2m 1.11. Integrals of the form ∫ z erf (α z ) erf (α1z ) dz ...... 17 1.12. Integrals of the form ∫ z 2 m+1erf 3 (α z ) dz ...... 18
n m 2 2 1.13. Integrals of the forms ∫ z sin (α z + β z + γ ) dz, ∫ znmsinh (α22 z++ βγ z) dz,
n m α 2 2 + β + γ ∫ z cos ( z z ) dz , ∫ znmcosh (α22 z++ βγ z) dz ...... 19 n 2 2 1.14. Integrals of the form ∫ z sin (α z + β z + γ )exp (β1 z ) dz ...... 27
n 2 2 1.15. Integrals of the form ∫ z exp (− α z + β z ) sin (β1 z + γ ) dz ...... 30
iv n −α22 +β α +β +γ 1.16. Integrals of the form ∫ zexp ( z z) sin( 11 z z) dz ...... 33
1.17. Integrals of the form n αβ++ β βγ ...... ∫ zerf( z ) exp( 12z) sin ( z) dz 36
1.18. Integrals of the form 21n+ αβ++ α2 α 2 γ ...... ∫ zerf( z ) exp( 12z) sin ( z) dz 40
Part 2. Definite integrals ...... 49
2.1. Integrals of z n exp [ (α z + β)2 ] ...... 49
2.2. Integrals of z n exp ( α 2 z 2 + β z + γ )...... 51
2.3. Integrals of erf n (α z + β) exp [− (α z + β)2 ] ...... 54
2.4. Integrals of z nerf (α z + β) exp [− (α z + β)2 ] ...... 55
n 2.5. Integrals of z erf (α z + β) exp (β1z + γ ) ...... 5 8
21m+ 2 2.6. Integrals of zzerf(α + β) exp ( −α1 z) ...... 60
n 2 ddddv 2.7. Integrals of zexp(−+α zz β) erf ( αβ11 z +) , vvd n 2 zzexp(−α) erf( αβ11 z ++) erf ( αβ 2 z 2) ...... 63
2.8. Integrals of sin 2m+1 (α 2 z 2 + β z + γ ), sinh2m+ 1(α 22zz++ βγ) ,
2m+1 2 2 2m+ 1 22 cos (α z + β z + γ ), cosh (αzz++ βγ) …… 73
n 2 2 2.9. Integrals of z sin (α z + β z + γ )exp (β1 z ) ...... 7 7 n 2 2 2 2.10. Integrals of z exp (− α z + β z )sin (α1 z + β1 z + γ ) ...... 80 n 2.11. Integrals of z erf ( α z + β ) exp ( β1 z ) sin(β 2 z + γ ) ...... 8 6 2 n+1 2 2 2.12. Integrals of z erf (α z + β) exp(− α1 z )sin (α 2 z + γ ) ...... 89 n 2 2 2.13. Integrals of z erf (α z + β) exp ( −α1 z + β1 z )sin(α 2 z + β 2 z + γ ),
n 2 z erf (α z ) exp ( −α1 z )sin(βz ),
n 2 z erf (α z ) exp ( −α1 z ) cos(βz ) ...... 96
2.14. Integrals of z n [±1− erf (α z + β)],
n z [erf (α1 z + β1 ) erf (α 2 z + β 2 )] ...... 109
v
2.15. Integrals of z n [±1− erf (α z + β)]2 , z n [1− erf 2 (α z + β)] ,
n (α + β ) 2 (α + β ) z [erf 1 z 1 erf 2 z 2 ], n 22α +β − α +β zzerf ( 11) erf ( 2 z 2) ...... 114
n 2.16. Integrals of z exp(β z ) [±1 − erf (α z + β1 )],
n z exp(β z ) [ erf (α1 z + β1 ) erf (α 2 z + β 2 )] ...... 12 2 n 22 2.17. Integrals of zexp(−α zz +β) ±1 − erf ( α11 z +β) ,
n 2 2 z exp(− α z + β z )[erf (α1 z + β1 ) erf (α 2 z + β 2 )] ...... 126
n 2 2.18. Integrals of z erf (α z + β1 ) exp [− ( α z + β2 ) ] ,
n 2 z [±1− erf (α z + β1 )] exp [− ( α z + β2 ) ] ,
n 2 zzexp(β) erf ( α z +β1 ) ,
n 2 zzexp(β) 1 − erf ( α z +β1 ) , n β22 α +β − α +β 141 zzexp( ) erf ( 11 z) erf ( 2 z 2) ......
2 2.19. Integrals of (± 1)nn − erf( αzz +β) exp −( α +β) ,
2m+1 3 2m+1 3 z [±1 − erf (α z )] , z [±1 − erf (α z )], 2m+1 3 3 z [erf (α1 z ) erf (α 2 z )] ,
z 2merf 2 (α z )exp(− α 2 z 2 ),
z 2m [1 − erf 2 (α z )]exp( − α 2 z 2 ) ...... 15 9
n (β + γ ) ± − (α + β ) 2.20. Integrals of z sin z [ 1 erf z 1 ], n z sin (β z + γ )[erf (α1z + β1 ) erf (α 2 z + β 2 )],
n 2 zsin (α zz +β+γ±−)1 erf ( α11 z +β) ,
n 2 163 zsin (α zz +β+γ)erf ( α+β11 z) erf ( α 2 z +β 2) ......
Appendix. Some useful formulas for obtaining other integrals ...... 177
vi NOTATIONS and DEFINITIONS
θ 2 erf (θ= )∫ exp( −z2 ) dz is the error function (probability integral). π 0 θ 2 erfi(θ= )∫ exp(z2 ) dz=− i erf ( i θ ) is the imaginary error function. π 0 m, n, k, l, r, q are nonnegative integers (0, 1, 2, …). a, b, c are real numbers. α, β, γ, µ, ν, θ are complex numbers (can be real as well). x is a real integration variable. z is a complex integration variable (can be real as well). i is the imaginary unit. π and e have their usual meaning. Reθ and Imθ are real and imaginary parts of θ , θ = Re θ + i Imθ. θ= Reθ − i Im θ is the complex conjugate of θ .
θ = Re 2 θ + Im 2 θ is the absolute value of θ . arg θ is the argument of a complex number θ , −<ππarg θ ≤ .
∞ is the infinity symbol used on the complex plane. − ∞ and + ∞ are the infinity symbols for real numbers (or the real axis). θ → ∞(T ) means θ goes to infinity on the complex plane along trajectory T . n ∑ θk denotes summation; equals 0 if n < m . k =m m! is the factorial of m (with usual convention 0!1!1= = ). E (c) is the integer part (floor function) of a real number c.
∞ ()T2 ∫ ϕ()z dz is an improper integral on the complex plane with trajectories ∞ ()T1 T1 and T2 both going to infinity. lnθ= ln θ +i arg θ is the principal value of natural logarithm.
11+θi arctanθ= arctg θ= atan θ= ln is the principal value of inverse tangent. 21ii−θ
1 INTRODUCTION
Formulas in this book are numbered by part, section, subsection, and formula number. When a formula is too long, it is written in terms of abbreviated expressions, which are listed at the ends of sections. Most of the presented formulas assume complex parameters and integration on the complex plane. However, the formulas which use letters a, b, x are valid only for real values of the corresponding variables (see the list of notations). In particular, integrals ∫ f() x dx assume integration only over the real axis. The symbol “ ∞ ” denotes infinity on the complex plane, while for the real axis we use notations “ +∞ ” (positive infinity) or “ −∞ ” (negative infinity). For improper ∞ integrals on the complex plane of the type f() z dz (usually µ = 0 ), the conditions ∫µ for how the trajectory approaches infinity are listed after the formula. When both ∞()T complex limits of integration are infinite, we use notation 2 f() z dz and list ∫∞()T1 conditions of approaching infinity for both trajectories T1 and T2 . Note that since the integrands considered in this book are analytic functions, the results of integration do not depend on the integration contours; it is only important how these trajectories approach infinity. The validity conditions for formulas are listed in curly brackets after them. For improper integrals, we also include in the curly brackets the conditions for approaching infinity, which are necessary and sufficient for the integral convergence. Most integrals with complex parameters presented in this book can also be used for real values of these parameters. However, conversion of the formulas to this case is not always simple. This is why we also present integrals with real parameters (as the most practically interesting case) for convenience. The square root θ in formulas is defined with a positive real part (or positive imaginary part if the real part is zero), as consistent with our convention −<ππarg θ ≤ . When the left-hand side of a formula contains θ2 but does not contain
θ , we assume θθ= 2 . Multiple double-sign notations ± or (in formulas and validity conditions) assume that either all upper signs or all lower signs are used.
Part 1 of this book contains indefinite integrals, Part 2 contains definite (mainly improper) integrals. Appendix contains some formulas with relations between integrals, which can be used to obtain integrals not presented in this book. Definite integrals with finite limits are presented in the Part 2 only in the case when there are no corresponding indefinite integrals. Improper integrals are presented independently of whether the corresponding indefinite integrals are presented or not.
2 ∞ Most of improper integrals have the form f() z dz ; however, some integrals have ∫0 ∞ ∞()T the form f() z dz . The integrals 2 f() z dz with two infinite complex limits are ∫µ ∫∞()T1 presented only when we are unable to obtain corresponding improper integrals with +∞ one infinite limit of integration. If an integral f() z dz with two real infinite limits ∫−∞ exists, it is presented (except for odd functions). If from an integral containing the function erf it is easy to obtain the corresponding integral containing the function erfi (using the first two formulas in Appendix), then we present only the integral with erf . However, if such a conversion is not easy (and in some other cases), we also present the integral with erfi .
The formulas in this book contain the error function erf (θ) and the imaginary error function erfi(θ) defined as θ θ 2 2 erf (θ= )∫ exp( −z2 ) dz , erfi(θ= )∫ exp(z2 ) dz . π 0 π 0 We do not use notation of the complementary error function erfc(θ)= 1 − erf (θ) . Let us briefly discuss some properties of the functions erf (θ) and erfi(θ) . 1. These functions are related to each other as erfi(θ)= −ii erf ( θ) , erf (θ)= −ii erfi( θ). 2. Both functions are odd, erf (−=−θ) erf (θ) , erfi(−=−θ) erfi(θ) . Obviously, erf(0)= erfi(0) = 0 . 3. For complex conjugate arguments, these functions are also conjugate, erf (θ)= erf (θ) , erfi(θ)= erfi(θ) . 4. For a real θ , both erf (θ) and erfi(θ) are real and have the same sign as θ . For an imaginary θ , both erf (θ) and erfi(θ) are imaginary, and their imaginary parts have the same sign as the imaginary part of θ . 5. For |θ| 1 (in practice, for at least θ3> ) the error function erf (θ) can be approximated by using a few terms in its asymptotic expansion, e.g.,
exp(−θ2 ) 1 3 3⋅ 5 θ≈ θ − − + − erf ( ) sign[Re( )] 1 2 22 23 , πθ 2θθ (2 ) (2 θ ) where sign[Re(θ)] is 1 for Re(θ)≥ 0 and −1 for Re(θ)< 0 . The discontinuity of this approximation at Re(θ)= 0 is irrelevant because in this case the approximation becomes applicable only at exponentially large erf (θ) . The addition of more terms
3 into this expansion requires larger θ for accuracy improvement. Numerical calculations show that the absolute value of error in using the formula above is always less than three times the absolute value of the first neglected term of the expansion, i.e., the term −exp( −θ2 )( π θ ) ⋅ 357/(2 ⋅ ⋅ θ)24.
As mentioned above, all improper integrals in this book are given with the necessary and sufficient conditions of their convergence. These conditions have been obtained using in particular the following properties of erf (θ) for θ →∞. The error function erf (θ) has a limit at θ →∞ (i.e., converges) if and only if
lim Re (θ2 )+ ln θ = +∞ ; θ→∞ in this case lim erf (θ)= ± 1 for lim Re(θ) = ±∞ . θ→∞ θ→∞ For θ →∞, there is an inequality ±−1 erf ( θ ) N < for ±Re(θ) > 0 , exp[−θ Re(2 )] θ where N is some positive number (it is possible to use N =1.4 / π for any θ ). As a consequence, for trajectories, for which Re(θ2 )+ ln θ →−∞ or at least Re(θ2 )+ ln θ remains bounded from above as θ →∞, there is another similar inequality: erf (θ ) 1 NM < +< , exp[−θ Re(22 )] exp[ −θ Re( )] θθ where M is also some positive number (which depends on the trajectory). The asymptotic properties of the error function erf (θ) at θ →∞ and their use to find the necessary and sufficient conditions for convergence of the improper integrals presented in this book, are discussed in detail in our Russian-language book: Коротков Н.Е., Коротков А.Н, Интегралы, связанные с интегралом вероятностей. Под ред. В.И. Борисова. – Воронеж: изд. ОАО «Концерн «Созвездие», 2012. – 276 с. – ISBN 978-5-900777-18-4. The methods to derive and check the presented integrals are described in another book, also published in Russia: Коротков Н.Е., Интегралы для приложений интеграла вероятностей. Под ред. В.И. Борисова. – Воронеж: изд. ФГУП «ВНИИС», 2002. – 800 с. – ISBN 5- 900777-10-3.
4 PART 1 INDEFINITE INTEGRALS
2 1.1. Integrals of the form zn exp α z +β dz ∫ ( )
1.1.1. π 1. exp [ − (α z + β)2 ] dz = erf (α z + β) . ∫ 2α
π 2. exp [ (α z + β)2 ] dz = erfi(α z + β). ∫ 2α
1.1.2.
221 1.∫ z exp −( α z +β) dz = – exp −( αz +β) + π β erf ( αz +β) . 2α2 { }
1 2. ∫ zexp[(α z + β)2 ] dz = {exp[(α z + β)2 ] − π β erfi (α z + β) } . 2α 2
1.1.3.
2 1.z2 exp −( α z +β) dz = ∫
2 1 2 π (2β + 1) = (β − α z )exp [ − (α z + β) ] + erf (α z + β) . 2α 3 2
2 2.z2 exp (α z +β) dz = ∫
π21 β−2 1 2 ( ) =( α−βzz)exp ( α+β) + erfi (α+βz) . 3 2α 2
1.1.4.
π E (n / 2) n! (− β)n−2k 1. z n exp − (α z + β)2 dz = erf (α z + β) − ∫ [ ] n+1 ∑ k 2α k =0 4 k ! (n − 2k )!
5 ( ) − − 1 E n 2 n!( −β)n 2 k k l!(α z + β)2l 1 − exp[ −(α z + β)2 ] + n+1 ∑ ∑ k −l 2α k =1 k!(n − 2k )! l =1 4 (2l )!
nk+−12 2l− 2 nEn− ( /2) nk!( − 1!) ( −β) k ( αz +β) + ∑∑ . 2kn− 1! +− 1 2 k ! l − 1! kl=11( ) ( ) = ( )
(2m)! π erf (α z ) exp ( −α 2 z 2 ) m k! z 2k −1 2. z 2m exp (− α 2 z 2 ) dz = − ∑ . ∫ 2m+1 α 2m+1−2k m! (2α) k =1 (2k )!(2α)
m! m z 2k 3. z 2m+1 exp (− α 2 z 2 ) dz = − exp (− α 2 z 2 ) . ∫ 2 ∑ m−k 2α k =0 k!(α 2 )
π E (n / 2) n! (− β)n−2k 4. z n exp (α z + β)2 dz = erfi(α z + β) + ∫ [ ] n+1 ∑ k 2 α k =0 (− 4) k ! (n − 2 k )!
( −1)n E (n / 2) n!βn−2k k l! (α z + β)2l −1 + exp (α z + β)2 − n+1 [ ] ∑ ∑ k −l 2 α k =1 k ! (n − 2 k )! l =1 (− 4) (2l )!
22l− nEn− ( /2) nk!( − 1!) βnk+−12 k (αz +β) − ∑∑ . 2kn− 1! +− 1 2 k ! kl− kl=11( ) ( ) = (−−1) (l 1!)
(2m)! π erfi(α z) 5. z 2m exp (α 2 z 2 ) dz = − exp (α 2 z 2 )× ∫ m! m 2α (− 4α 2 )
m kz! 21k− × . ∑ mk+−1 k=1 22mk+− 12 2!k −α2 ( ) ( )
m! m z2 k 6.z2m+ 1 expα= 22z dz exp α22z . ∫ ( ) 2 ( ) ∑ mk− 2α k=0 k!(−α2 )
6 1.2. Integrals of the form ∫ zn exp ( α22 z +β z +γ) dz
1.2.1.
π β 2 β − α 2 2 + β + γ = + γ α − 1. ∫ exp ( z z ) dz exp erf z . 2α 4α 2 2α
π β2 β α 2 2 + β + γ = γ − α + 2. ∫ exp( z z ) dz exp erfi z . 2α 4α 2 2α
1.2.2.
π β β 2 β − α 2 2 + β + γ = + γ α − − 1. ∫ z exp ( z z ) dz exp erf z 4α 3 4α 2 2α 1 − exp (− α 2 z 2 + β z + γ ). 2α 2
1 π β β 2 α 2 2 + β + γ = α 2 2 + β + γ − γ − × 2. ∫ z exp ( z z ) dz exp ( z z ) exp 2α 2 4α 3 4α 2
β × erfi α z + . 2α 1.2.3.
π 2α 2 + β2 β2 β 2 − α 2 2 + β + γ = ( ) + γ α − − 1. ∫ z exp( z z ) dz exp erf z 8α5 4α 2 2α
2α 2 z + β − exp (− α 2 z 2 + β z + γ ). 4α 4
π β2 − 2α 2 β2 β 2 α 2 2 + β + γ = ( ) γ − α + + 2. ∫ z exp( z z ) dz exp erfi z 8α5 4α 2 2α
2α 2 z − β + exp (α 2 z 2 + β z + γ ). 4α 4 1.2.4.
π ββ2 1.zn exp −α22 zzdz +β +γ = exp+γ erf αz − × ∫ ( ) n+12 24αα2α
7 ( ) − ( ) − E n 2 n!β n 2k 1 E n 2 n! β n 2k × ∑ − exp (− α 2 z 2 + β z + γ ) ∑ × n−k n k!(n − 2k )! k=0 k! (n − 2k )! (α 2 ) 2 k=1
2l−1 2l−2 k l ! (2α2 z − β) n−E(n 2) n! (k − 1)! β n+1−2k k 4 k−l (2α2 z − β) × ∑ + ∑ ∑ . 2 n−k+l (2k − 1)! (n + 1 − 2k)! 2 n−k+l l=1 (2l)! (α ) k=1 l=1 (l − 1)! (α )
π ββ2 2.zn exp α+β+γ=22 zzdz exp γ− erfi α+z × ∫ ( ) n+12 24αα2α
( ) n−2k ( ) n−2k E n 2 n! β 1 E n 2 n! β × ∑ + exp (α 2 z 2 + β z + γ ) ∑ × n−k n k! (n − 2k )! k =0 k! (n − 2k )! (− α 2 ) 2 k =1
2l −1 − ( ) + − − 2l −2 k l! (2α 2 z + β) n E n 2 n!(k −1)!β n 1 2 k k 4 k l (2α 2 z + β) × ∑ − ∑ ∑ n−k +l ( − ) ( + − ) n−k +l l =1 ( ) −α 2 k =1 2k 1 ! n 1 2k ! l =1 ( − ) − α 2 2l ! ( ) l 1 ! ( ) .
n! n z k 1.2.5. z n exp( β z + γ ) dz = exp( β z + γ ) . ∫ ∑ n−k β k =0 k!(− β)
2 1.3. Integrals of the form erfn αz +β exp − αz +β dz ∫ ( ) ( )
2 π 1.3.1. erf(α+βz ) exp −α+β( z) dz =erf 2 ( α+βz ) . ∫ 4α
2 π 1.3.2. erf23(α+βz ) exp −α+β( z) dz =erf ( α+βz ) . ∫ 6α
2 π 1.3.3. erfnn(αz +β) exp −( αz +β) dz = erf +1 (αz +β) . ∫ (22n +α)
2 zn erfα z +β exp − αz +β dz 1.4. Integrals of the form ∫ ( ) ( )
2 1 1.4.1. ∫ zerf(α z +β) exp −( αz +β) dz = { 2 erf 2 (αz +β) − 4α2
8 1 2 −πβerf 2 ( α+β−z)} erf( α+β zz) exp −α+β( ) . 2α2 22β−αz 1.4.2. ∫ zzzd2 erf(α+β) exp −α+β( ) z =erf( α+β zz) exp −α+β+( ) 2α3
π (2β2 +1) β 1 + erf 2 (α z + β) − erf [ 2 (α z + β)]− × 8α3 2α3 4 πα3
× exp [ − 2(α z + β)2 ] . 1.4.3. π 1. ∫ z n erf (α z + β) exp [ − (α z + β)2 ] dz = erf 2 (α z + β) × 4α n+1 ( ) − ( ) E n 2 n! (− β)n 2k n! E n 2 (− β)n−2k × − erf (α z + β) × ∑ k n+1 ∑ k k =0 4 k ! (n − 2k )! α k =1 4 k ! (n − 2k )!
+ k−1 4l l ! (α z + β)2l 1 1 k−1 (l ! )2 × exp [ − (α z + β)2 ] ∑ + exp [ − 2(α z + β)2 ] ∑ × l =0 (2l + 1)! 2 π l =0(2l + 1)!
+ 2r − ( ) n+1−2k l 2l r (α z + β) n! n E n 2 (k −1)! (−β) 1 × + × ∑ n+1 ∑ r =0 r ! 2α k=1 (2k −1)! (n +1− 2k)! 2
k−1 (2l )! k−1(α z + β)2l × erf [ 2 (α z + β)] − erf (α z + β) exp[ −(α z + β)2 ] − ∑ l 2 ∑ l =08 (l! ) l =0 l!
k −1 l −1 2r +1 2 2 (2l )! r ! (α z + β) − exp − 2(α z + β) . [ ] ∑ 2 ∑ l −r π l =1 (l ! ) r =0 8 (2r + 1)!
(2m)! π α 2. z 2m erf (α z ) exp ( − α 2 z 2 ) dz = erf 2 (α z ) − ∫ m! m+1 (4α 2 )
+ m−1 k ! z 2k 1 1 − erf (α z ) exp ( − α 2 z 2 ) − exp ( − 2α 2 z 2 )× ∑ m−k k =0 (2k + 1)! (4α 2 ) 2 π α
− 2 m 1 (k! ) k z2l × ∑ ∑ . m−k + 2 m−l k=0 2 (2k 1)! l=0 l ! (2α )
9 m + m! 1 (2k )! 3. z 2m 1 erf (α z ) exp ( − α 2 z 2 ) dz = erf ( 2 α z ) − ∫ 2m+2 ∑ k 2 2α 2 k =0 8 (k ! )
m (α z )2k 2 m (2k )! − erf (α z ) exp ( − α 2 z 2 ) − exp ( − 2α 2 z 2 ) × ∑ ∑ 2 k =0 k ! π k =1(k ! )
k−1 l ! (α z)2l+1 × . ∑ k−l l=0 8 (2l +1)!
n 1.5. Integrals of the form ∫ zerf(α z +β) exp( β1 z +γ) dz
1 1.5.1. erf(α+βzzd) exp( β+γ) zzz =erf( α+β) exp( β+γ−) ∫ 11β 1
β 2 − 4αββ β − exp 1 1 + γ erf α z + β − 1 . 2 4α 2α
β−z 1 1.5.2. zzzderf(α+β) exp( β+γ) z =1 erf( α+β zz) exp( β+γ+) ∫ 112 β1 2 1 β 1 β − 4α ββ1 β 1 + + − exp 1 + γ erf α z + β − 1 + × β2 α β α 2 α 2 2α π α β 1 1 2 4 1
2 × exp[ −(α z + β) + β1 z + γ ] .
n! n zk 1.5.3. zn erf(α+β z) exp( β+γ z) dz =erf( α+βz ) exp ( β+γz ) − ∫ 11 ∑ nk− β1 k=0 −β k!( 1 )
2 n β − 4α ββ1 β 1 − exp 1 + γ erf α z + β − 1 ∑ × 2 2α k n−k 4α k =0 2 (−β1 )
E (k 2 ) k−2l n ( β1 − 2α β) 2α 2 1 × ∑ + exp [ − (α z + β) + β1z + γ ] ∑ × 2 k−l π k n−k l =0 l! (k − 2l )! (α ) k=1 2 (−β1 )
− 2r −1 E (k 2 ) ( β − 2αβ)k 2l l r! (2α 2 z + 2αβ − β ) × ∑ 1 ∑ 1 + l!(k − 2l )! 2 k −l +r l =1 r =1 (2r )! (α )
10 2r −2 k −E (k 2 ) k +1−2l l l −r 2 (l −1)!( β1 − 2αβ) 4 (2α z + 2αβ − β1 ) + ∑ ∑ . (2l −1)!(k +1− 2l )! 2 k −l +r l =1 r =1 (r −1)! (α )
+ 1.6. Integrals of the form ∫ z2m 1erf(α z +β) exp( α22z +γ) dz
1.6.1.
1 β 2 β α + β − α 2 2 + γ = γ − α + − 1. ∫ z erf ( z ) exp ( z ) dz exp erf 2 z 2 2 α 2 2 2
1 − erf (α z + β) exp ( − α 2 z 2 + γ ). 2α 2
exp(γ) erf ( 2α z) 2.∫ z erf(α z) exp( −α22 z + γ) dz = −erf( αz) exp( −α22 z ) . 2α2 2
22 1 22 3.∫ zzzd erf(α+β) exp( α +γ) z =erf( α+β zz) exp( α +γ+) 2α2
1 2 + exp (γ −β − 2α β z ) . π β
22 exp(γ) 1 22 z 4.∫ zzzdz erf(αα+γ=) exp( ) erf( αα− zz) exp( ) . αα2 π
1.6.2.
+ m! 1. z 2m 1 erf (α z + β) exp ( − α 2 z 2 + γ ) dz = − erf (α z + β) exp ( − α 2 z 2 + γ )× ∫ 2
m z 2k m! m (2k )! 1 β 2 β × ∑ + ∑ exp γ − erf 2 α z + × m+1−k m+1 k ! 2 k =0 k ! (α 2 ) 2 (α 2 ) k =0 2 2 − − k β 2k 2l 1 k β 2k 2l × + exp (γ − β 2 − 2 α β z − 2α 2 z 2 ) × ∑ 2k +l ∑ l =0 2 l ! (2k − 2l )! π l =1 (2k − 2l )!
− l (2α z + β)2r 2 (l −1)!β r!(2α z + β) × − . ∑ 2k −l +r l −r r =1 2 (2k +1− 2l ) (2l −1)!(r −1)! 4 l!(2r )!
11 m + m! exp(γ ) 1 2. z 2m 1erf (α z )exp ( − α 2 z 2 + γ ) dz = × ∫ m+1 ∑ (α 2 ) k =0 k!
− (2 k )! erf ( 2 α z ) exp( − 2 α 2 z 2 ) k 8l l! (α z )2l 1 × − − k ∑ 8 k ! 8 4 π l =1 (2l )!
2k (αz ) 2 2 − erf (αz )exp (− α z ) . 2 3.∫ z 2m+1 erf (α z + β)exp (α 2 z 2 + γ ) dz =
m ! m z 2k = erf ( α z + β)exp ( α 2 z 2 + γ ) + 2 ∑ m−k 2α k =0 k !(− α 2 )
exp (γ − β 2 − 2αβ z ) m (2k )! 2k z l + ∑ ∑ . π β 2 m−k 2k −l k =0 k ! ( − α ) l =0 l !(2αβ)
m ! exp (γ ) m z 2k 4. z 2m+1 erf (α z )exp ( α 2 z 2 + γ ) dz = ∑ × ∫ α m−k k =0 k ! ( − α 2 )
erf (α z ) 2 2 z × exp (α z ) − . 2 α (2 k + 1) π
2 m+1 2 1.7. Integrals of the form ∫ z erf (α z + β) exp (α 1 z + γ ) dz
1.7.1. 2 1 2 1. ∫ z erf (α z + β) exp (α1z + γ ) dz = erf (α z + β) exp (α1z + γ )− 2 α1
αβ2 −αexp1 + γV ( β) . 2 α −α1
22erf (αz) αγexp( ) 2.∫ z erf(α z) exp( α11 z +γ) dz = exp(αz +γ) − V (0) 22αα11
2 { α1 ≠α }.
12 1.7.2.
m! m ( −1) m−k z 2 k 1. z 2m+1 erf (α z + β) exp(α z 2 + γ ) dz = erf (α z + β) exp(α z 2 + γ ) + ∫ 1 1 ∑ m+1−k 2 = α k 0 k! 1
m!α m (2!k ) αβ2 + ∑ exp1 +γV ( β) × 2 mk+−12α −α k=0 k!( −α1 ) 1
− k (α β)2 k 2l 1 × + exp − (α z + β)2 + α z 2 + γ × ∑ 2 k −l [ 1 ] l =0 l 2 π 4 l! (2 k − 2l )! (α − α1 )
2 r −2 k (l − 1)! (αβ)2 k+1−2l l (α2 z − α z + αβ) k (αβ)2 k−2l × 1 − × ∑ ∑ 2 k−l+r ∑ l=1 (2 l − 1)! (2 k + 1 − 2 l)! r =1 2 l=1 l ! (2 k − 2 l)! (r − 1)! (α − α1 )
2 r −1 l r ! (α2 z − α z + αβ) × 1 . ∑ 2k−l+r = l−r 2 r 1 4 (2 r )! (α − α1 )
mk− 2k m! m ( −1) z 2.z21m+ erf(α z) exp α z2 +γ dz =erf( αz) exp α z2 +γ + ∫ ( 11) ( ) ∑ mk+−1 2 k=0 k!α1
mk!αγ exp( ) m (2) ! 1 +− ∑ +− V (0) 2 21mk k 2 k k=0 (k! ) ( −α1 ) 4 α −α ( 1 )
2 2 k 2l −1 exp[(α1 − α ) z ] l! z − . ∑ k +1−l π l =1 k −l 2 4 (2l )! (α − α1 )
Introduced notation:
1 2 αβ Vz(β) = erf α −α1 + = 22 α −α11α −α
1 2 α β = erfi α1 − α z − . 2 2 α1 − α α1 − α
13 1.8. Integrals of the form ∫ zn erf2(αα z) exp ( 22 z) dz
2 erf (αz) 2 1.8.1. ∫ zerf2(α z) exp ( α22 z) dz =exp ( α−×22z ) 2α22πα
2 z × exp(− α 2 z 2 )− erf (αz ) . π α
παerf 3 ( zz) erf( 3αz) erf (α) 1.8.2. ∫ z2erf 2(α z) exp( −α22 z) dz = + − × 12α3 2 3πα 33 2 πα
z erf 2 (αz ) × exp(− 2α 2 z 2 )− exp(− α 2 z 2 ). 2α 2
3 (2mz) ! πα erf ( ) 1.8.3. z2m erf 2(α z) exp −α22 z dz = − ∫ ( ) m! 21m+ 32( α)
+ m−1 k ! z 2k 1 1 m−1 (k ! )2 − erf 2 (αz ) exp(− α 2 z 2 ) ∑ + ∑ × m−k (2k +1)! k =0 (2k +1)! (4α 2 ) π k =0
erf ( 3 αz ) k (2l )! × − erf (αz )exp(− 2α 2 z 2 )× 2m+1 ∑ 2m−k +l l 2 3α l =0 2 3 (l!)
k z 2l exp(− 3α 2 z 2 ) m−1 (k ! )2 k (2l )! × − × ∑ 2m−k −l 2m+1−2l ∑ ∑ 2 l=0 2 l! α π k =1 (2k +1)! l=1(l! )
2 r −1 l r ! z × ∑ . − + − + − m+1−r r =1 2 2 m k l 2 r 3 l 1 r ( 2 r )! (α 2 )
2 erf (αz) m mz! 2 k 1.8.4. z2m+ 1erf 2(αα z) exp 22 z dz =exp α22z − ∫ ( ) 2 ( ) ∑ mk− 2α k=0 k! −α2 ( )
− 2 m (−1)m k m! z 2k +1erf (αz ) exp(− α 2 z 2 ) k z 2l − ∑ + ∑ . π 2k +1 2m+1−2k π 2 m+1−l k =0 k ! α l =0 l! (α )
14 1.9. Integrals of the form ∫ zn erf(α z +β) dz
β 1 2 1.9.1. ∫erf(α+βz) dz =+ zerf ( α+β+ z ) exp −α+β( z ) . α πα z 2 2β 2 + 1 α z − β (α + β) = − (α + β) + − (α + β) 2 1.9.2. ∫ z erf z dz erf z exp[ z ]. 2 4α 2 2 πα 2 z3 23β+β3 α22zz −αβ +β2 +1 1.9.3. z2 erf(α z +β) dz = + erf (αz +β) + × ∫ 33 3 63α πα × exp[− (αz + β)2 ]. 1.9.4. − ( ) + − z n+1 n! n E n 2 (− β)n 1 2k 1. z n erf (α z + β) dz = erf (αz + β) − + ∫ n+1 ∑ k n +1 α k =0 4 k! (n +1− 2k )!
( ) − n! E n 2 k!(− β)n 2k k (αz + β)2l + exp[− (α z + β)2 ] ∑ ∑ + n+1 + − π α k =0 (2k 1)!(n 2k )! l =0 l!
n−E ( n 2 ) (− β) n+1−2 k k l! (αz + β) 2l−1 + . ∑ ∑ k−l k=1 k! (n + 1 − 2k )! l=1 4 (2l )!
z 2m+1 m! m z 2k 2m (α ) = (α ) + − α 2 2 2. ∫ z erf z dz erf z exp( z ) ∑ • 2m + 1 (2m + 1) π α 2m+1−2k k=0 k !
z 2m+2 (2m +1)! (2m +1)! 3. ∫ z 2m+1 erf (α z ) dz = erf (α z ) − + × 2m + 2 2 m+1 ( + ) π α (m +1)! (4α ) m 1 !
m (k +1)! z 2k +1 × exp(− α 2 z 2 ) . ∑ m−k k =0 (2k + 2)! (4α 2 )
1.10. Integrals of the form ∫ zn erf2 (α z +β) dz
22β 2 1.10.1. ∫erf(α+βz) dz =+ z erf ( α+β+z ) erf ( α+β×z ) α πα 2 × exp[− (αz + β)2 ]− erf [ 2 (αz + β)]. π α
15 z2 21β2 + αz −β 1.10.2. zerf22(α z +β) dz = − erf (αz +β) +erf ( αz +β) × ∫ 22 2 4α πα 2β 1 × exp [− (α z + β)2 ]+ erf [ 2 (αz + β)]+ exp[− 2(αz + β)2 ]. πα 2 2πα 2
z3 23β+β3 α−βz 2 1.10.3. z22erf(α z +β) dz = + erf 2(αz +β) + × ∫ 33 3 63α πα 2(α 2 z 2 − α β z + β2 + 1) × exp [− 2(α z + β)2 ]+ exp[− (α z + β)2 ]erf (α z + β) − 3 π α3
2 (12β 2 + 5) − erf [ 2 (αz + β)]. 12 π α 3 1.10.4. − ( ) + − z n+1 n! n E n 2 (− β) n 1 2k 1. z n erf 2 (α z + β) dz = erf 2 (α z + β) − ∑ + ∫ + n+1 k n 1 α k =0 4 k ! (n + 1 − 2k )!
E (n 2 ) n−2k n! k !(− β) 2 + 2erf (αz + β)exp − (αz + β) × n+1 ∑ [ ] π α k =0 (2k +1)!(n − 2k )!
k (αz + β)2l k (2l )! 4 × − 2 erf [ 2 (αz + β)] + exp[− 2(αz + β)2 ]× ∑ ∑ l 2 l =0 l ! l =0 8 (l ! ) π
+ − ( ) + − k (2l )! l−1 r ! (αz + β) 2 r 1 n! n E n 2 (− β) n 1 2 k × + × ∑ 2 ∑ l−r n+1 ∑ l=1 (l! ) r =0 8 (2r + 1)! π α k=1 k! (n + 1 − 2k )!
k 2l −1 2 (l −1)!(αz + β) × erf (αz + β)exp[− (αz + β) ] + ∑ k −l l =1 4 (2l −1)!
k −1 2 l 2r 2 2 (l! ) (αz + β) + exp[− 2(αz + β) ] . ∑ ∑ 2k −l −r π l =0(2l +1)! r =0 2 r!
z 2m+1 m! 2. ∫ z 2m erf 2 (αz ) dz = erf 2 (αz ) + 2erf (αz)× 2m +1 (2m +1) π α 2m+1
m (αz )2k m (2k )! 1 × exp(− α 2 z 2 ) − 2 erf ( 2 αz ) + × ∑ ∑ k 2 k =0 k ! k =0 8 (k ! ) 2 π
m (2k )! k (l −1)! (αz )2l −1 × exp(− 2α 2 z 2 ) . ∑ 2 ∑ k −l k =1 (k ! ) l =1 8 (2l −1)!
16 z 2m+2 (2m + 1)! 3. z 2m+1 erf 2 (αz )dz = erf 2 (αz ) − + ∫ 2m + 2 m+1 (m + 1)! (4α 2 )
(2m +1)! m k ! (αz )2k + z erf (αz ) exp(− α 2 z 2 ) + 2m+1 ∑ m−k (m +1)! π α k =0 4 (2k +1)!
1 m (k ! )2 k (αz )2l + exp − 2α 2 z 2 . ( ) ∑ ∑ 2m−k −l 2 π α k =0 (2k +1)! l=0 2 l!
2 m 1.11. Integrals of the form ∫ zerf(αα z) erf( 1 z) dz
1.11.1. 1 erf (αz) 22 1. ∫erf(αα=αα+z) erf( 11 z) dz zerf( z) erf ( z) exp( −α+ 1z ) π α1 2 2 erf (α z ) α + α + 1 exp(− α 2 z 2 )− 1 erf α 2 + α 2 z . α αα 1 1 1 2. ∫erf(αα=αα+×zzdzzzz) erfi( ) erf( ) erfi( ) πα ×erfi αz exp −α22 z − erf α zz exp α22 ( ) ( ) ( ) ( ) . 1.11.2. 3 2 z z 1. ∫ z erf (α z )erf (α1 z ) dz = erf (α z )erf (α1 z ) + × 3 3π α α1
2 2 α 2 2 + 1 α z +1 z 1 × − α 2 + α 2 2 + 1 α − α 2 2 + × exp[ ( 1 )z ] erf ( z )exp( 1 z ) 3 π α3 α3 1
4 4 2 2α + 2α + (αα1 ) × erf (α z)exp(− α 2 z2 )− 1 erf α 2 + α 2 z . 1 1 α 2 + α 2 αα 3 2 1 ( 1 )
z3 1 2.∫ zzzdzzz2 erf(αα=αα+×) erfi( ) erf( ) erfi ( ) 3 3 πα
112 22 2 × +zerfi( α z) exp( −α z) + − zzerf ( α) × αα22
17 4 z ×αexp( 22z ) − . πα 1.11.3. 2m+1 m 2m z 1 m! 1. ∫ z erf (α z )erf (α1 z ) dz = erf (α z )erf (α1 z ) + ∑ × 2m +1 (2m +1) π k = 0 k!
2 erf (αz ) 2 2 erf (α1z ) 2 2 (2k )! × z k exp (− α z )+ exp (− α z ) − × 2m+1−2k 1 2m+1−2k k α α k α 2 + α 2 1 4 k ! ( 1 )
erf α 2 + α 2 z α α 1 1 × + 1 − exp(− α 2 z 2 − α 2 z 2 )× 2m+1−2k 2m+1−2k 1 α α α 2 + α 2 π 1 1
l−1 k l 2 2 2l−1 4 l! (α + α1 ) z × ∑ (2l )! l=1
22 { α +α1 ≠0 for m > 0 }.
1! + m 2. ∫ z2mmerf(αα= zzdz) erfi( ) erf( αα zzz) erfi( ) 21 +× 21m + πα
m 2k m−k z (−1) −1 2αz + − × ∑ ⋅ + (−1)m 1 k erf (αz )exp(α 2 z 2 )+ m−k 2k +1 k =0 k ! (α 2 ) π
2 2 + erfi(αz )exp(− α z ) .
1.12. Integrals of the form ∫ z21m+ erf 3(αz) dz
2 z2 1 3zz erf (α ) 1.12.1. zerf3(α z) dz = −erf 3( αz) + exp −α22z + ∫ 2 ( ) 2 4α 2 πα 3erf (αz) 3 erf ( 3αz) + exp (− 2 α 2 z2 )− . 2πα 2 2πα 2
18 z22m+ (2mm++ 1!) ( 2 1!) 21m+ 3α=α3 − + × 1.12.2. ∫ z erf( z) dzerf ( z)+ 22m + 2 m 1 m +α1! 4 (m +π1!) ( ) ( ) m k!z 2k+1 1 m (k!)2 × 3erf 2 (α z)exp(− α 2 z 2 ) + × ∑ 2m+1−2k ∑ k=0 (2k +1)!(2α) π k=0 (2k +1)!
k z 2l × 3erf (α z )exp(− 2α 2 z 2 ) ∑ − + − − m+1−l l =0 2 2m 1 k l l! (α 2 )
3 erf ( 3αz) k (2l )! exp (− 3α 2 z 2 ) − ∑ + × m+1 2m+1−k+l l 2 2πα (α 2 ) l =0 2 3 (l!)
− m (k!)2 k (2l )! l r ! z 2r 1 × ∑ ∑ ∑ . (2k +1)! 2 − + − − m+1−r k=1 l =1 (l! ) r =1 2 2m k l 2r 3l r (2r )!(α 2 )
1.13. Integrals of the forms ∫ z n sin m (α 2 z 2 + β z + γ ) dz ,
∫ znmsinh (α22z +β z +γ) dz,
∫ znmcos (α22 z +β z +γ) dz , ∫ znmcosh (α22z +β z +γ) dz
1.13.1.
22 22 21π+ b b ib 1. sin a x+ bx +γ dx = sinγ− +cos γ− Re erf ax + + ∫ ( ) 42aa22 44aa2 2 2 b b 1 + i b + sin γ − − cos γ − Im erf ax + . 4a 2 4a 2 2 2a
2πβ 2 2. sin α22z +β z +γ dz =(1 +i) exp i−γ × ∫ ( ) 8α 2 4α
11+−iiβ ββ2 ×erf αz + +(1 − ii) exp γ− erf αz + . 22αα2 22 4α
2 22 πβ β 3. sinh α+β+γ=z z dz exp γ− erfi α+z − ∫ ( ) αα2 42 4α
19 ββ2 −exp −γ erf αz + . 2 α 4α 2 1.13.2.
2π bb22 1. 22+ +γ = γ− − γ− × ∫cos(a x bx) dx cossin 4a 44aa22 11++i b b22 b ib × + + γ− + γ− + . Reerf a x sin 22cos Im erf ax 2 22aa 44aa2
21πβ 2 + i β 2. cos α22z++ βγ z dz =1 −i exp i −γerf αz + + ∫ ( ) ( ) 2 84αα 2 2 α
2 β 1− i β + (1+ i )exp i γ − erf αz + . 4α 2 2 2α
2 22 πβ β 3. cosh α+β+γ=z z dz exp γ− erfi α+z + ∫ ( ) αα2 42 4α
ββ2 +exp −γ erf αz + . 2 α 4α 2 1.13.3.
n! b 2 n 2 2 + + γ = γ − (13) − 1. ∫ x sin (a x bx ) dx sin Re V1 (n,1, a,b, x) 2n 4a 2
b 2 − γ − (13) − 2 2 + + γ (13) + cos Im V1 (n,1, a,b, x) sin (a x bx )ReV2 (n,1, a,b, x) 4a 2
+ 2 2 + + γ (13) cos(a x bx )ImV2 (n,1, a,b, x) .
n! β2 2. zn sin α22 z +β z +γ dz = iexp i−γ V(13) ( n ,1, α , β , z ) − ∫ ( ) n+12 1 24 α
β2 −exp i γ− V(13) ( n ,− 1, α , β , z ) + exp iz α22 +β z +γ × 2 1 ( ) 4α
×(13) − αβ − − α22 +β +γ(13) αβ Vn22( , 1, , , z ) exp izz ( ) Vn ( ,1, , , z ) .
20 n! β2 3. znsinh α22 z +β z +γ dz =exp γ− V(13) ( n ,1,β ) + ∫ ( ) n+123 24 α
β2 +Vn(13) ( ,1, β ) − Vn(13) ( , − 1, β ) − exp −γVn(13) ( ,1, β ) . 44 2 5 4α 1.13.4.
n! b2 n 2 2 + + γ = γ − (13) + 1. ∫ x cos(a x bx ) dx sin Im V1 (n,1,a,b, x) 2n 4a 2
b 2 + γ − (13) − 2 2 + + γ (13) − cos ReV1 (n,1,a,b, x) sin(a x bx ) ImV2 (n,1,a,b, x) 4a 2
−cos(a 2 x 2 + bx + γ )ReV (13) (n,1,a,b, x) . 2
2 n! β (13) 2. z n cos α 2 z 2 + βz + γ dz = exp i − γ V (n,1,α,β, z) + ∫ ( ) + 1 2 n 1 4α 2
β2 + γ − (13) − α β − α 2 2 + β + γ (13) − α β − exp i V1 (n, 1, , , z) exp[i( z z )]V2 (n, 1, , , z) 4α 2
− − α 2 2 + β + γ (13) α β . exp [ i( z z )]V2 (n,1, , , z)
n! β 2 3. zncosh α2 z 2 ++ βγ z dz =exp γ − V(13) ( n ,1,β ) + ∫ ( ) n+12 3 24 α
β 2 +(13) β +(13) −+ β − γβ(13) Vn44( ,1, ) Vn ( , 1, ) exp 2 Vn5( ,1, ) . 4α
1.13.5.
(2mx )!nk+1 n !(2 m )!m (− 1) 1. xnmsin2 a 22 x+ bx +γ dx = + × ∫ ( ) m 2nm−+ 12 ∑ 4 (nm+ 1)( !) 2 k=1 (mk−+ )!( mk )!
bb22 ×sin 2k γ− ImV(13) ( n ,2 kabx , , , )+ cos 2 k γ− × 221 44aa
× (13) − 22+ +γ (13) − ReV12 ( n ,2 kabx , , , ) sin 2 ka ( x bx) Im V ( n ,2 kabx , , , )
21 − 22+ +γ (13) cos 2k ( a x bx) Re V2 ( n ,2 k , a , b , x ) .
n!(2m +1)! m (−1) k 2. x n sin 2m+1 (a 2 x 2 + bx + γ )dx = × ∫ n+2m ∑ 2 k =0 (m − k )!(m +1+ k )!
bb22 ×sin(21)k + γ− Re(,21,,,)cos(21)V(13) n k+ abx − k + γ− × 221 44aa (13) 22 (13) ×ImV12 ( n ,2 k +−+++γ++ 1, abx , , ) sin[(2 k 1)( a x bx)]Re V ( n ,2 k 1, abx , , )
22 (13) +cos[(2k + 1)( a x + bx +γ)] Im V2 ( n ,2 k+ 1, a , b , x ) .
(2m)!z n+1 n!(2m)! m (−1) k 3. z n sin 2m (α 2 z 2 + βz + γ )dz = + × ∫ m 2 n+2m ∑ 4 (n +1)(m!) 2 k =1(m − k )!(m + k )!
ββ22 × exp 2k i−γ V(13) (,2, n kαβ ,,) z + exp2 ki γ− × 221 44αα ×Vnk(13) (, − 2, αβ ,,) z − exp2 kizz α22 +β +γ Vnk(13) (, − 2, αβ ,,) z − 12( )
−exp − 2ki α22 z +β z +γ V(13) ( n ,2 k ,α , β , z ) . ( ) 2
n!(2m + 1)!i m (−1) k 4. z n sin 2m+1 (α 2 z 2 + βz + γ )dz = × ∫ n+1+2m ∑ 2 k =0(m − k )!(m + 1 + k )!
ββ22 ×exp21i( k +) −γ V(13) (,21,,,)exp21 nk+ αβ z − i( k +) γ− × 221 44αα ×Vnk(13) ( , − 2 − 1, αβ , , z ) + exp ik( 2 + 1) α22 zzVnk +β +γ(13) ( , − 2 − 1, αβ , , z ) − 12( ) −exp21 −i( k +) α22 z +β z +γ V(13) (,21,,,) nk+ α β z . ( ) 2
(2mz )!n+1 n !(2 m )! 5. ∫ znmsinh2(α 22z +β z +γ) dz = + × (−+ 4)m (nm 1)( !)22 2nm+
22 m mk− 2 (−β 1) (13) (13) ×∑ exp 2kγ− V( nk ,2 ,β+ ) V ( nk ,2 , β+ ) −+ 2 34 k=1 (mk )!( mk )! 4α
β2 +Vnk(13) (, − 2,) β + exp2 k−γ Vnk(13) (,2,)β . 4 2 5 4α
nm!(2+− 1)!m ( 1)mk− 6. znmsinh2+ 1α 22z +β z +γ dz = × ∫ ( ) nm++12 ∑ 2 k=0 (mk− )!( m ++ 1 k )!
β2 ×exp21( k +) γ− V(13) (,21,) nk+ β+ V(13) (,21,) nk + β− 2 34 4α
β2 −Vnk(13) ( , − 2 − 1, β ) − exp( 2 k + 1) −γ Vnk(13) ( , 2+ 1, β ) . 4 2 5 4α 1.13.6. + (2m)!x n 1 n!(2m)! m 1 1. x n cos 2m (a 2 x 2 + bx + γ ) dx = + × ∫ m 2 n−1+2m ∑ 4 (n +1)(m!) 2 k =1(m − k )!(m + k )!
bb22 ×sin 2k γ− ImV(13) ( n ,2 kabx , , , )+ cos 2 k γ− × 221 44aa
×ReV(13) ( n ,2 kabx , , , )− sin 2 k a22 x+ bx +γ Im V(13) ( n ,2 kabx , , , )− 12( )
−cos 2k a22 x+ bx +γ Re V(13) ( n ,2 k , a , b , x ) . ( ) 2
n!(2m +1) ! m 1 2. x n cos 2m+1 (a 2 x 2 + bx + γ ) dx = × ∫ n+2m ∑ 2 k=0(m − k )!(m +1+ k )!
bb22 ×sin21( k +) γ− Im(,21,,,)cos(21)V(13) n k+ abx + k + γ− × 221 44aa ×Re(,21,,,)sin(21)V(13) n k +−+ abx k a22 x ++γ bxIm(,21,,,) V(13) n k +− abx 12( )
−cos(21)k + a22 x + bx +γ Re(,21,,,) V(13) n k+ a b x . ( ) 2
(2mz )!n+1 n !(2 m )! 3. ∫ znmcos2(α 22 z +β z +γ) dz = + × 4m (nm+ 1)( !)22 2nm+
23 m 2 1 β (13) ×∑ exp 2ki−γ V( n ,2 k ,αβ , , z ) + −+ 2 1 k=1 (mk )!( mk )! 4α β2 +exp 2ki γ− V(13) (, n− 2,,,) k α β z − exp2 k i α22 z +β z +γ × 2 1 ( ) 4α
×(13) − αβ − − α22 +β +γ (13) αβ Vnkz22(, 2, ,,) exp2 kizz( ) Vnkz(,2, ,,) .
n!(2m +1)! m 1 4. z n cos 2m+1 (α 2 z 2 + βz + γ ) dz = × ∫ n+1+2m ∑ 2 k=0(m − k )!(m +1+ k )!
ββ22 ×exp(21)ik + −γ V(13) (,21,,,)exp(21) nk+ αβ z + ik + γ− × 221 44αα ×Vnk(13) ( , − 2 − 1, αβ , , z ) − exp ik( 2 + 1) α22 zzVnk +β +γ(13) ( , − 2 − 1, αβ , , z ) − 12( )
−exp(21) −ik + α22 z +β z +γ V(13) (,21,,,) nk+ α β z . ( ) 2
(2mz )!n+1 n !(2 m )!m 1 5. znmcosh 2α 22 z++ βγ z dz = + × ∫ ( ) m 22nm+ ∑ 4 (n+ 1)( m !) 2k =1 (mk−+ )!( mk )!
2 β (13) (13) (13) × exp 2k γ − V (n, 2k ,β) +V (n, 2k ,β) +V (n, − 2k ,β) + 2 3 4 4 4α β2 + − γ (13) β exp 2k V5 (n,2k , ) . 4α 2
nm!( 2+ 1)! m 1 6. znmcosh2+ 1α 22z +β z +γ dz = × ∫ ( ) nm++12 ∑ 2 k=0 (mk− )!( m ++ 1 k )!
β2 ×exp(21)k + γ− V(13) (,21,) nk+ β+ V(13) (,21,) nk + β+ 2 34 4α β2 +Vnk(13) ( , − 2 − 1, β ) + exp (2 k + 1) −γ Vnk(13) ( , 2+ 1, β ) . 4 2 5 4α 1.13.7.
(2mz )!nk+1 n ! (2 m )!m (− 1) 1. znmsin2 β z +γ dz = + V(13) ( n ,2 k ) ∫ ( ) mm2 21− ∑ 6 4 (nm+ 1)( !) 2 k=1 (mk−+ )!( mk )! {β≠0 }.
24 nm!(2+− 1)!m ( 1) k+1 2. znmsin21+ (β z +γ ) dz = V(13) ( n , 2 k + 1) ∫ m ∑ 7 4 k=0 (mk− )!( m ++ 1 k )!
{β≠0 }.
(2mz )!n+−1 n !(2 m )!m (− 1)mk 3. znmsinh2 (βγ z+= ) dz + × ∫ mm2 21+ ∑ (−+ 4) (n 1)( m !) 2β k =1 (mk− )!( mkk + )!
× (13) − (13) − [V8 (n,2k ) V8 (n, 2k )].
nm!(2+− 1)! m ( 1) mk− 4. znmsinh21+ (βγ z+= ) dz × ∫ 21m+ ∑ 2β k =0 (mk− )!( m ++ 1 k )!(2 k + 1)
× (13) + + (13) − − [V8 (n,2k 1) V8 (n, 2k 1)].
1.13.8.
+ (2mz )!n 1 n !(2 m )! m 1 1. znmcos2 (β z +γ ) dz = + V(13) ( n ,2 k ) ∫ mm2 21− ∑ 6 4 (nm+ 1)( !) 2 k=1 (mk−+ )!( mk )!
{β ≠ 0} .
nm!(2+ 1)! m 1 2. znmcos21+ (β z +γ ) dz = V(13) ( n , 2 k + 1) ∫ m ∑ 6 4 k=0 (mk− )!( m ++ 1 k )!
{β ≠ 0} .
(2mz )!n+1 n !(2 m )!m 1 3. znmcosh2 (βγ z+= ) dz + × ∫ mm2 21+ ∑ 4 (n+ 1)( m !) 2β k =1 (mk−+ )!( mkk )!
× (13) − (13) − V8 (n,2k ) V8 (n, 2k ) .
nm!(2+ 1)! m 1 4. znmcosh21+ (βγ z+= ) dz × ∫ 21m+ ∑ 2β k =0 (mk− )!( m ++ 1 k )!(2 k + 1)
× (13) + − (13) − − V8 (n,2k 1) V8 (n, 2k 1) .
25 Introduced notations:
(13) πβ(1 is ) 1) V1 ( ns ,± , αβ , , z ) = erf (1±i ) α z + × 22sα 22α
l En( /2) ()±β nl−2 i × ∑ (s > 0), 2 nl− s l=0 ln!(−α 2 l )!( )
n1 (13) π±(1 ii ) 1 i V(2 n1 ,±α s , , 0, z ) = erf szα (s > 0) , 1 21n1+ ns1!8α s 2
(13) + α = (13) + − α = V1 (2n1 1, s, , 0, z) V1 (2n1 1, s, , 0, z) 0 ; − E (n / 2) n−2l l 2 2r 1 l+1−r (13) (−β) r!(2α z + β) i 2) V (n, s, α, β, z) = − + 2 ∑ ∑ 2n−2l+2r l=1 l!(n − 2l )! r=1 (2r )!α s
n−E (n / 2) n+1−2l − 2r−2 + − (l −1)!(−β) l 4l r (2α 2 z + β) i l 1 r + − , ∑ ∑ 2n−2l+2r l=1 (2l −1)!(n +1− 2l )! r=1 (r −1)!α s
n − n +1−l 1 1 l! (2 z) 2l 1 i 1 V (13) (2n , s, α, 0, z) = − , 2 1 ∑ 2 n1 ! l=1 (2l )! sα
n n n +1−l 4 1 n ! 1 z 2l i 1 V (13) (2n +1, s, α, 0, z) = 1 − ; 2 1 ∑ 2 (2n1 +1)! l=0 l! sα
E (n / 2) n−l n−2l (13) π β (−1) β 3) V (n, s, β) = erfi s αz + , 3 ∑ l 2n−2l 2 sα 2α l=0 l!(n − 2l )!s α
π erfi ( sα z) (13) = ⋅ (13) V3 (2n1 , s, 0) , V (2n1 + 1, s, 0) = 0 ; n1 2n1 +1 3 n1!(−s) α 2 s
4) V(13) ( ns , ,β ) = exp s α22 z +β z +γ × 4 ( )
En( /2) βnl−−2 l rz!(2α2 +β ) 21r ×− ∑∑l+−12 r nlr −+ lr=11ln!(− 2 l )! = (2rs )! (−α )
nEn− ( /2) (l −β 1)! nl+−12 l 4lr−− (2α2z +β ) 22r − ∑∑ , − +− l+−12 r nlr −+ lr=11(2ln 1)!( 1 2 l )! = (rs− 1)! ( −α )
− − 1 n1 (−1)n1 l l!(2z)2l 1 V (13) (2n , s, 0) = exp s(α 2 z 2 + γ) , 4 1 [ ] ∑ n +1−l 2n +2−2l n1! l=1 (2l )!s 1 α 1
26 nn112l (13) 4!n1 2 2 z V41(2 n+=− 1, s , 0) exp szαγ+ ; ( ) ∑ 2 nl1 +−1 (2n1 +− 1) ! l=0 ls!(α )
E (n / 2) n−2l (13) π β (−β) 5) V (n, s, β) = erf s αz + , 5 ∑ l 2n−2l 2 sα 2α l=0 l!(n − 2l )!s α
π erf ( s α z) (13) = ⋅ (13) + = V5 (2n1 , s, 0) , V5 (2n1 1, s, 0) 0 ; n1 2n1 +1 n1!s α 2 s
En( /2) (− 1)lnz −2 l 6) V(13) ( ns , )= sin s ( β z +γ ) −coss ( β z +γ ) × 6 [ ] ∑ 21l+ [ ] l=0 (n−β 2 ls )!( ) +− n− En( /2) (− 1)lnz 12 l 1 × , V (13) (0, s) = sin[s(βz + γ) ]; ∑ 2l 6 l=1 (n+− 1 2 ls )!( β ) sβ
En( /2) (− 1)lnz −2 l 7) V(13) ( ns , )= cos sz ( β +γ ) +sinsz ( β +γ ) × 7 [ ] ∑ 21l+ [ ] l=0 (n−β 2 ls )!( )
+− n− En( /2) (− 1)lnz 12 l 1 × , V (13) (0, s) = cos [s(βz + γ)]; ∑ 2l 7 l=1 (n+− 1 2 ls )!( β ) sβ
n zl 8) V(13) ( ns , )= exp[ s ( β z +γ )] . 8 ∑ nl− l=0 ls!(−β )
n 2 2 1.14. Integrals of the form ∫ z sin(α z + βz + γ ) exp(β1z ) dz
1.14.1.
2πbb 1. sin a22 x+ bx +γ exp b x dx =exp −1 × ∫ ( ) ( 1 ) 2 4a 2a 22 22 bb−−11 bb 1+ i bib + 1 × sin γ− +cos γ− Re erf ax + + 22 2a 44aa2
22 22 bb−−11 bb 1+ i bib + 1 +sin γ− −cos γ− Im erf ax + . 22 2a 44aa2
2 2 2 2π (β1 − i β) 2. ∫sin (α z + β z + γ ) exp (β1z ) dz = (i +1)exp − i γ × 8α α 2 4i
27 ii+−11β+i β ()β + ii β 2 β− β ×erf α+z11 +(1 − ii )exp γ− erf α+z 1 . αα 2 2222 4iα 1.14.2.
2πbb 1. xsin a22 x+ bx +γ exp ( b x ) dx = exp −1 × ∫ ( ) 1 32 82aa
22 22 bb−−11bb 1+ i bib + 1 × (b11 + b )sin −γ +(b − b )cos −γ Re erf ax + + 22 2a 44aa 2
22 22 bb−−11bb 1+ i bib + 1 +(b − b11 )sin −γ +(b + b )cos −γ Im erf ax + − 22 2a 44aa 2 1 22 − cos (a x+ bx +γ) exp( b1 x ) . 2a2
2 2 2 2π (β1 − iβ) 2. ∫ z sin (α z + β z + γ )exp (β1z) dz = (1− i)(β1 − iβ)exp − i γ × 16α3 4iα 2
i +1 β+βii()β+β2 ×erf αz +11 +(1 + ii ) ( β + β )exp i γ− × 1 2 2 2α 4iα 1− i β−i β 1 × α+1 − α22 +β +γ β . erf z 2 cos ( zz)exp(1 z ) 2 2α 2α
1.14.3.
n! 1. xn sin a22 x+ bx +γexp( b x ) dx = exp (b x ) cos a22 x+ bx +γ × ∫ ( ) 11n ( ) 2
bb ×ImV(14) ( nabbx ,1, , , , )− sin ax22 + bx +γ Re V(14) ( nabbx ,1, , , , ) − exp −1 × 1111( ) 2 2a
22 bb− 1 (14) ×sin −γ ReV2 ( n ,1, abb , ,1 , x ) + 4a2
2 − 2 b b1 (14) + cos − γ ImV (n,1, a,b,b , x) . 2 2 1 4a
28 ni! 2. n α22 +β+γ β = β+α22 +β+γ× ∫ zsin ( z z ) exp (11z ) dz + exp z i( z z ) 2n 1
×Vn(14) ( , − 1, αββ , , , z ) − exp β zizz − α22 +β +γ Vn(14) ( ,1,αββ , , , z ) + 1111( ) 1
()β−βii22 ()β+ β +exp 11−iV γ(14) (,1, n αββ ,, ,) z − exp i γ− × 222 1 44iiαα
(14) ×Vn2 ( , − 1, αββ , ,1 , z ) . 1.14.4.
n+1 n+1−l n l!z 1. ∫ z sin(β z + γ)exp(β1 z) dz = n! exp (β1 z) ∑ × + − β 2 + β 2 l l=1(n 1 l )!( 1 )
El( /2) (−− 1) lr+−12ββ rlr − 2 l− El( /2) ( 1) lr+−12112β r − β l +− r ×sin (βzz + γ ) ∑∑11++cos (βγ ) = rr=01(2rl )!(− 2 r )! = (2r− 1)!( l +− 1 2 r )! nz!n (− 1) nk− kexp (β+β+γzizi) exp ( β−β−γ zizi) = 11− . ∑ nk+−11nk+− 2!ikk=0 ()β+β11ii ()β−β
izn+1 2. zn sin(β z +γ )exp ( ± i β z ) dz =± exp(i γ ) − ∫ 22n +
n! n ()±iznl− l −exp ±iz (2 β +γ ) . [ ] ∑ nl+−1 2 l=0 l!(2β )
Introduced notations:
2r−1 E (n / 2) (β − isβ)n−2l l r! [i s(2α 2 z + β) − β ] 1) V (14) (n, s, α, β, β , z) = 1 1 + 1 1 ∑ ∑ 2 n−l+r l=1 l!(n − 2l )! r=1 (2r )!(isα )
−− − nl+−12 lrα2 +β −β 22r nEn( /2) (l− 1)!( β− is β ) l 4is (2 z ) 1 + 1 , ∑∑2 nlr−+ lr=11(2lnl− 1)! ( +− 1 2 )! = (r−α 1)!( is )
− 1 n1 l!(2z)2l 1 V (14) (2n , s, α, β, i sβ, z) = , 1 1 ∑ 2 n +1−l n1!l=1 (2l )! (isα ) 1
29 n n n !4 1 1 z 2l V (14) (2n +1, s, α, β, i sβ, z) = 1 ; 1 1 ∑ 2 n +1−l (2n1 +1)! l=0 l!(i s α ) 1
(14) 2π− (1is ) 1 + is β+is β1 2) V2 (,,,,,) nsαββ1 z = erf αz + × 42αα2
En( /2) ()β−is βnl−2 × 1 , ∑ 2 nl− l=0 l!( n−α 2 l )!( is )
2π (1− is) 1+ i s (14) α β β = α V2 (2n1 , s, , , i s , z) erf z , 2 n1 n1!4α(i s α ) 2
(14) + α β β = V2 (2n1 1, s, , , i s , z) 0 .
n 22 1.15. Integrals of the form ∫ zexp (−α z +β z)sin ( β1 z + γ) dz
1.15.1.
πb22− b bb 1. exp −a22 x + bxsin ( b x +γ) dx = exp11 sin +γ × ∫ ( ) 1 22 2a 42aa
b++ ib11 bb b ib1 ×Reerf ax − +cos +γ Imerf ax − . 22aa2a2
22 22 πb11− b bb 2. ∫exp(a x+ b x)sin( b1 x +γ) dx = exp cos−γ × 2a 42aa22
b++ ib11 bb b ib1 ×Imerfia x + −sin −γ Reerfi ax + . 22aa 2a2
2 π i β−i β 22 ( 1) 3. ∫exp (−αz + β z)sin ( β1 z + γ) dz = exp −i γ × 4α 4α2 2 β−ii β (β+i β1) β+ β ×erf α−z11 −exp +γizerf α− . 22αα2 4α 1.15.2.
π bb22− 1. xexp − a22 x + bxsin ( b x +γ) dx = exp 1 × ∫ ( ) 1 32 44aa
30 bb bb b+ ib × bsin 11+γ +b cos +γ Reerf ax − 1 + 221 22aa2a
bb bb b+ ib +bcos 11 +γ −b sin +γ Imerf ax − 1 − 221 22aa2a
sin (b x + γ) − 1 exp (− a 2 x 2 + bx). 2a 2 π bb22− 2. xexp a22 x+ bx sin( b x +γ) dx = exp 1 × ∫ ( ) 1 32 44aa
bb bb b+ ib × bsin 11−γ −b cos −γ Reerfi ax + 1 − 221 22aa2a
bb bb b+ ib −b sin 11 −γ +bcos −γ Imerfi ax + 1 + 1 22 22aa2a
sin (b x + γ) + 1 exp (a 2 x 2 + bx). 2a 2
π i ()β−i β 2 3. zexp −α22 z +β zsin ( β+γ z ) dz = ( β−βi )exp 1 −i γ × ∫ ( ) 1132 84αα
β−i β ()β+ ii β 2 β+ β ×erf α−z1 −β+β( i )exp 11+γ izerf α− − αα 1 2 22 4α sin (βz +γ ) − 1 exp (−α22zz + β ) . 2α2 1.15.3.
n! πb22− b bb 1. xn exp − a22 x + bxsin ( b x +γ ) dx = exp 11sin +γ × ∫ ( ) 1 n+12 2 242aa a
bib++11(15) 22bib (15) ×Reerf ax −ReV11 ( n , a , b , b11 )−− Imerf ax ImV ( n , a , b , b ) + 22aa
bb11 b++ ib (15) 2 b ib1 +cos +γ Reerf ax − ImV1 ( n , a , b , b1 )+ Imerf ax − × 2a2 22aa n! × (15) 2 − −22 +× ReV1 ( n , a , b , b1 ) exp ( a x bx) 2n ×sin (bx +γ )Re V(15) ( na ,22 , b , b , x )+ cos( bx +γ )Im V(15) ( na , , b , b , x ) . 12211 1
31 n! π b22− b bb 2. xn exp a22 x+ b xsin( b x +γ) dx = exp11 sin−γ × ∫ ( ) 1 n+12 2 24aa 2 a b++ ib (15) b ib × +11 −2 − +× Imerfia x ImV1 ( n , a , b , b1) Reerfi ax 22aa (15) 22bb11 b+ ib (15) ×ReV n , − a , bb , + cos −γ Reerfiax+ ImV n ,− a , bb , + 11( 11) 2 2a ( ) 2a b+ ib (15) n! + + 1 −2 −22 +× Imerfiax ReV1 ( n , a , b , b1) exp (a x bx) 2a 2n
×sin (bx +γ )Re V(15) ( n , − a22 , b , b , x ) + cos( bx +γ )Im V(15) ( n , − a , b , b , x ) . 12211 1
π n! ()β+ii β 2 β+ β 3. zn exp −α22 z + β zsin ( β z + γ ) dz = exp 11+i γerf α z − × ∫ ( ) 1 n+22α 24iαα 2
2 (15) ()β−ii β β− β (15) ×Vn( , αββ−22 , , ) exp 11−γierf α− z Vn(,,,)αβ−β+ 11112 α 4α 2
n! + −α+β22 −β−γ(15) αβ−β−2 + exp ( z z) exp( izi11 ) V2 (,,, n ,) z 2n 1i
(15) 2 −exp(izi β11 + γ ) V2 (,,,,) n α ββ z . Introduced notations:
E (n / 2) (β + iβ ) n−2l 1) V (15) (n,α 2 , β, β ) = 1 , 1 1 ∑ n−l l=0 l!(n − 2l )!( α 2 ) 1 (15) α 2 β β = (15) + α 2 β β = V1 (2n1 , , , i ) , V1 (2n1 1, , , i ) 0 ; 2 n1 n1 ! (α )
En( /2) (β+i β )nl−−2l r ! (2α 2 zi −β− β ) 21r 2) Vn(15) (,α2 , ββ , ,) z = 11+ 2 1 ∑∑2 nlr−+ lr=11ln! (− 2 l )! = (2r )! (α )
n− En( /2) (l− 1)! ( β+ i β )n+−12 ll 4 lr − (2α2zi −β− β ) 2r− 2 + 11, ∑∑2 nlr−+ lr=11(2ln− 1)!( +− 1 2 l )! = (r −α 1)!( )
n − 1 1 l!(2 z) 2l 1 V (15) (2n ,α 2 , β, iβ, z) = , 2 1 ∑ 2 n +1−l n1 ! l=1 (2l )! (α ) 1
n n n !4 1 1 z 2l V (15) (2n +1,α 2 , β, iβ, z) = 1 . 2 1 ∑ 2 n +1−l (2n1 +1)! l=0 l!(α ) 1
32 n 2 2 1.16. Integrals of the form ∫ z exp (− αz + β z )sin (α1z + β1z + γ ) dz
1.16.1.
2 − 2 − 2 2 π a b a b1 2a1bb1 1. ∫ exp (− ax + bx)sin (a1 x + b1 x + γ ) dx = exp × 2 2 + 2 4a 4a1
ab22−+ ab2 abb − 1111 1 1 b ib ×sin +γ Re erf a+ ia1 x − − 22+ a++ ia 2 a ia 44aa1 11
ab22−+ ab2 abb − 1111 1 1 b ib −cos +γ Im erf a+ ia1 x − 22+ a++ ia 2 a ia 44aa1 11
{ a + a1 > 0} .
π i 2. exp −α+βz22 zsin α z +β+γ z dz = × ∫ ( ) ( 11) 4
2 (β−i β ) β− β 111 i 1 × exp −i γerf α+ iz α1 − − × α+i α 44α+i α 2 α+ii α α− α 11 11
2 ()β+ii β11β+ β 2 2 ×exp +i γerf α− iz α − {α ≠ −α }. 44α−i α 1 1 1 2 α−i α1 2 2 2 π i (β iβ1 ) 3. ∫ exp(− α z + βz )sin ( i αz + β1 z + γ )dz = ± exp i γ × 4 2α 8α
β iβ1 1 × erf 2αz − − exp[β z ± i (β1 z + γ)] {β1 ≠ ±iβ} . 2 2α 2β1 2iβ
π i β2 − α 2 + β α 2 ± β + γ = ± γ × 4. ∫ exp( z z )sin ( i z i z )dz exp i 4 2α 2α
β i z × erf 2αz − exp (±iγ) . 2α 2 1.16.2.
2 − 2 − 2 2 π ab ab1 2a1bb1 1. ∫ x exp (− ax + bx)sin (a1 x + b1 x + γ )dx = exp × 4 2 + 2 4a 4a1
33 ab22−+ ab2 abb b− ib b− ib ×sin 1 11 1+γ Re 1 erf a+ ia x − 1 − 22 ++1 + 44aa+ 1 ()aiaaia112aia1 ab22−+ ab2 abb b− ib b− ib −cos 1 11 1+γ Im 1 erf a+ ia x − 1 − 22 ++1 + 44aa+ 1 ()aiaaia112aia1
2 exp (− ax + bx) 2 2 − [a sin (a1x + b1x + γ )+ a1 cos (a1x + b1x + γ ) ] 2 + 2 2a 2a1
{ a + a1 > 0}.
2 2 π i β − iβ1 2. ∫ z exp (− αz + βz )sin (α1z + β1z + γ ) dz = × 8 (α + iα1 ) α + iα1
()β−i β 2 β−ii β β+ β × 1− γ α+ α −11 − × exp ierf iz1 44α+i α1 2α+i α1 () α− ii α 11 α− α
2 exp −αzz2 + β (β+i β1) β+i β ( ) ×exp +i γerf α− iz α − 1 − × 44α−i α 1 α− α 22 1 2 i 1 22α+α1
×αsin αzz22 +β+γ+αcos α zz +β+γ {α 2 ≠ −α 2 }. ( 11) 1( 11) 1
22 πβ±β()1 i 3. ∫ zexp(−α z +β z)sin ( i α z +β1 z +γ) dz = × 16αα 2
()ββi 2 ββi() β izi β± ×exp 1iγerf 2 α z −11 + exp[βzi ± ( β z +γ ) ] α 2 1 8 22α 2(β±i β1 )
i 2 exp− 2 αz +β zi ( β z +γ ) {β ≠ ±iβ} . 8α 1 1
πβi β2 4. zexp −α+β z22 zsin i α±β+γ=± z i z dz exp i γ× ∫ ( ) () 82αα 2α
2 β i 2 iz ×erf 2 αz − exp(−α 2z + 2 β zi γ) exp (±γi ) . 2α 84α 1.16.3.
2 − 2 − n 2 2 n! ab ab1 2a1bb1 1. ∫ x exp (− ax + bx)sin (a1 x + b1 x + γ ) dx = exp × n 2 + 2 2 4a 4a1
a b22−+ a b2 abb ×sin1 11 1+γ ReV(16) ( naa , , , bb , , x ) + 22 1 11 44aa+ 1
34 a b22−+ a b2 abb +cos 1 11 1+γ ImV(16) ( naa , , , bb , , x ) − 22 1 11 44aa+ 1
−exp −axbx22 + sin axbx+ +γ Re V(16) ( naabbx , , , , , ) + ( ) ( 1 1 ) 2 11
+ 2 + + γ (16) + > . cos (a1x b1x ) ImV2 (n, a, a1 , b, b1 , x)] { a a1 0}
2 n 2 2 n!i (β − iβ1 ) 2. z exp(− α z + β z ) sin (α z + β z + γ ) dz = exp − i γ × ∫ 1 1 n+1 2 4α + 4iα1
(β + iβ )2 × (16) α − α β − β − 1 + γ (16) α α β β + V1 (n, , 1 , , 1 , z) exp i V1 (n, , 1 , , 1 , z) 4α − 4iα1
+ − α − α 2 + β + β + γ (16) α α β β − exp[ ( i 1 ) z ( i 1 ) z i ]V2 (n, , 1 , , 1 , z)
− − α + α 2 + β − β − γ (16) α − α β − β exp [ ( i 1 )z ( i 1 ) z i ]V2 (n, , 1 , , 1 , z)
α 2 ≠ −α 2 { 1 }.
2 n 2 2 n!i (β i β ) 3. z exp(− αz + β z ) sin ( i α z + β z + γ )dz = ± exp 1 i γ × ∫ 1 n+1 2 8α
× (16) α α β β − − α 2 + β β γ (16) α α β β ± V1 (n, , i , , 1 , z) exp [ 2 z ( i 1 ) z i ]V2 (n, , i , , 1 , z)
− n! n (−1) n k z k ± exp [β z ± i(β z + γ)] ∑ {β1 ≠ ±iβ} . 2i 1 n+1−k k =0 k!(β ± iβ1 )
n!i β2 4. z n exp − αz 2 + βz sin iαz 2 ± iβz + γ dz = ± exp i γ × ∫ ( ) ( ) + 2n 1 2α
×(16) α αβ−β − −α2 + β γ(16) α αβ−β V12(, ni , , , iz , ) exp( 2 z 2 ziV) (, ni , , , iz , )
izn+1 exp (±γi ) . 22n +
35 Introduced notations:
π β+i β 1) (16) α α β β = α− α −1 × V1 (,,,,,) n11 z erf iz1 22α−ii α11α− α
En( /2) ()β+i β nl−2 × ∑ 1 , − α− α nl− l=0 ln!( 2 l )!( i1 ) π (16) α α β β = α− α V1 (2 n11 , , , , iz , ) erf ( i1 z) , n1 2 ni11!2(α−α ) α −α i
(16) + α α β β = V1 (2n1 1, , 1 , , i , z) 0 ;
E (n / 2) (β + iβ ) n−2l l r!(2α z − 2i α z − β − iβ ) 2r−1 2) (16) α α β β = 1 1 1 + V2 (n, , 1 , , 1 , z) ∑ ∑ l!(n − 2l )! α − α n−l+r l=1 r=1 (2r )!( i 1 ) n−E ( n / 2) (l −1)!(β + iβ ) n+1−2l l 4l−r (2α z − 2i α z − β − iβ ) 2r−2 + ∑ 1 ∑ 1 1 , (2l −1)!(n +1 − 2l )! n−l+r l=1 r=1 (r −1)!(α − iα1 )
− 1 n1 l!(2 z) 2l 1 (16) α α β β = V2 (2n1 , , 1 , , i , z) ∑ , n ! n1 +1−l 1 l=1 (2l )!(α − iα1 )
n n !41 n1 z2l (16) + αα β β = 1 V2 (2 n11 1, , , , iz , ) ∑ . (2n + 1)! nl1 +−1 1 l=0 li!(α− α1 )
n 1.17. Integrals of the form ∫ z erf (α z + β) exp (β1z ) sin (β 2 z + γ ) dz
1.17.1. erf ( a x + b ) 1. ∫ erf ( a x + b ) exp (b1x)sin( b2 x + γ ) dx = exp (b1 x )× b 2 + b 2 1 2 2 − 2 − 1 b1 b2 4a bb1 ×[ b1 sin (b2 x + γ ) − b2 cos (b2 x + γ )] + exp × 2 + 2 2 b1 b2 4 a b b − 2 a b b2 b + i b × cos 1 2 + γ b Re erf a x + b − 1 2 − 2 2 2 a 2a b + ib b b − 2 a b b2 − b Im erf a x + b − 1 2 − sin 1 2 + γ × 1 2a 2 2a
36 ++ b12 ib b12 ib ×bRe erf ax+− b +bIm erf ax+− b . 1222aa
erf i (a x + b) 2. ∫ erf i(a x + b) exp(b1 x) sin (b2 x + γ ) dx = exp(b1 x)[b1 sin (b2 x + γ )− 2 + 2 b1 b2
b2 − b2 − 4 a b b 1 2 1 1 b1b2 + 2 a b b2 − b2 cos (b2 x + γ )] + exp cos − γ × 2 + 2 2 2 b1 b2 4 a 2 a
b1 + i b2 b1 + i b2 × b2 Re erfi a x + b + − b1 Im erfi a x + b + + 2 a 2 a
b b + 2 a b b b + i b + 1 2 2 − γ + + 1 2 + sin b1 Re erfi a x b 2 a 2 2 a
b1 + i b2 + b2 Im erfi a x + b + . 2 a
2 1 b bb 2ab − ib 3. erf (ax + b)sin (b x + γ )dx = exp − 1 cos 1 − γ Reerf ax + 1 + ∫ 1 b 2 a 2a 1 4a
bb1 2ab − ib1 1 + sin − γ Imerf ax + − erf (ax + b)cos(b1 x + γ ) . a 2a b1
2 1 b bb 2ab + i b 4. erf i (ax + b)sin (b x + γ ) dx = exp 1 cos 1 − γ Re erfi ax + 1 + ∫ 1 2 b1 4a a 2a
bb1 2ab + ib1 1 + sin − γ Imerfi ax + − erfi (ax + b)cos (b1x + γ) . a 2a b1
1 1 5. ∫ erf (α z + β) exp( β1z ) sin ( β2 z + γ ) dz = erf (α z + β) × 2 iβ1 − β2
1 × exp[ β1 z + i (β 2 z + γ ) ]− exp[ β1 z − i (β 2 z + γ ) ] + iβ1 + β 2
1 1 (β1 − iβ 2 )(β1 − 4αβ − iβ 2 ) β1 − iβ 2 + exp −i γ erf α z + β − − 2 2 iβ1 + β 2 4α 2α
37 1 (β1 + i β 2 )(β1 − 4 α β + i β 2 ) β1 + i β 2 − exp + i γ erf α z + β − 2 i β1 − β 2 4 α 2 α
β22 +β ≠ { 120 }.
2 1 4 i αββ1 − β iβ 6. erf (α z + β)sin (β z + γ)dz = exp 1 − i γ erf α z + β + 1 + ∫ 1 β 2 α 2 1 4α 2
2 β + 4i αββ1 iβ 1 + exp i γ − 1 erf αz + β − 1 − erf (αz + β) cos (β z + γ) . 2 α β 1 4α 2 1
iiexp( γ) β 1.17.2. ∫erf(αz +β) exp( ± i β11 z) sin( β z +γ) dz =± z+erf ( α z +β) + 2 α
1 2 1 + exp [ − (α z + β) ] − erf (α z + β) exp[ ± i (2β1z + γ ) ] − π α 4β1
β2 ±2ii αββ β −exp − 11±iz γerf α +β 1 . 2 α α 1.17.3. 2 2 b − b − 4abb1 1. x n erf (ax + b)exp(b x)sin (b x + γ )dx = exp 1 2 × ∫ 1 2 2 4a
b b − 2a bb b + i b × 1 2 2 + γ + − 1 2 (17) + sin Re erf ax b V1 (a, b, b1 , b2 ) 2a 2 2a
b b − 2abb b + ib + 1 2 2 + γ + − 1 2 (17) − × cos Im erf ax b V1 (a, b, b1 , b2 ) exp (b1 x) 2a 2 2a
(17) 2a (17) × sin (b2 x + γ ) erf (ax + b) Re V (b1 , b2 , x) + Re V (a, b, b1 , b2 , x) + 2 π 3
(17) 2a (17) + cos (b2 x + γ )erf (ax + b) ImV (b1 , b2 , x) + ImV (a, b, b1 , b2 , x) 2 π 3
{ b1 + b2 > 0} . 2 2 b − b − 4abb1 2. x n erf i (ax + b)exp (b x) sin (b x + γ) dx = exp 2 1 × ∫ 1 2 2 4a
38 2abb + b b b + ib × 2 1 2 − γ + + 1 2 (17) − cos Im erfi ax b V1 (i a, ib, b1 , b2 ) 2a 2 2a 2abb++ b b b ib − 2 12−γ ++1 2 (17) − × sin 2 Re erfi ax b V1 ( i a , ib , b12 , b ) exp ( b1 x ) 22aa
(17) 2a (17) × sin (b2 x + γ ) erfi (ax + b) ReV (b1 , b2 , x) + ReV (i a, i b, b1 , b2 , x) + 2 π 3
(17) 2a (17) + cos (b2 x + γ ) erfi (ax + b)ImV (b1 , b2 , x) + Im V (i a, ib, b1 , b2 , x) 2 π 3
{ b1 + b2 > 0} .
2 2 1 β − β − 4αββ1 3. z n erf (α z + β)exp(β z )sin (β z + γ ) dz = exp 1 2 × ∫ 1 2 2i 2 4α
β β − 2αββ β + i β × 1 2 2 + γ α + β − 1 2 (17) α β β β − exp i erf z V1 ( , , 1 , 2 ) 2α 2 2α
2αββ − β β β −i β − 2 12−γ α+β−1 2 (17) αββ−β+ exp ierf zV1 (,,,)12 2α2 2α
exp (β z ) 2α + 1 −β+γ α+β(17) β−β +(17) αββ−β − exp iz( 2 ) erf ( z ) V23 (12 , , z ) V(,,,,)12z 2i π
(17) 2α (17) − exp [i (β2 z + γ ) ] erf (α z + β) V (β1 , β2 , z) + V (α, β, β1 , β2 , z) 2 π 3
β2 + β2 ≠ { 1 2 0}.
β2 ± αββ n 1 1 2i 1 1.17.4. z erf (α z + β)exp (±iβ z)sin (β z + γ )dz = ± exp − ± i γ × ∫ 1 1 2 2i α
i β1 (17) ×erf αz +β V(,,αβ±β±β− i11 , )exp[ ±iz (2 β+γ×1 )] α 1 2α × α+β(17) ±β±β +(17) αβ±β±β ± erf (zV )23 ( i11 , , z ) V ( , , i11 , , z ) π nEn− ( /2) +− iz n+1 (− 1)n n ! βnk12 ±exp (iz γ ) erf ( α +β ) + ∑ + 21n + nk+1 α k=0 4kn !(+− 1 2 k )! (−1) n n! E (n / 2) k!β n−2k k (αz + β) 2l + exp [− (αz + β) 2 ] − n+1 ∑ ∑ πα k =0 (2k +1)!(n − 2k )! l=0 l!
39 − − n E (n / 2) β n+1−2k k l! (αz + β) 2l 1 − ∑ ∑ {β1 ≠ 0} . + − k −l k =1 k!(n 1 2k )!l=1 4 (2l )!
Introduced notations:
nk+−1 − n n!1Ek( /2) (2β − αβ +i β )kl2 1) V (17) (,,αββ , β ) = − 12, 1 12 ∑∑k 22kl− kl=002 β12 +βi = lk!(−α 2 l )!
n! E (n / 2) 1 V (17) (α,β, β , iβ − 2i αβ) = ; 1 1 1 n+1 ∑ n+1−2k (2α) k=0 k!(−β)
n+1−k n n!z k 1 2) (17) β β = − ; V2 ( 1 , 2 , z) ∑ k=0 k! β1 + iβ2
n+1−k n n! 1 3) V (17) (α,β, β , β , z) = exp [− (αz + β) 2 ] − × 3 1 2 ∑ k k=12 β1 + iβ2
E (k / 2) (β − 2αβ + iβ ) k −2l l r!(2α 2 z + 2αβ − β − iβ ) 2r−1 × ∑ 1 2 ∑ 1 2 + − 2k −2l+2r l=1 l!(k 2l )! r=1 (2r )!α
k− Ek( /2) (l− 1)!( β − 2 αβ + i β )k+−12 ll 4 lr − (2α2 zi + 2 αβ − β − β ) 2r− 2 + 1 2 12 , ∑∑2klr−+ 22 lr=11(2lk− 1)!( +− 1 2 l )! = (r −α 1)!
(17) α β β β − αβ = − α +β2 × V3 (,,,11 i 2 iz ,)exp( z )
En( /2) n!!k−1 lz21l+ ×+ ∑∑nk+−12 nl+−12 kl=10kl!(−β )= (2+ 1)!(2 α )
− n E (n / 2) n!k! k−1 z 2l + . ∑ n+2−2k ∑ n+2−2l k=1 (2k )!(−2β) l=0 l!α
21n+ 2 2 1.18. Integrals of the form ∫ zerf (α z +β )exp( α12z) sin ( α z +γ) dz
1.18.1. 2 2 2 erf (ax + b)exp(a1 x ) 1. ∫ x erf (ax + b)exp (a1 x )sin (a2 x + γ )dx = × 2 + 2 2a1 2a2
40 2 2 − 2 − 2 2 2 a b (a a1 a1 a2 ) × [a1 sin (a2 x + γ )− a2 cos(a2 x + γ ) ]− exp × 2 + 2 4 + 2 + 2 − 2 2a1 2a2 a a1 a2 2a a1
aab22 × sin2 +γ × 422 2 aaa++−122 aa 1
a12− ia 2 ab ×Re erf a−− a12 ia x + + 22 a−− a12 ia a−− a12 ia
aab22 +cos 2 +γ × 422 2 aaa++−122 aa 1
a− ia ab ×Im 12erf a2 −− a ia x + 12 22−− −− a a12 ia a a12 ia
2 { a1 + a2 > 0, a − a1 + a2 > 0}.
2 2 2 erfi (ax + b)exp (a1 x ) 2. ∫ x erfi (ax + b)exp(a1 x ) sin (a2 x + γ )dx = × 2 + 2 2a1 2a2
2 2 + 2 + 2 2 2 a b (a a1 a1 a2 ) × [a1 sin (a2 x + γ )− a2 cos (a2 x + γ ) ]− exp × 2 + 2 4 + 2 + 2 + 2 2a1 2a2 a a1 a2 2a a1
a− ia ab ×Re 12erfia2 ++ a ia x + × 12 22++ ++ a a12 ia a a12 ia
aab22 aab22 ×sin22+γ +cos+γ × 422 2 422 2 aaa+++1222 aa 1 aaa+++12 aa 1 a− ia ab ×Im 12erfia2 ++ a ia x + 12 22++ ++ a a12 ia a a12 ia
2 { a1 + a2 > 0, a + a1 + a2 > 0}.
2 2 2 2 2 a a1 b a a1b 3. ∫ x erf (ax + b)sin (a1 x + γ))dx = exp − cos + γ × 2a 4 + 2 4 + 2 1 a a1 a a1
22 1 ab a ab ×Re erf a2 − ia x + −sin 1 +γ × 1 42 22aa+ a−− ia11a ia 1
41 1 ab erf (ax+ b ) ×Im erf a22− ia x + − cos a x +γ 11 ( ) 22−− 2a1 a ia11a ia
{ a1 > 0}.
2 2 2 2 2 a a1 b a a1b 4. ∫ x erfi (ax + b)sin (a1 x + γ ) dx = exp cos + γ × 2a 4 + 2 4 + 2 1 a a1 a a1
2 2 1 ab a a b × Re erfi a 2 + ia x + − sin 1 + γ × 1 2 2 a 4 + a 2 a + ia1 a + ia1 1
+ 1 2 ab erfi (ax b) 2 × Im erfi a + ia1 x + − cos (a1x + γ ) 2 2 2a1 a + ia1 a + ia1
{ a1 > 0}.
erf (αzz +β )exp α 2 22 ( 1 ) 5. ∫ zerf (α z +β )exp( α12z) sin ( α z +γ ) dz = × α+α22 2212
2 2 α × [ α1 sin (α 2 z + γ )− α 2 cos (α 2 z + γ ) ]+ × 2 4(α 2 + iα1 ) α − α1 + iα 2 β2 () α −αi αβ ×exp 12−i γerf α2 −α +iz α + + 2 12 α −α +i α 2 12 α −α12 +i α
α β2 (α + iα ) + exp 1 2 + i γ × 2 α 2 − α − iα 4(α 2 − iα1 ) α − α1 − iα 2 1 2 2 αβ × erf α − α1 − iα 2 z + 2 α − α1 − iα 2
α 2 ≠ −α 2 α 2 − α 2 ≠ −α 2 { 2 1 , ( 1 ) 2 } .
2 α 1 α β 6. z erf (α z + β)sin (α z 2 + γ )dz = exp 1 − i γ × ∫ 1 2 4α1 2 i α − α α + iα1 1 αβ 1 αβ2 ×erf α2 +iz α + + exp iγ−1 × 1 2 22α +αi α+αii11 α−α 1 αβ erf (αz + β) × α 2 − α + − α 2 + γ α 2 ≠ −α 4 erf i 1 z cos ( 1z ) { 1 }. 2 2α1 α − iα1
42 1.18.2.
1 1 β2 ± α + β α 2 2 + γ = − γ × 1. ∫ z erf [(1 i ) z ]sin (2 z ) dz exp i 8α 2 2 2
β 1 2 × erf (1 ± i ) 2 α z + − exp[− 2(1 ± i )αβz − β ± iγ ]− 2 π β
− exp[± i(2α 2 z 2 + γ )]erf [(1 ± i )αz + β] − exp[ i(2α 2 z 2 + γ )]×
× erf [(1± i )αz + β] .
(1 ± i )exp(± iγ ) exp( iγ ) 2. ∫ z erf [(1 ± i )αz]sin(2α 2 z 2 + γ ) dz = z + × 4 πα 8 2α 2
1 × erf [(1 ± i ) 2 αz ] − erf [(1 ± i )αz]× 8α 2
2 2 2 2 × exp[± i(2α z + γ )]+ exp[ i(2α z + γ )] .
2 2 1 α 3. ∫ z erf (αz + β)exp(± iα1z )sin (α1z + γ ) dz = × 8α1 2 α 2iα1
2 2α β αβ × ± 1 + γ α 2 α + − exp i erf 2i 1 z α 2 2iα 2 1 α 2iα1
2 2 2 2 i 2α z − 2β −1 − exp[± i(2α z + γ )]erf (αz + β) ± × 1 2 2 4α αz − β 2 ×exp( iγ )erf (αz + β) + exp[ iγ − (αz + β) ] . π (1 ± i )αz − β 4. ∫ z erf [(1 ± i )αz + β]exp(± iα 2 z 2 )sin (α 2 z 2 + γ ) dz = × 8 πα 2
2 exp(± iγ ) × exp iγ −[(1 ± i )αz + β] − × 8α 2
43 1 × erf [(1 ± i )αz + β]exp(± 2iα 2 z 2 )+ exp[− 2αβz(1 ± i ) − β 2 ] − π β
exp( iγ ) − (1 + 2β 2 4iα 2 z 2 ) erf [(1 ± i )αz + β]. 16α 2
(1 ± i )z 5. ∫ z erf [(1 ± i )αz]exp(± iα 2 z 2 )sin (α 2 z 2 + γ ) dz = × 4 πα
1 2 2 1 × exp(± iγ ) + exp[ i(2α z + γ ) ] − erf [(1 ± i )αz]× 2 8α 2
exp( iγ ) × exp[± i(2α 2 z 2 + γ )]+ (1 4iα 2 z 2 ) . 2
2 2 2 i exp( iγ ) 6. ∫ z erf (αz + β)exp[(α ± iα1 )z ]sin (α1z + γ ) dz = ± × 4α 2
1 × exp(− 2αβz − β 2 )+ erf (αz + β)exp(α 2 z 2 ) − π β
erf (αz + β) (1 i )α − exp (α 2 ± 2iα )z 2 ± iγ − × 2 [ 1 ] 2 4(2α1 iα ) 8 α1 (α ± 2iα1 )
α 2 ± 2iα αβ × exp± i β 2 1 + γ erf (1 i ) α z + . α 1 2 1 (1 i ) α1
2 2 2 i exp( iγ ) 7. ∫ z erf (αz )exp[(α ± iα1 )z ]sin (α1z + γ ) dz = ± × 2α
erf (αz ) 2 2 z exp(± iγ ) × exp(α z )− − × 2α π 4
erf (αz ) 2 2 (1 i )α × exp (α ± 2iα )z + erf [(1 i ) α z ] . 2 [ 1 ] 2 1 2α1 iα 2 α1 (α ± 2iα1 )
1.18.3.
2 2 − 2 − 2 2n+1 2 2 n!a b (a a1 a1 a2 ) 1. ∫ x erf (ax + b)exp (a1 x )sin (a2 x + γ ) dx = exp × 2 4 + 2 + 2 − 2 a a1 a2 2a a1
44 aab22 ×sin 2 +γ ReV(18) ( aa , , a , bx , ) + 422 2 1 12 aaa++−122 aa 1
aab22 +cos 2 +γ × 422 2 aaa++−122 aa 1
n! × (18) − 2 2 + γ + × ImV1 (a, a1 , a2 , b, x) exp (a1x ) sin (a2 x ) erf (ax b) 2
a × (18) + (18) +2 +γ × ReV23 (,,) aa12 x Re V (,,,,) aaabx12 cos ( ax2 ) π
(18) a (18) ×+erf (axb ) Im V23 ( aa12 , , x )+ Im V ( aaabx ,12 , , , ) π
2 { a1 + a2 > 0, a − a1 + a2 > 0}.
2 2 + 2 + 2 2n+1 2 2 n!a b (a a1 a1 a2 ) 2. ∫ x erfi (ax + b) exp (a1x )sin (a2 x + γ ) dx = exp × 2 4 + 2 + 2 + 2 a a1 a2 2a a1
aab22 ×sin 2 +γ ReV(18) ( i a , a , a , ib , x ) + 422 2 1 12 aaa+++122 aa 1
aab22 +cos 2 +γ × 422 2 aaa+++122 aa 1
n! × (18) − 2 2 + γ + × ImV1 (ia, a1 , a2 , ib, x) exp(a1 x ) sin (a2 x ) erfi (ax b) 2
(18) a (18) × + +2 +γ × ReV23( aa12 , , x) Re V( iaaaibx ,12 , , ,) cos( ax2 ) π
(18) a (18) ×+erfi(axb) Im V23 (,,) aa12 x+ Im V (,,,,) iaaaibx12 π
2 { a1 + a2 > 0, a + a1 + a2 > 0} .
2 2n+1 2 2 n!i α β (α1 − iα 2 ) 3. z erf (α z + β) exp (α1z )sin (α 2 z + γ ) dz = exp − i γ × ∫ 4 2 α − α1 + iα 2 2 (18) β (α1 + i α 2 ) (18) n!i ×V (α, α1 , − α 2 , β, z) − exp + i γ V (α, α1 , α 2 , β, z) + × 1 2 1 4 α − α1 − iα 2
45 2 2 (18) α (18) × exp [α1z + i(α 2 z + γ)] erf (α z + β) V (α1 , α 2 , z) + V (α, α1 , α 2 , β, z) − 2 π 3
22(18) α (18) −exp α1zi − ( α 2 z +γ ) erf ( α z +β ) V23 ( α12 , −α , z ) + V ( α , α12 , −α , β ,z ) π
α 2 ≠ −α 2 α 2 ≠ − α 2 − α 2 { 2 1 , 2 ( 1 ) } . 1.18.4.
2 2n+1 n! i β 1. ± α + β α 2 2 + γ = ± ± α − γ × ∫ z erf [(1 i) z ] sin (2 z ) dz (1 i) exp i 4 2
2α 2α n!i × (18) α 2 β − ± γ (18) β α 2 2 + γ × V1 ( , 0, 2 , , z) exp ( i ) V4 ( , ) exp [ i (2 z )] 1 i 1 i 4
(18) 2 (1± i)α (18) 2α 2 × erf [ (1± i)α z + β] V (0, 2α , z) + V ( , 0, 2α , β, z) 2 π 3 1 i {α ≠ 0}.
2 21n+ n! 2αβ1 2. zerf (α+β z )exp ±α i z22 sin α+γ=± z dz αexp ±γ−i × ∫ ( 11) ( ) 4i 2 2α1 ±αi
×V(18) (, α ± i α , ±α ,,)exp β z − ± i (2 α z2 +γ )erf( α zV +β )(18) ( ± i α , ±α ,) z + 1211 1 11
α (18) i (18) + V (α, ± i α1 , ± α1 , β, z) ± exp (i γ)V (α, β) π 3 2 5
iα 2 {α ≠ 0, α1 ≠ 0, α1 ≠ }. 2
2n+1 n! 3. z erf [(1 ± i)α z + β] exp (± iα 2 z 2 )sin (α 2 z 2 + γ ) dz = ± exp (±i γ) × ∫ 4i
(18) 2α i (18) 2α ×V ( , β) ± exp(iγ)V ( , β) {α ≠ 0} . 4 1 i 2 5 1 i
21n+ 4. zerf (α z +β )exp α22 ±i α ) z sin α z 2 +γ dz = ∫ ( 11) ( )
46 n!α α±2 2i α =±exp ±i β221 +γ V(18) (,,,,) α α ± iz α ±α β ± α 1 11 42i 1 ni! ±exp α22zi ± (2 α z 2 +γ ) × 4 1
(18) 22α (18) ×erf ( αzV +β ) ( α ± i α11 , ±α , z ) + V ( α , α ± i α11 , ±α , β , z ) + 23π 2 (18) iα +exp (iV γ )4 ( αβ , ) {α ≠ 0, α1 ≠ ± }. 2
Introduced notations:
(18) 1 2 αβ 1) V (α, α1 , α 2 , β, z) = erf α − α1 − i α 2 z + × 1 2 2 α − α1 − iα 2 α − α1 − i α 2
n+1−k − n (2k )! 1 k (αβ) 2k 2l × − = ∑ ∑ 2k −l k =0 k! α1 + iα 2 l=0 l 2 4 l!(2k − 2l )!(α − α1 − iα 2 ) 1 2 αβ = erfi α1 − α + iα 2 z − × 2 2 α1 − α + iα 2 α1 − α + iα 2
n+1−k − n (2k )! 1 k (αβ) 2k 2l × − , ∑ ∑ 2k −l k =0 k! α1 + iα 2 l=0 l 2 (−4) l!(2k − 2l )! (α1 − α + iα 2 )
(18) 1 2 V(α , α12 , α , 0,z ) = erf α −α1 −iz α 2 × 1 2 α −α12 −i α
n (2k )! ×× ∑ k k=0 k 22 4 (ki !) (α −α12 − α )
n+1−k 1 1 2 × − = erfi α1 − α + i α 2 z × α1 + iα 2 2 α1 − α + iα 2
n+1−k n (2k )! 1 × − ; ∑ k k =0 k 2 2 α1 + iα 2 (−4) (k!) (α1 − α + iα 2 )
n+1−k n z 2k 1 2) (18) α α = − ; V2 ( 1 , 2 , z) ∑ k =0 k! α1 + iα 2
47 n+1−k n (2k )! 1 3) (18) α α α β = − α + β 2 − × V3 ( , 1 , 2 , , z) exp[ ( z ) ] ∑ k=1 k! α1 + iα 2 2r−1 k (αβ) 2k −2l l r!(α 2 z + αβ − α z − iα z ) × 1 2 − ∑ ∑ 2k −l+r l=1 l!(2k − 2l )! r=1 l−r 2 4 (2r )!(α − α1 − iα 2 ) 2r−2 k (l −1)!(αβ) 2k+1−2l l (α 2 z + αβ − α z − iα z)) − 1 2 , ∑ ∑ 2k−l+r l=1 (2l −1)!(2k +1− 2l )! r=1 2 (r −1)! (α − α1 − iα 2 )
n+1−k n (2k )! 1 V (18) (α,α , α , 0, z) = exp (− α 2 z 2 ) − × 3 1 2 ∑ 2 k =1 (k!) α1 + i α 2
k l!z 2l−1 × ; ∑ k +1−l l=1 k −l 2 4 (2l )! (α − α1 − iα 2 ) 1 n z 2k 4) V (18) (α, β)= erf (αz + β)exp (α 2 z 2 ) + 4 2 ∑ n−k α k =0 k! (− α 2 )
2 exp(−− 2αβz β ) nk(2kz )! 2 l + , ∑∑−α 22nk−−αβ kl πβ kl=00kl!( )= !(2 )
n 2k (18) 1 z erf (α z) 2 2 2 z V (α,0)= ∑ exp (α z )− ; 4 α n−k α k =0 k!(− α 2 ) (2k + 1) π
2n+2 n+1 2n+2−2k (18) z (2n + 1)! β 5) V (α, β)= erf (α z + β) − + 5 2n+2 ∑ k 2n + 2 α k =0 4 k!(2n + 2 − 2k )!
(2n +1)! n β2n−2k k (l +1)! (α z + β) 2l+1 + exp − (αz + β)2 − 2n+2 [ ] ∑ ∑ k−l πα k=0 (k +1)!( 2n − 2k )! l=0 4 (2l + 2)!
n k! β 2n+1−2k k (α z + β) 2l − ∑ ∑ , k =0 (2k + 1)! (2n +1 − 2k )! l=0 l!
2n+2 (18) z (2n + 1)! V (α, 0)= erf (α z) − + 5 + 2n + 2 (n + 1)!(4α 2 ) n 1
(2n + 1)! n (k + 1)!z 2k +1 + exp(− α 2 z 2 ) . ∑ 2 n−k (n + 1)! πα k =0 (2k + 2)! (4α )
48 PART 2 DEFINITE INTEGRALS
2 zzn exp α +β 2.1. Integrals of ( ) 2.1.1. +∞ 2 π 1. exp −a z +β dz = 1 − erf β { a > 0 }. ∫ ( ) ( ) 0 2a
+∞ 2 π 2. exp −(a z +β) dz = { a > 0 }. ∫ a −∞ ∞ π 3. exp[− (α z + β)2 ] dz = − [erf (β) 1] ∫ 2α 0
{ lim Reα22z +αβ+ 2 zz ln =+∞ , lim Re ( α=±∞z) }. zz→∞ ( ) →∞
∞ π 4. exp[(α z + β)2 ] dz = − [erfi (β) i] ∫ 2α 0
{ lim Reα22z + 2 αβ zz − ln = −∞ , lim Im (αz) = ±∞ }. z→∞ ( ) z→∞
2.1.2. +∞ exp −β2 2 πβ ( ) 1.z exp −( a z +β) dz = erf( β) − 1 + { a > 0 }. ∫ 22 0 22aa
+∞ 2 πβ 2.z exp −( a z +β) dz =− { a > 0 }. ∫ 2 −∞ a
∞ π β exp(− β 2 ) 3. z exp[− (α z + β)2 ] dz = [erf (β) 1]+ ∫ 2 2 0 2α 2α
{ lim Re (α 2 z 2 + 2αβ z )= +∞, lim Re (α z ) = ±∞ }. z→∞ z→∞
∞ π β exp (β 2 ) 4. z exp [(α z + β) 2 ] dz = [erf i (β) i] − ∫ 2 2 0 2α 2α
{ lim Re(α 2 z 2 + 2αβ z ) = −∞, lim Im(α z ) = ±∞ }. z →∞ z →∞
49 2.1.3. +∞ π β22 + β −β 2 (2 1) exp( ) 1.z2 exp −( a z +β) dz = 1 − erf ( β) − { a > 0 }. ∫ 33 0 42aa
+∞ π21 β+2 2 ( ) 2.z2 exp −( a z +β) dz = { a > 0 }. ∫ 3 −∞ 2a
∞ π (2β 2 +1) βexp (− β 2 ) 3. z 2 exp [ − (α z + β) 2 ] dz = − [ erf (β) 1] − ∫ 3 3 0 4α 2α
{ lim [Re (α 2 z 2 + 2αβ z )− ln z ] = +∞ , lim Re (α z ) = ±∞ }. z→∞ z→∞
∞ π (1 − 2β 2 ) β exp (β 2 ) 4. z 2 exp [(α z + β) 2 ] dz = [erf i (β) i] + ∫ 3 3 0 4α 2α
{ lim [Re(α 2 z 2 + 2αβ z )+ ln z ] = −∞, lim Im (α z ) = ±∞ }. z→∞ z→∞
2.1.4.
+∞ 2 1 (1) (1) 1.zn exp − a z +β dz = W n, a ,β 1 − erf β + W na,,β ∫ ( ) 1 ( )( ) 2 ( ) 0 a { a > 0 }. +∞ 2 2 (1) 2.zn exp − a z +β dz = W n,, a β > ∫ ( ) 1 ( ) { a 0 }. −∞ a
∞ 2 (11) 1 ( ) 3.zn exp −α( z +β) dzWn =( , αβ ,) − erf( β) 1 Wn( , αβ , ) ∫ 21α 0
{ lim Reα22z +αβ−− 2 zn( 1) ln z =+∞ , lim Re ( α=±∞z) }. zz→∞ ( ) →∞
∞ 2 (11) 1 ( ) 4.zn exp (α+β z) dz = W( ni, αβ− , i ) erf i ( β) iW ( ni,, αβ i ) ∫ 21α 0
{ lim Reα22z +αβ+− 2 zn( 1) ln z =−∞ , lim Im ( α=±∞z) }. zz→∞ ( ) →∞
50 Introduced notations:
En( 2) nk−2 (1) n! π β 1) Wn( ,,αβ) = , 1 nk∑ 2(−α) k=0 4kn !( − 2! k)
(1) (2!m) π (1) Wm1 (2 ,α= ,0) , Wm1 (2+α 1, ,0) = 0 ; 2!21mm+ m α 2
n!exp −β2 n− En( 2) (1) ( ) (k −1!) 2) Wn( ,,αβ) = ∑ × 2 n+1 − +− 2(−α) k=1 (2kn 1!) ( 1 2 k) !
− − + ( ) n−1−2k + 2l k β n 1 2k 2l E n 2 1 k l! β × ∑ − ∑ ∑ , − − k −l l =1 (l 1)! k =1 k ! (n 2k )! l =1 4 (2l )!
(1) (1) m! Wm2 (2 ,α= ,0) 0, Wm2 (2+α 1, ,0) = . 2 α22m+
2.2. Integrals of zn exp (α22 zz +β +γ)
2.2.1. +∞ π β 2 β 1. exp(− a 2 z 2 + β z + γ ) dz = exp + γ 1+ erf . ∫ 2a 2 2a 0 4a
+∞ π β 2 2. exp (− a 2 z 2 + β z + γ ) dz = exp + γ . ∫ 2 −∞ a 4a ∞ 2 22 π ββ 3. exp −αz +β z +γ dz = exp +γerf ± 1 ∫ ( ) 22αα2 0 4α
{ lim Re α22zz − β +ln z = +∞ , lim Re (αz) = ±∞ }. z→∞ ( ) z→∞ ∞ 2 22 π ββ 4. exp αz +β z +γ dz =−exp γ− erfi i ∫ ( ) 22αα2 0 4α
{ lim Re α22zz + β −ln z = −∞ , lim Im(αz) = ±∞ }. z→∞ ( ) z→∞
51 2.2.2. +∞ πβ ββ2 exp(γ) 1.z exp − a22 z +β z +γ dz = exp +γ1 + erf + . ∫ ( ) 32 2 0 44aa2a 2a
+∞ π β β 2 2. zexp(− a 2 z 2 + β z + γ ) dz = exp + γ . ∫ 3 2 −∞ 2a 4a
∞ π β β 2 β 3. z exp (− α 2 z 2 + β z + γ ) dz = exp + γ erf ±1 + ∫ 3 2 2α 0 4α 4α exp(γ ) + { lim Re (α 2 z 2 − β z )= +∞, lim Re (α z ) = ±∞ }. 2α 2 z→∞ z→∞ ∞ 2 22 πβ ββ 4.z exp α z +β z +γ dz =exp γ− erfi i − ∫ ( ) 322α 0 44αα exp(γ ) − { lim Re(α22zz + β) = −∞, lim Im( αz) = ±∞ }. 2α 2 zz→∞ →∞
2.2.3. +∞ π (2a 2 + β 2 ) β 2 1. z 2 exp(− a 2 z 2 + β z + γ ) dz = exp + γ × ∫ 5 2 0 8a 4a β β × 1+ erf + exp(γ ). 2a 4a 4 +∞ π (2a 2 + β 2 ) β 2 2. z 2 exp (− a 2 z 2 + β z + γ ) dz = exp + γ . ∫ 5 2 −∞ 4a 4a ∞ π (2α 2 + β 2 ) β 2 3. z2 exp(− α 2 z2 + β z + γ ) dz = exp + γ × ∫ 5 2 0 8α 4α β β × erf ±1 + exp (γ ) 2α 4α 4
{ lim Re α22zz −β−ln z =+∞ , lim Re( α=±∞z) }. zz→∞ ( ) →∞
∞ π (2α 2 − β 2 ) β 2 4. z 2 exp (α 2 z 2 + β z + γ ) dz = exp γ − × ∫ 5 2 0 8α 4α ββ ×erfi i +γexp( ) 2α 4α4
52 { lim Re α22zz +β+ln z =−∞ , lim Im ( α=±∞z) }. zz→∞ ( ) →∞ 2.2.4.
+∞ n 22 1 β (22) 2 ( ) 2 1.z exp − a z +β z +γ dz =1 + erf W na,, β + W na,,. β ∫ ( ) aa2 12( ) ( ) 0
+∞ n 22 2 (2) 2 2.∫ z exp(− a z +β z +γ) dz = W1 ( n,, a β) . −∞ a
∞ 1 β (2) 3.zn exp −α22 z +β z + γ dz = erf± 1W n , α2 , β + ∫ ( ) αα 1 ( ) 0 2
(2) 2 +Wn2 ( ,, αβ)
{ lim Re α22z −β−− zn( 1) ln z =+∞ , lim Re( α=±∞z) }. zz→∞ ( ) →∞
∞ (2) 1 β 4.zn exp α22 z +β z + γ dz = W n, −α2 , β − erf i i × ∫ ( ) 2 ( ) αα 0 2
(2) 2 ×Wn1 ( ,, −α β)
{ lim Re α22z +β+− zn( 1) ln z =−∞ , lim Im( α=±∞z) }. zz→∞ ( ) →∞
2.2.5. +∞ 0 n n! 1.znn exp(− b z +γ) dz =( −1) z exp( b z+γ) dz =exp( γ) { b > 0 }. ∫∫n+1 0 −∞ b
∞ n! 2.zn exp(β z +γ) dz = exp(γ) { lim [Re(β z ) + nln z ] = −∞ }. ∫ n+1 →∞ 0 (−β) z
Introduced notations:
22En( 2) nk− (2) n! π ββ 1) Wn ,α2 , β = exp+γ , 1 ( ) n+−12∑ nk 24α k=0 kn!( −α 2! k) ( 2 )
53 (2) (2!m) π (2) Wm2 ,α=2 ,0 exp(γ) , Wm2+α 1,2 ,0 = 0 ; 1 ( ) m 1 ( ) 2!21m+ m (α 2)
n− En( 2) (2) nk!exp(γ−) ( 1) ! 2) Wn ,,α2 β= ∑ × 2 ( ) n − +− 2 k=1 (2kn 1!) ( 1 2 k) !
k−l n−1−2 k+2 l E ( n 2 ) n−1−2 k+2 l k 4 β 1 k l !β × ∑ − ∑ ∑ , 2 n−k+l k! ( n − 2 k )! 2 n−k+l l=1 ( l − 1 )! (α ) k=1 l=1 ( 2 l )! (α )
(2) (2) m! Wm2 ,α=2 ,0 0, Wm2+α 1,2 ,0 = exp(γ) . 2 ( ) 2 ( ) m+1 2(α2 )
2.3. Integrals of erf n (α z + β) exp[− (α z + β)2 ]
2.3.1. +∞ 2 π 1. erfaz+β exp −az +β dz =1 − erf 2 β { a > 0 }. ∫ ( ) ( ) ( ) 0 4a
+∞ 2 2. erfaz+β exp −az +β dz =0 ≠ ∫ ( ) ( ) { a 0 }. −∞
∞ 2 π 3. erf(αz +β) exp −( αz +β) dz =1 − erf 2 ( β) ∫ α 0 4
{ lim Reα22z + 2 αβ zz + ln = +∞ }. z→∞ ( ) 2.3.2. +∞ 2 π 1. erf2 az+β exp −az +β dz = −β3 { a > 0 }. ∫ ( ) ( ) 1 erf ( ) 0 6a
+∞ 2 2 π 2. erfaz+β exp −az +β dz = > ∫ ( ) ( ) { a 0 }. −∞ 3a
∞ 2 π 3. erf23(αz +β) exp −( αz +β) dz =−erf( β) 1 ∫ α 0 6
{ lim Reα22z + 2 αβ zz + ln = +∞ , lim Re(αz) = ±∞ }. z→∞ ( ) z→∞
54 2.3.3. +∞ 2 π + 1.∫ erf2mm(az+β) exp −( az +β) dz = 1− erf 21( β) { a > 0 }. (42ma+ ) 0 +∞ ++2 π 2. erf 21mm(az+β) exp −( az +β) dz = 1− erf 22( β) { a > 0 }. ∫ 44ma+ 0 ( )
+∞ 2 π 3.∫ erf2m (az+β) exp −( az +β) dz = { a > 0 }. (21ma+ ) −∞
+∞ 2 4. erf 21m+ az+βexp − az +β dz =0 ≠ ∫ ( ) ( ) { a 0 }. −∞
∞ 21π n+ 5. erfnn(αz +β) exp −( αz +β) dz = (±1) − erf +1 ( β) ∫ +α 0 (22n )
{ lim Reα22z + 2 αβ zz + ln = +∞ , lim Re(αz) = ±∞ }. z→∞ ( ) z→∞
2.4. Integrals of z n erf (α z + β) exp[− (α z + β)2 ]
2.4.1. +∞ 2 πβ 1.z erf( az+β) exp −(az +β) dz =erf2 ( β) − 1 + ∫ 2 0 4a
erf (β) 2 +exp( −β2 ) − erf( 2β) − 1 { a > 0 }. 24aa22 +∞ 2 1 2.z erf( az+β) exp −( az +β) dz = { a > 0 }. ∫ 2 −∞ 2 a
∞ 2 πβ erf (β) 3.z erf(α+β z ) exp −α+β( z) dz =erf22( β−+) 1 exp −β− ∫ 22 ( ) 0 42αα
2 −βerf( 2) 1 { lim Re(α22zz +αβ=+∞ 2) , lim Re( α z) =±∞ }. 4α2 zz→∞ →∞
55 2.4.2. +∞ 2 π 1 1.z2 erf( az+β) exp −+β( az) dz = β+221 − erf ( β+) ∫ 3 0 4a 2
2 exp(−β 2 ) β + − 2 1− erf 2 β +erf( β) exp −β2 { a > 0 }. 3 ( ) ( ) π 2a { }
+∞ 2 2 β 2.z2 erf( az+β) exp −( az +β) dz =− { a > 0 }. ∫ 3 −∞ a
∞ 2 2 π 2 1 2 3. z erf (α z + β) exp [− (α z + β) ] dz = β + [1− erf (β)]+ ∫ 3 2 0 4α 2 exp(− 2β ) β 2 + + 2 [erf ( 2 β)1]− erf (β)exp(− β ) π 2α 3
{ lim [Re (α 2 z 2 + 2αβ z )− ln z ]= +∞, lim Re (α z ) = ±∞ }. z→∞ z→∞
2.4.3. +∞ 2 1.zn erf az+β exp −az +β dz =erf 2 β − 1 × ∫ ( ) ( ) ( ) 0
(4) 1 (44) ( ) ×W( na,, β+) 1erf −2 ( β) W( na,, β+) W( na ,, β) { a > 0 }. 1a { 23}
+∞ 2 (4) 2.zn erf az+β exp − az +β dz =−2 W n , a , β > ∫ ( ) ( ) 1 ( ) { a 0 }. −∞
∞ 2 (4) 3.zn erfα z +β exp − αz +β dz =erf 2 β 1W n ,α , β + ∫ ( ) ( ) ( ) 1 ( ) 0
1 (44) ( ) + 1− erf 2 ( β) Wn( ,,,, αβ) + Wn( αβ) α {23}
{ lim Reα22z +αβ−− 2 zn( 1) ln z =+∞ , lim Re ( α=±∞z) }. zz→∞ ( ) →∞
56 Introduced notations:
n n− En( 2) nk+−12 k−1 (4) (−1) nk ! ( −β1!) (2 l) ! 1)Wn( ,,αβ) = , 1 n+12∑∑l 22α kl=10(2kn− 1!) ( +− 1 2 k) ! = 8!(l )
m (4) (4) m! (2!k ) Wm(2 ,α= ,0) 0, Wm(2+α 1, ,0) =− ; 1 1 22m+ ∑ k 2 22α k=0 8!(k )
En( 2) nk−2 (4) n! π β 2) Wn( ,,αβ) = , 2 nk∑ +1 (−α) k=0 4kn !( − 2! k)
(4) (2!m) π (4) Wm2 (2 ,α= ,0) , Wm2 (2+α 1, ,0) = 0 ; 4!mm+12m α
En( 2) nk−2 (4) n! β 3) Wn( ,α , β) = erf(β) exp −β2 × 3 nk ∑ ( ) (−α) k=1 4kn !( − 2! k)
+ k −1 4l l ! β 2l 1 exp(− 2β 2 ) k −1 (l ! )2 l 2l +r β 2r × ∑ + ∑ ∑ − l =0 (2l + 1)! 2 π l =0 (2l + 1)! r =0 r !
− ( ) + − n E n 2 (k −1)! β n 1 2k erf (β) k −1 β 2l − ∑ exp(− β 2 )∑ + k =1 (2k −1)! (n + 1 − 2k )! 2 l =0 l !
+ exp(− 2β 2 ) k −1 (2l )! l −1 r ! β 2r 1 + , ∑ 2 ∑ l −r π l =1 (l ! ) r =0 8 (2r + 1)!
m−1 2 (4) (2!mk) ( !) (4) Wm(2 ,α= ,0) , Wm(2+α 1, ,0) = 0 . 3 2m∑ 21 mk+− 3 mk!πα k=0 2( 2+ 1!)
57
n 2.5. Integrals of z erf (α z + β)exp(β1z + γ )
2.5.1. +∞ 0 erf (β) 1 1.∫∫ erf(a z+β) exp( − b z +γ) dz =−erf( a z −β) exp( b z +γ) dz =exp( γ) − × 0 −∞ bb
bb2 βb × γ+ + β+ − > > exp erf 1 { a 0, b 0 }. 4a2 aa2
∞ 2 1 β11 ββ β1 2. erf(αz +β) exp( β1 z +γ) dz = exp − +γerf β−1 − ∫ βα2 α 0 1 4α 2 erf (β) − (γ ) { lim Reα22z + 2 αβ zz − β +2ln z = +∞ , exp ( 1 ) β1 z→∞
lim Re (β1 z ) = −∞, lim Re (αz ) = ±∞ }. z→∞ z→∞
2.5.2. +∞ 0 1.∫∫z erf( a z+β) exp ( − b z +γ) dz = zerf( a z −β) exp( b z +γ) dz = 0 −∞
erf (β) 22a22−−β b ab =exp ( γ−) × b2 2ab22
2 bb2 βb exp(γ−β ) ×exp γ+ +erf β+ − 1 + { a > 0, b > 0 }. 2 4a aa2 π ab
∞ erf (β) 22α22 −β + αββ 2.z erf(α z +β) exp( β z +γ) dz =exp( γ) − 11× ∫ 1 2 22 0 β112αβ
2 exp γ−β2 β11 ββ β1 ( ) ×exp − +γerf β− 1 − 2 αα 4α 2 παβ1
22 { lim Reαz + 2 αβ zz − β1 +ln z = +∞ , z→∞ ( )
lim [Re(β1z ) + ln z ] = −∞ , lim Re(αz) = ±∞ }. z→∞ z→∞
58 2.5.3. +∞ 0 n+1 1.∫∫znn erf( a z+β) exp( − b z +γ) dz =( −1) z erf( a z−β) exp( b z +γ) dz = 0 −∞
n!exp(γ) erf ( β) b = −erf β+ − 1 × ba bn 2
(55) 2a ( ) ×Wa12( ,, β− b) − Wa( ,, β− b) { a > 0 , b > 0 }. π
n+1 ∞ n!exp(γ−) ( 1) 2.zn erf(α z +β) exp ( β z +γ) dz = erf (β) + ∫ 1 β n 0 1 β1
β1 (55) 2α ( ) +erf β− 1WW(αββ , , 11) +( αββ,, ) 2α 12 π
22 { lim Reαz +αβ−β 2 z1 zn −−( 2) ln z =+∞ , z→∞ ( )
lim [Re (β1z ) + n ln z ] = −∞ , lim Re(αz) = ±∞ }. z→∞ z→∞
Introduced notations:
− 2 n Ek( 2) kl2 (5) β11 ββ 1 (β1 −2 αβ) 1) W (αββ , ,1 ) = exp − ∑ ∑ , 1 α2 α k nk− −α22kl− 4 k=0 2 (−β1 ) l=0 lk!( 2! l)
exp −β2 En( /2) (5) ( ) 1 Wn( ,α ,2 αβ) = ; 1 n ∑ nk−2 (2!α) k=0 k (−β)
n Ek( 2) 5 11 ( ) 2 2) W2 (α , β , β1 ) = exp( −β ) ∑∑× k nk− lk!( − 2! l) kl=112 (−β1 ) =
− − + l r ! ( β − 2 αβ ) k 1 2 l 2 r k−E ( k 2 ) ( l − 1 )! × ∑ 1 − ∑ × 2 k−l+r ( 2 l − 1 )! ( k + 1 − 2 l )! r =1 ( 2 r )! (α ) l=1
l 4 l−r ( β − 2 αβ ) k−1−2 l+2 r × ∑ 1 , k−l+r r =1 ( r − 1 )! (α 2 )
59 n n− En( /2) (5) (−1) k! W (α, β ,2 αβ) =exp −β2 . 2 n+1 ( ) ∑ nk+−12 α k=1 (2k ) !2( β)
2m+1 2 2.6. Integrals of z erf (αz + β)exp(− α1z )
2.6.1.
+∞ erf (β) a a β2 1. zerf( az+β) exp − a z2 dz = + exp−1 × ∫ ( 1 ) 2 2a1 2 aa+ 0 2aa11+ a 1
aβ × 1− erf {a ≥ 0, a1 > 0}. 2 a + a1
+∞ a a β2 2 1 2.∫ z erfi( az+β) exp( − a1 z) dz = exp × 2 aa− 2 0 2aa11− a 1
aβ erfi(β) ×+1 erf + { a≥>0, aa2 }. 1 2 2a1 aa1 −
+∞ 0 3. ∫ z erfi (a z + b)exp(− a 2 z 2 )dz = ∫ z erfi (a z − b)exp(− a 2 z 2 ) dz = 0 −∞
erf i(b) exp(b 2 ) = − {ab><0, 0}. 2a 2 2 πa 2b +∞ a a β 2 4. ( + β) − 2 = − 1 { ≥ > }. ∫ z erf az exp( a1 z ) dz exp a 0, a1 0 2 a 2 + a −∞ a1 a + a1 1
+∞ a a β2 2 1 > 2 5. ∫ zerfi( az+β) exp( − a1 z) dz = exp { aa1 }. 2 aa− 2 −∞ aa11− a 1
∞ erf (β) α αβ2 6. zerf(α z +β) exp −αz2 dz = − exp −1 × ∫ ( 1 ) 2 2α1 2 α +α 0 2α11 α +α 1
αβ 22 2 × erf 1 { lim Re(αzz + α1 +2 αβ zz) + ln = 2 z→∞ α + α1
60 = α2 = +∞ 2 lim Re( 1 z ) , lim Re α + α1 z = ±∞ }. z→∞ z→∞ ( )
∞ 11 7. zerf(α z +β) exp α22z dz = −erf( β) + exp −β2 ∫ ( ) 2 ( ) 0 2α πβ
{ lim Re α22zz = −∞, lim Re( αβ) = +∞ }. zz→∞ ( ) →∞
2.6.2.
+∞ 21m+ 2 ma! erf (β) 1. ∫ zerf( az+β) exp( − a1 z) dz = + × 2 am+1 2 0 1 aa+ 1
aβ (66) ( ) ×1 − erf W( aa,, β+) aW( aa,, β) {a ≥ 0, a > 0}. 1211 1 2 + aa1
+∞ 21m+ 2 ma! erfi(β) 2. ∫ zerfi( az+β) exp( − a1 z) dz = + × 2 am+1 2 0 1 aa1 −
aβ (66) ( ) ×1 + erf W( iaai,, β+) aW( iaai,, β) a≥>0, aa2 . 1211 { 1 } − 2 aa1
+∞ 0 3. ∫ z 2m+1 erfi (az + b)exp(− a 2 z 2 ) dz = ∫ z 2m+1 erfi (az − b)exp(− a 2 z 2 )dz = 0 −∞ m! 1 = ( ) − (6 ) 2 erfi b W3 (b ) {ab><0, 0}. 2a 2m+2 b +∞ 2m+1 2 m!a (6 ) 4. ∫ z erf (az + β)exp(− a1 z )dz = W (a,a1 , β) {a ≥ 0, a1 > 0}. 2 1 −∞ a + a1 +∞ 2m+1 2 m!a (6 ) 2 5. ∫ z erfi (az + β)exp(− a1 z )dz = W (i a,a1 , iβ) { aa1 > }. 2 1 −∞ a1 − a
∞ 21m+ 2 m!erf (β) α 6. ∫ zerf(α z +β) exp( −α1z) dz = − × 2 αm+1 2 0 1 α +α1
61 αβ × (6 ) (α α β) + α (6 ) (α α β) erf 1 W1 , 1 , W2 , 1 , 2 α + α1
22 2 2 { lim Re α+α+αβ−z11 z2 z( 2 m − 1) ln z = lim Re αz − 2 mz ln =+∞ , zz→∞ ( ) →∞ ( )
2 lim Re α + α1 z = ±∞ }. z→∞
∞ m+1 (−1!) m 1 (6) 7. z2m+ 1erf(α z +β) exp α22z dz = erf (β) +W −β2 ∫ ( ) 22m+ β 3 ( ) 0 2α
22 { lim Reαz + 2 mz ln = −∞ , lim Re(αβz) − mz ln = +∞ }. zz→∞ ( ) →∞
Introduced notations:
2 2k −2l ( ) α β m (2k )! k (αβ) 1) W 6 (α,α , β)= exp − 1 , 1 1 2 ∑ m+1−k ∑ 2k −l α + α k =0 k!α l=0 l 2 1 1 4 l!(2k − 2l )!(α + α1 )
m (6) (2!k ) W (αα, ,0) = ; 1 1 ∑ k k=0 k2 mk+−12 4!(k ) α11( α +α )
2 −− + exp −β mk l2k 12 lr 2 (6) ( ) (2!k ) 1 (αβ) 2) W (αα,, β) = × 2 1 ∑∑∑m+−12 k klr−+ π klr=11k!α = (2kl− 2!) = 12 1 (α +α1 )
r! (l −1)! (6 ) × − , W (α,α , 0) = 0 ; − 2 1 4l r l!(2r )! (2k +1 − 2l )(2l −1)!(r −1)!
( ) 1 m (2k )! 3) W 6 (β 2 )= exp(β 2 ) . 3 ∑ k π k=0k!(4β 2 )
62 n 2 2.7. Integrals of z exp(− αz + βz ) erf (α 1 z + β 1 ),
n 2 z exp(− αz ) erf (α 1 z + β 1 )erf (α 2 z + β 2 )
2.7.1. +∞ 1 a 1. −=2 1 {a > 0}. ∫ exp( az) erf ( a1 z) dz arctan 0 π aa
+∞ 2 1 a + a1 2 2. ∫ exp(− az )erfi (a1 z )dz = ln { aa> 1 }. 0 2 πa a − a1 +∞ +∞ 1 3. exp−=az22 erf( a z) erf ( a z) dz exp−=az erf( a z) erf ( a z) dz ∫∫( ) 122 ( ) 12 0 −∞ 1 aa = arctan 12 {a > 0}. π 222 a a++ aa12 aa
+∞ +∞ 1 4. exp(− az 2 )erf (a z )erfi (a z )dz = exp(− az 2 ) erf (a z )erfi (a z )dz = ∫ 1 2 2 ∫ 1 2 0 −∞
2 + 2 − 2 + 1 a aa1 aa2 a1a2 2 = ln { aa> 2 }. 2 πa 2 + 2 − 2 − a aa1 aa2 a1a2 +∞ +∞ 1 5. exp(− az 2 )erfi (a z )erfi (a z )dz = exp(− az 2 ) erfi(a z )erfi (a z )dz = ∫ 1 2 2 ∫ 1 2 0 −∞
1 aa12 22 = arctan { aa>+12 a}. π 222 a a−− aa12 aa
+∞ π β 2 2aβ + a β − 2 + β + β = 1 1 { > } 6. ∫ exp( az z ) erf (a1 z 1 )dz exp erf a 0 . −∞ a 4a 2 + 2 2 a aa1 +∞ π β 2 2aβ + a β − 2 + β + β = 1 1 7. ∫ exp( az z )erfi (a1 z 1 )dz exp erfi −∞ a 4a 2 − 2 2 a aa1
2 { aa> 1 }. ∞ 2 1 α1 22 8. exp−=ααz erf ( z) dz arctan { α1 ≠ −α, lim Re αzz + ln = ∫ ( ) 1 z→∞ ( ) 0 πα α
2 22 =lim Re αzz + α1 +2 ln z = +∞ }. z→∞ ( )
63 ∞ 2 1 αα12 9. ∫ exp(−αz) erf( α12 z) erf ( α z) dz = ± arctan πα 22 0 α α+α12 +α
2 2 22 { lim Reαz + ln z = lim Re α zz +α1 +2ln z = zz→∞ ( ) →∞ ( )
2 22 2 22 22 =lim Re αzz +α2 +2ln z = lim Re α zzz +α12 +α +3ln z =+∞ , zz→∞ ( ) →∞ ( )
2 2 22 α1 ≠ −α, α 2 ≠ −α , lim Re α+α12 +αz = ±∞ }. z→∞ ( )
∞(T ) 2 π β 2 2αβ + α β 10. exp(− αz 2 + βz ) erf (α z + β )dz = A exp erf 1 1 ∫ 1 1 α ∞( ) α 4 α α + α 2 T1 2 1
{ lim Re αz2 −β z +ln z = lim Re αz2 +α 22 z +2 α β zz −β +2ln z = ( ) ( 1 11 ) zT→∞( 11) zT→∞( )
= α 2 − β + = α 2 + α 2 2 + α β − β + = +∞ ; lim [Re( z z ) ln z ] lim [Re( z 1 z 2 1 1 z z ) 2ln z ] z→∞(T2 ) z→∞(T2 )
= ± α+α22 =− α+α =±∞ A 1 if lim Re( 11zz) lim Re( ) , zT→∞( 21) zT→∞( )
= α+α22 ⋅ α+α =+∞ A 0 if lim Re( 11zz) lim Re( ) }. zT→∞( 12) zT→∞( )
ν 2 11. ∫ exp(− α z )erf (α1 z )dz = 0 . −ν
2.7.2.
+∞ aa 1. zexp−= a z2 erf( a z) erf ( a z) dz 12arctan + ∫ ( ) 12 22 0 π aaa++11 aa
aa + 21arctan {a > 0}. 22 π+aaa22 aa +
+∞ + 2 + 2 1 a1 a a1 a2 2. ∫ z exp(− az )erf (a1 z ) erfi (a2 z )dz = ln + πa + 2 + 2 − 0 2 a a1 a a1 a2 aa212 + arctan { aa> 2 }. 22 aa−−22 aa
64 +∞ − 2 + 2 1 a1 a a1 a2 3. ∫ z exp(− az )erfi (a1 z )erfi (a2 z )dz = ln + 2πa − 2 − 2 − 0 a a1 a a1 a2
−+2 a2 aa21 a 22 + ln { aa>+12 a}. 22 aa−2 aa −− 21 a
+∞ 2 πβ β 2aaβ+ β 4. zexp − az2 +β zerf a z +β dz = exp erf 11 + ∫ ( ) ( 11) −∞ 2aa 4a 22+ 2 a aa1
2 2 a β − 4aβ − 4a1ββ1 + 1 exp 1 {a > 0}. 2 + 2 4a + 4a a a a1 1 +∞ πβ β 2 2aβ + a β − 2 + β + β = 1 1 + 5. ∫ z exp( az z )erfi (a1 z 1 )dz exp erfi −∞ 2a a 4a 2 − 2 2 a aa1 2 2 a β + 4aβ + 4a1ββ1 + 1 exp 1 { aa> 2 }. 2 1 − 2 4a − 4a a a a1 1
2 +∞ a aβ 6. z exp(− az 2 ) erf (a z + β )erf (a z + β )dz = 1 exp − 1 × ∫ 1 1 2 2 2 −∞ + 2 a + a a a a1 1
2 22 aβ2 + a β 2 − aa 121 β a aβ a β+1 a β− 1 aa 122 β ×erf 1+− 22exp erf 2 2 +2 ++ 22 + 2 aa+ +2 ++ 22 aa1 aa 12 a aaa 2 2 aa2 aa 12 a
{a > 0}.
2 +∞ a aβ 7. z exp(− az 2 ) erf (a z + β )erfi (a z + β ) dz = 1 exp − 1 × ∫ 1 1 2 2 2 −∞ + 2 a + a a a a1 1
2 22 aβ2 + a β 2 − aa 121 β a aβ a β−1 a β+ 1 aa 122 β ×+erfi 1 22exp erf 2 2 +2 +− 22 − 2 aa− −2 +− 22 aa1 aa 12 a aaa 2 2 aa2 aa 12 a 2 { aa> 2 }.
2 +∞ a aβ 8. z exp(− a z 2 )erfi (a z + β )erfi (a z + β )dz = 1 exp 1 × ∫ 1 1 2 2 2 −∞ − 2 a − a a a a1 1
65 2 22 aβ2 − a β 2 + aa 121 β a aβ a β−1 a β+ 1 aa 122 β ×+erfi 1 22exp erfi 2 2 −2 −− 22 − 2 aa− −2 −− 22 aa1 aa 12 a aaa 2 2 aa2 aa 12 a 22 { aa>+12 a}.
∞ 1 αα 2 12 9. zexp(−=α z) erf( αα12 z) erf ( z) dz arctg + ∫ πα 22 0 αα++11 αα α α + 2 1 α2 = α2 +α 22 + = arctg { lim Re( z) lim Re( zz1 ) ln z α + α 2 α + α 2 zz→∞ →∞ 2 2 = α 2 + α 2 2 + = α 2 + α 2 2 + α 2 2 + = +∞ lim [Re( z 2 z ) ln z ] lim Re ( z 1 z 2 z ) 2ln z , z→∞ z→∞
2 2 22 α1 ≠ −α,, α 2 ≠ −α α 12 +α ≠ −α }. ∞( ) T2 π β β 2 − α 2 + β α + β = × 10. ∫ z exp( z z ) erf ( 1 z 1 )dz A exp 2α α 4α ∞(T1 )
2 2 2αβ + α β α β − 4αβ − 4α1ββ1 × erf 1 1 + A 1 exp 1 2 α α + α 2 α α + α 2 4α + 4α 2 1 1 1
{ lim Re(αzz22 −β ) = lim Re( α zz −β ) = zT→∞( 12) zT→∞( )
= α 2 + α 2 2 + α β − β + = lim [Re( z 1 z 2 1 1 z z ) ln z ] z→∞(T1 )
= α 2 + α 2 2 + α β − β + = +∞ = ± lim [Re( z 1 z 2 1 1 z z) ln z ] ; A 1 if z→∞(T2 )
α+α22 =− α+α =±∞ lim Re( 11zz) lim Re( ) , zT→∞( 21) zT→∞( )
= α+α22 ⋅ α+α =+∞ A 0 if lim Re( 11zz) lim Re( ) }. zT→∞( 12) zT→∞( )
∞(T ) 2 α αβ2 11. zexp−α z2 erf( α z +β) erf ( αz +β) dz = A 11exp− × ∫ ( ) 11 2 2 2 ∞ α α+α2 α+α (T1 ) 1 1
22 αβ2 + α β 2 − α 1 α 21 β α αβ ×erf 1+A 22exp −× 2 α+α2 α+α 22 +α α α+α2 α+α 1 12 22
66 2 αβ+αβ−ααβ1 1 122 × 2 α 2 ≠ −α α 2 ≠ −α erf { 1 , 2 , α+α2 α+α 22 +α 2 12 lim Re(α=zz22) lim Re( α=) zT→∞( 12) zT→∞( )
= lim Re αz2 +α 22 z +2 α β zz + ln = ( 1 11) zT→∞( 1 )
= lim Re αz2 +α 22 z +2 α β zz + ln = ( 1 11) zT→∞( 2 )
2 22 = lim Re αz +α z +2 α22 β zz + ln = →∞ ( 2 ) zT( 1 )
= α 2 + α 2 2 + α β + = lim [Re( z 2 z 2 2 2 z ) ln z ] z→∞(T2 ) = α 2 + α 2 2 + α 2 2 + α β + α β + = lim [Re( z 1 z 2 z 2 1 1 z 2 2 2 z ) 2ln z ] z→∞(T1 ) = α 2 + α 2 2 + α 2 2 + α β + α β + = +∞ ; lim [Re( z 1 z 2 z 2 1 1 z 2 2 2 z ) 2ln z ] z→∞(T2 )
= ± α+α22 +α =− α+α22 +α =±∞ A 1 if lim Re( 12zz) lim Re( 12) , zT→∞( 21) zT→∞( )
= α+α22 +α ⋅ α+α22 +α =+∞ A 0 if lim Re( 12zz) lim Re( 12) }. zT→∞( 12) zT→∞( )
ν 2 12. ∫ z exp(− αz ) erf (α1 z )erf (α 2 z )dz = 0 . −ν
2.7.3. +∞ (2m)! ( ) 2m − 2 = 7 > 1. ∫ z exp( az ) erf (a1 z )dz W1 (m, a, a1 ) {a 0}. 0 m! π
+∞ (2!m) (7) 22m −= > 2 2. ∫ zexp( az) erfi ( a11 z) dz W2 ( m,, a a ) { aa1 }. 0 m! π
+∞ +∞ 1 3. z 2m exp(− az 2 ) erf (a z )erf (a z )dz = z 2m exp(− az 2 ) erf (a z )erf (a z )dz = ∫ 1 2 2 ∫ 1 2 0 −∞ (−1)m ∂ m 1 a a = ⋅ arctg 1 2 {a > 0}. m π ∂a a 2 + 2 + 2 a aa1 aa2
67 +∞ +∞ 1 4. z 2m exp(− az 2 )erf (a z )erf i(a z )dz = z 2m exp(− az 2 )erf (a z )erfi (a z )dz = ∫ 1 2 2 ∫ 1 2 0 −∞
m m 2 2 2 (−1) ∂ 1 a + aa − aa + a1a2 = ⋅ ln 1 2 { aa> 2 }. m 2 2 π ∂a a 2 + 2 − 2 − a aa1 aa2 a1a2 +∞ +∞ 2m 2 1 2m 2 5. ∫ z exp(− az ) erfi (a1z ) erfi (a2 z )dz = ∫ z exp(− az ) erfi (a1z ) erfi (a2 z )dz = 0 2 −∞ (−1)m ∂ m 1 a a = ⋅ arctg 1 2 { aa>+22 a}. m 12 π ∂a a 2 − 2 − 2 a aa1 aa2
+∞ m! m (2!k ) 6. z21m+ exp−=× az 2 erf( a z) erf ( a z) dz ∑ ∫ ( ) 12 π 2 mk+−1 0 k=0 (ka!)
(77) 22( ) ×aW111( ka,, ++ a1 a 2) aW 2 ( ka ,, + a21 a) {a > 0}.
+∞ m! m (2!k ) 7. z21m+ exp−=× az 2 erf( a z) erfi ( a z) dz ∑ ∫ ( ) 12π 2 mk+−1 0 k=0 (ka!)
(77) 22( ) { > 2 }. ×aW121( ka,, ++ a1 a 2) aW 2 ( ka ,, − a21 a) aa2
+∞ m! m (2!k ) 8. z21m+ exp−=× az 2 erfi( a z) erfi ( a z) dz ∑ ∫ ( ) 12π 2 mk+−1 0 k=0 (ka!)
× (7 ) − 2 + (7 ) − 2 >+22 [a1W2 (k , a a1 , a2 ) a2W2 (k , a a2 , a1 )] { aa12 a}. +∞ n! ( ) 9. z n exp(− az 2 + βz ) erf (a z + β )dz = W 7 (n, a, a , β, β ) {a > 0}. ∫ 1 1 n 3 1 1 −∞ 2
+∞ n! (7 ) 10. z n exp(− az 2 + βz )erfi (a z + β )dz = W (n, a, a , β, β ) { aa> 2 }. ∫ 1 1 n 4 1 1 1 −∞ 2
+∞ m! m (2!k ) 11. z21m+ exp− az 2 erf( a z+β) erf ( a z +β) dz = × ∫ ( ) 11 2 2 ∑ k mk+−1 −∞ π k=0 4!ka
(7) ×aexp −β22 W2 ka ,+ a , a , − 2 a β , β + 1 ( 13) ( 12 11 2)
(7) +aexp −β22 W2 ka ,+ a , a , − 2 a β , β {a > 0}. 2 ( 23) ( 21 22 1)
68 +∞ m! m (2k )! 12. z 2m+1 exp(− az 2 )erf (a z + β )erfi (a z + β ) dz = × ∫ 1 1 2 2 ∑ k m+1−k −∞ π k=04 k!a
× − β2 (7 ) + 2 − β β + [a1 exp( 1 )W4 (2k , a a1 , a2 , 2a1 1 , 2 ) ( ) + β 2 7 − 2 β β > 2 a2 exp( 2 ) W3 (2k , a a2 , a1 , 2a2 2 , 1 ) ] { aa2 }. +∞ m! m (2k )! 13. z 2m+1 exp (− az 2 )erfi (a z + β )erfi (a z + β )dz = × ∫ 1 1 2 2 ∑ k m+1−k −∞ π k=04 k!a ( ) × a exp(β2 )W 7 (2k , a − a 2 , a , 2a β , β )+ [ 1 1 4 1 2 1 1 2 ( ) + β 2 7 − 2 β β >+22 a2 exp( 2 )W4 (2k , a a2 , a1 , 2a2 2 , 1 ) ] { aa12 a}.
∞ (2!m) (7) 22m −α α = α α α ≠ α2 ≠ −α 14. ∫ zexp( z) erf ( 11 z) dz W1 ( m,, ) { 0,1 , 0 m! π
2 2 22 lim Reα−−zmz( 2 1) ln = lim Re α+α−− zzmz1 (2 2) ln =+∞}. zz→∞ ( ) →∞ ( )
∞ m 22m A ∂ 1 αα12 15. ∫ zexp(−α z) erf( α12 z) erf ( α z) dz = ⋅ arctan πα∂αm 22 0 α α+α12 +α
2 2 22 { lim Reα−−zmz( 2 1) ln = lim Re α+α−− zzmz1 (2 2) ln = zz→∞ ( ) →∞ ( )
= α 2 + α 2 2 − − = α 2 + α 2 2 + α 2 2 − − = +∞ , lim [Re ( z 2 z ) (2m 2)ln z ] lim [Re ( z 1 z 2 z ) (2m 3)ln z ] z→∞ z→∞
22 m 22 α12 ≠ −α, α ≠ −α; A = (− 1) if lim Re α+α12 +αz =+∞, z→∞ ( )
m+1 22 A = (−1) if lim Re α+α12 +αz =−∞}. z→∞ ( )
∞ m 21m+ 2 m! (2!k ) 16. z exp−αz erf( α12 z) erf ( α z) dz = ∑ × ∫ ( ) π 2 mk+−1 0 k=0 (k!) α
× α (7 ) α + α 2 α + α (7 ) α + α 2 α 1W1 (k , 1 , 2 ) 2W1 (k , 2 , 1 )
α 2 ≠ −α α 2 ≠ −α α 2 + α 2 ≠ −α { 1 , 2 , 1 2 ,
α 2 − = α 2 + α 2 2 − − = lim [Re( z ) 2m ln z ] lim [Re( z 1 z ) (2m 1)ln z ] z→∞ z→∞
= α 2 + α 2 2 − − = lim [Re( z 2 z ) (2m 1)ln z ] z→∞
69 2 22 22 =lim Re αzzz +α12 +α −(2 m − 2) ln z =+∞}. z→∞ ( )
∞(T ) 2 n! n − α 2 + β α + β = (7 ) α α β β 17. ∫ z exp( z z ) erf ( 1z 1 )dz A W3 (n, , 1 , , 1 ) 2n ∞(T1 )
{α≠ 0, lim Re αz2 −β zn − −1 ln z = ( ) ( ) zT→∞( 1 )
= lim Re α+αz2 22 z +αβ−β−−2 zz n2 ln z = ( 1 11 ) ( ) zT→∞( 1 )
= lim Re αz2 −β zn − −1 ln z = ( ) ( ) zT→∞( 2 )
= lim Re α+αz2 22 z +αβ−β−2 zz n −2 ln z =+∞ ; ( 1 11 ) ( ) zT→∞( 2 )
= ± α+α22 =− α+α =±∞ A 1 if lim Re( 11zz) lim Re( ) , zT→∞( 21) zT→∞( )
= α+α22 ⋅ α+α =+∞ A 0 if lim Re( 11zz) lim Re( ) }. zT→∞( 12) zT→∞( )
∞(T ) 2 m! m (2k )! 18. z 2m+1 exp(− αz 2 ) erf (α z + β )erf (α z + β )dz = A × ∫ 1 1 2 2 ∑ k m+1−k π = 4 k!α ∞(T1 ) k 0
(7) × αexp −β22Wk2 ,α + α , α , − 2 α β , β + 1( 1) 3 ( 1 2 11 2)
(7) +αexp −β22Wk2 ,α + α , α , − 2 α β β 2( 2 ) 3 ( 2 1 2 2, 1 )
{ lim Reα−z22 2 mz ln = lim Re α−z 2 mz ln = ( ) ( ) zT→∞( 12) zT→∞( )
= lim Re αz2 +α 22 z +2 α β zm − 2 − 1 ln z = ( 1 11) ( ) zT→∞( 1 ) = lim Re αz2 +α 22 z +2 α β zm − 2 − 1 ln z = ( 1 11) ( ) zT→∞( 2 )
= lim Re αz2 +α 22 z +2 α β zm − 2 − 1 ln z = ( 2 22) ( ) zT→∞( 1 ) = lim Re αz2 +α 22 z +2 α β zm − 2 − 1 ln z = ( 2 22) ( ) zT→∞( 2 )
= lim Re αzzz2 +α 22 +α 22 +2 α β z + 2 α β zmz − 2 − 2 ln = ( 1 2 11 2 2 ) ( ) zT→∞( 1 )
70 = α 2 + α 2 2 + α 2 2 + α β + α β − − = +∞ , lim Re( z 1 z 2 z 2 1 1z 2 2 2 z ) (2m 2)ln z z→∞(T2 )
α 2 ≠ −α α 2 ≠ −α 1 , 2 ;
= ± α+α22 +α =− α+α22 +α =±∞ A 1 if lim Re( 12zz) lim Re( 12) , zT→∞( 21) zT→∞( )
= α+α22 +α ⋅ α+α22 +α =+∞ A 0 if lim Re( 12zz) lim Re( 12) }. zT→∞( 12) zT→∞( )
ν 2m 2 19. ∫ z exp(− αz ) erf (α1z )dz = 0 . −ν
ν 21m+ 2 20. ∫ z exp(−αz) erf( α12 z) erf( α z) dz = 0 . −ν
Introduced notations:
− 2 ( ) 1 α m 1 (l!) 1) W 7 (m, α, α )= arctg 1 + α ; 1 1 m m 1 ∑ l+1 4 α α α l=0 + α m−l α + α 2 (2l 1)!(4 ) ( 1 )
− 2 ( ) 1 α + α m 1 (l!) 2) W 7 (m, α, α )= ln 1 + α ; 2 1 2m+1 m 1 ∑ l+1 2 α α α − α1 l=0 + α m−l α − α 2 (2l 1)!(4 ) ( 1 )
2 E (n / 2 ) n−2l ( ) π β 2αβ + α β β 3) W 7 (n, α, α , β, β )= exp erf 1 1 ∑ + 3 1 1 α n−l α 4 α α + α 2 l=0 l!(n − 2l )!α 2 1 22 n− En( /2) nl+−12 l lr+−1 1 β−αβ−αββ4411 l!β 4( 2r − 2!) +×exp1 ∑∑ 2 α+ α2 (2!ln) ( +− 12! l) ( r− 1!) α+α 441 lr=11= 1
−− 2rq−− 22 rl−1 α2rq 12 (2 αβ + α β) En( /2) βnl−2 × 1 11 − (r −×1!) ∑ 22rq−− ∑∑ q=0 nlr−+ −22 q lr=11ln!( − 2! l) = qr!2( − 2 − 2 q) ! α( α+α1 )
− −− r−1 α22rq αβ + α β 2rq 12 1 (2 11) × , ∑ 21rq−− q=0 − − αnlr−+ −22 q α+α qr!2( 1 2 q) ! ( 1 ) (7) π αβ1 Wm3 (2 ,αα ,11 , 0, β) = erf − m!ααm 2 α+α1
71 2lr+− 22 2rr+− 12 1!αβ2 ml−1 l αβ(2 ) −−exp1 ∑∑11 , 2 2 ml− 21l+− r m! α+α α+α1 lr=00 α = 2 1 rl!2( + 1 − 2 r) !( α+α1 )
(7 ) α α = W3 (2m, , 1 , 0, 0) 0 , 2 ( ) (m +1)!α αβ W 7 (2m +1, α, α , 0, β )= 1 exp − 1 × 3 1 1 2 + α + α 2 α + α (2m 2)! 1 1
22lr− ml(2!l) 4mr+−1 (αβ) 11 , × +− 2lr− ∑∑ml1 2 lr=00l!α = rl!2( −+ 2 r) !(αα1 )
m m+1−l ( ) (m +1)!α 4 (2l )! W 7 (2m +1, α, α , 0, 0) = 1 ; 3 1 ∑ l (2m + 2)! α + α 2 l=0 2 α m+1−l α + α 2 1 (l!) ( 1 )
2 E (n / 2 ) n−2l ( ) π β 2αβ + α β β 4) W 7 (n, α, α , β, β )= exp erfi 1 1 ∑ + 4 1 1 α n−l α 4 α α − α 2 l=0 l!(n − 2l )!α 2 1
β+αβ+αββ22 n− En( /2) nl+−12 l lr+−1 1 441 11 l!β 4( 2r − 2!) +×exp ∑∑ 2 2 2!ln+− 12! l r− 1! α−α 44α− α lr=11( ) ( ) = ( ) 1 1 −− 2rq−− 22 rl−1 α2rq 12 (2 αβ + α β) En( /2) βnl−2 × 1 11 + (r −×1!) ∑ 22rq−− ∑∑ q=0 nlr−+ −22 q lr=11ln!( − 2! l) = qr!2( − 2 − 2 q) ! α( α−α1 )
2r−2q − − r−1 α (2αβ + α β)2r 1 2q × 1 1 1 , ∑ 2r−1−q q=0 − − α n−l+r−2q α − α 2 q!(2r 1 2q )! ( 1 ) (7) π αβ1 Wm4 (2 ,αα ,11 , 0, β) = erf i + m!ααm 2 α−α1
2lr+− 22 2lr+− 12 1!αβ2 ml−1 l αβ(2 ) + exp 1 ∑∑11 , 2 2 ml− 21l+− r m! α−α α−α1 lr=00 α = 2 1 rl!2( + 1 − 2 r) !( α−α1 )
(7 ) α α = W4 (2m, , 1 , 0, 0) 0 , 2 (m +1)!α αβ (7 ) + α α β = 1 1 × W4 (2m 1, , 1 , 0, 1 ) exp + α − α 2 α − α 2 (2m 2)!( 1 ) 1 22lr− ml(2!l) 4mr+−1 (αβ ) × 11 , ∑∑m+−12 l lr− lr=00l!α = 2 rl!2( − 2 r) !( α−α1 )
72 m m+1−l (7) (m +α1!) 4 (2l )! Wm(2+ 1, αα , ,0,0) = 1 . 4 1 ∑ l (2m + 2!) α−α2 l=0( )2 α m+1−l α − α 2 1 l! ( 1 )
+ + 2.8. Integrals of sin 2 m 1 (α 2 z 2 + β z + γ ), sinh2m 1(α 22zz +β +γ) ,
+ + cos 2 m 1 (α 2 z 2 + β z + γ ), cosh2m 1(α 22zz +β +γ)
2.8.1. +∞ 2π b 2 b 2 1. sin(a 2 z 2 + bz + γ )dz = sin γ − + cos γ − × ∫ 4a 2 2 0 4a 4a 2 2 i +1 b b b i +1 b × 1− Re erf ⋅ + cos γ − − sin γ − Im erf ⋅ . 2 2a 4a 2 4a 2 2 2a
+∞ +∞ 2.∫∫ sin(a22 z+ b z +γ) dz =2 sin(a22 z+ b z +γ) dz = −∞ −ba/2( 2 )
ba/2( 2 ) π bb22 =2 sin a22 z− b z +γ dz = cos−γ −sin −γ . ∫ ( ) 22 −∞ 2 a 44aa
∞ 2 2 2 2π β 3. sin (α z + β z + γ ) dz = (i −1)expi γ − × ∫ 8α 2 0 4α
11−+iiβ ββ2 ×erf ⋅−−+Ai( 1) exp i −γ erf ⋅−A αα122 22224α
22 { lim lnz −Im α zz + β = +∞ ; Al = ±1 z→∞ ( ) if lim[Re(α z)+ (3 − 2l)Im(α z)]= ±∞ }. z→∞
∞ 2 22 πβ β 4. sinh (αz++ βγ z) dz =exp −γerf −A1 + ∫ αα2 α 0 44 2
2 ββ 22 +exp γ−A2 −erfi { lim lnz− Re( α zz + β) = +∞ ; 2 α z→∞ 4α 2
A1 = 1 if lim Re(α z ) = +∞ , A1 = −1 if lim Re(α z) = −∞ , z→∞ z→∞
73 A2 = ±i if lim Im(αz) = ±∞ }. z→∞ 2.8.2.
+∞ 2π bb22 1. cos a22 z+ bz +γ dz = cosγ− −sin γ− × ∫ ( ) 4a 22 0 44aa
ib++11 b22 b ib ×− ⋅ −γγ − + − ⋅ 1 Re erf cos22sin Im erf . 222a 44 aa 2 a
+∞ +∞ 2.∫∫ cos (a22 z+ b z +γ) dz =2 cos(a22 z+ b z +γ) dz = −∞ −ba/2( 2 )
ba/2( 2 ) π bb22 =2 cos a22 z− b z +γ dz = sin −γ +cos −γ . ∫ ( ) 22 −∞ 2 a 44aa
∞ 2 22 21π− ββi 3. cos αz +β z +γ dz =(1 +i) exp i γ− A −erf ⋅ + ∫ ( ) 82αα2 1 0 4α 2 β2 1+ i β +(1 −ii) exp −γ A −⋅erf 2 2 4α 2 2α
{ lim lnz −Im α22 zz + β = +∞ ; z→∞ ( )
Al = ±1 if lim[Re(α z)+ (3 − 2l)Im(α z)]= ±∞ }. z→∞
∞ 2 22 πβ β 4. cosh (αz++ βγ z) dz =exp −γA1 −erf + ∫ αα2 α 0 44 2
ββ2 +exp γ−A −erfi 2 2 4α 2α
22 { lim lnz− Re( α zz + β) = +∞ ; A1 = 1 if lim Re(αz ) = +∞ , z→∞ z→∞
A1 = −1 if lim Re(αz ) = −∞ , A2 = ±i if lim Im(αz) = ±∞ }. z→∞ z→∞
2.8.3. +∞ (2m +π 1!) 2 1. sin2m+ 1a 22 z+ b z +γ dz = × ∫ ( ) m+1 0 4 a
74 k m (−1) b2 × sin( 2k + 1) −γ × ∑ 2 k=0 (mk−)!( m ++ 1 k) !2 k + 1 4a
11++ii21kb++21kb × ⋅+ ⋅−+ Re erf Im erf 1 2222aa
bi2 1+ 21kb+ +cos( 2k + 1) −γ Im erf ⋅ − 2 4a 2 2a
1+ i 21kb+ − ⋅+ Re erf 1 . 2 2a
+∞ +∞ 2.∫∫ sin2mm++ 1(a 22 z+ b z +γ) dz =2 sin2 1(a 22 z+ b z +γ) dz = −∞ −ba/2( 2 )
2 ba/2( ) k (2m +π 1!) m ( −1) =2 sin2m+ 1a 22 z− b z +γ dz = × ∫ ( ) m ∑ −∞ 4 a k=0 (mk−)!( m ++ 1 k) !4 k + 2
b2 b2 ×cos( 2k + 1) −γ −sin( 2k + 1) −γ . 2 2 4a 4a
∞ + (2m +1)! π 3. sin 2m 1 (α 2 z 2 + β z + γ ) dz = × ∫ m+1 0 4 α
m (−1)k β 2 × (i −1)exp i(2k +1) γ − × ∑ 2 k =0 (m − k )!(m +1+ k )! 4k + 2 4α
1− i 21k +β β2 ×erf ⋅ −A −( i +1) exp ik( 2+ 1) −γ × 1 2 2 2α 4α
1+ i 21k +β ×⋅erf −A α 2 2 2
{ lim lnz −(2 m + 1) Im α22 zz + β = +∞ ; z→∞ ( )
Al = ±1 if lim[Re(α z)+ (3 − 2l)Im(α z)]= ±∞ }. z→∞
75 mk− ∞ (2m +− 1!) π m ( 1) 4. sinh2m+ 1α 22z++ βγ z dz = × ∫ ( ) m+1 ∑ 0 4 α k =0 (mk−)!( mk ++ 1!) 2 k + 1
ββ2221k +β ×exp( 2k + 1) −γ erf −Ak +exp ( 2 + 1) γ− × 22α 1 44αα2
+β 21k 22 ×−A2 erfi { lim lnzm−( 2 + 1) Re( α zz + β) = +∞ ; 2α z→∞
A1 = 1 if lim Re(αz) = +∞ , A1 = −1 if lim Re(αz) = −∞ , z→∞ z→∞
A2 = ±i if lim Im(αz) = ±∞ }. z→∞
2.8.4. +∞ m + (2m +π 1!) 2 1 1. cos2m 1a 22 z+ b z +γ dz = ∑ × ∫ ( ) m+1 − ++ + 0 4 a k=0 (mk)!( m 1 k) !2 k 1
b2 11++ii21kb++21kb ×sin( 2k + 1) −γ 1+ Im erf ⋅−Re erf ⋅ + 2 4a 2222aa
bi2 1+ 21kb+ 1+ i 21kb+ +cos( 2k + 1) −γ1 − Re erf ⋅ − Im erf ⋅ . 2 4a 2 2a 2 2a
+∞ +∞ 2.∫∫ cos2mm++ 1(a 22 z+ b z +γ) dz =2 cos2 1(a 22 z+ b z +γ) dz = −∞ −ba/2( 2 )
ba/2( 2 ) (2m +π 1!) m 1 =2 cos2m+ 1a 22 z− b z +γ dz = × ∫ ( ) m ∑ −∞ 4 a k=0 (mk−)!( m ++ 1 k) !4 k + 2
b2 b2 ×sin( 2k + 1) −γ + cos( 2k + 1) −γ . 2 2 4a 4a
∞ + (2m +1)! π 3. cos 2m 1 (α 2 z 2 + β z + γ ) dz = × ∫ m+1 0 4 α
m 1 β2 × (1+i) exp ik( 2 + 1) γ− × ∑ 2 k=0 (mk−)!( m ++ 1 k) !4 k + 2 4α
76 1−ik 21 +β β2 ×A −erf ⋅ +1 −i exp ik 2 + 1 −γ × 1 ( ) ( )2 2 2α 4α
1+ i 21k +β ×−A erf ⋅ 2 α 2 2
{ lim lnz −(2 m + 1) Im α22 zz + β = +∞ ; z→∞ ( )
Al = ±1 if lim[Re(α z)+ (3 − 2l)Im(α z)]= ±∞ }. z→∞
∞ (2m +π 1!) m 1 4. cosh2m+ 1α 22z +β z +γ dz = ∑ × ∫ ( ) m+1 − ++ + 0 4 α k=0 (mk)!( m 1 k) !2 k 1
ββ2221k +β ×exp( 2kA + 1) −γ −erf +exp ( 2 k + 1) γ− × 221 α 44αα2
21k +β ×−A erfi 2 α 2
22 { lim lnzm−( 2 + 1) Re( α zz + β) = +∞ ; A1 = 1 if lim Re(αz ) = +∞ , z→∞ z→∞
A1 = −1 if lim Re(α z ) = −∞ , A2 = ±i if lim Im(αz) = ±∞ }. z→∞ z→∞
n 22 2.9. Integrals of zsin (α zz +β +γ) exp ( β1 z)
2.9.1. +∞ 0 22 22 1.∫∫ sin (a z+ b z +γ) exp ( − b11 z) dz =sin( a z − b z +γ)exp( b z) dz = 0 −∞
2πbb b22− b ii−−11b ++ ib b ib = exp11 cos−γ1 + Re erf ⋅1 −Im erf ⋅ 1 − 22 4a24aa 2222aa
22 b− b1ii−−11b 11 ++ ib b ib −sin −γ1 + Reerf ⋅ +Imerf ⋅ { b1 ≥ 0 }. 2 2222aa 4a
∞ 2 π (β −βi ) 22 2 1 2.∫ sin (αz +β z +γ) exp ( β1 z) dz =(1 +i) exp −i γ × 8α α2 0 4i
77 2 1− i β − iβ (β + iβ) × A + erf ⋅ 1 + (1− i )exp i γ − 1 × 1 2 2 2α 4iα
1+ i β1 + iβ 22 × A2 + erf ⋅ { lim Re(β1 z) + Im α zz + β −ln z = −∞ ; →∞ ( ) 2 2α z
Ak = ±1 if lim Re(ααzk) +( 2 − 3) Im ( z) = ±∞ }. z→∞ 2.9.2. +∞ 0 22 22 1.∫∫z sin( a z+ b z +γ)exp( − b11 z) dz =− zsin( a z− b z +γ)exp( b z) dz = 0 −∞
cos γ 21π−bb b22−+ b i b ib = + exp11 sin−γ(bb − )Re erf ⋅1 + 23 2 2 1 28aa 2 a 4 a 2 2a
i −1 b+− ib b22 b +(bb + )Im erf ⋅11 +bb − +cos −γ × 11 2 2 2a 4a
ii−−11b11++ ib b ib ×−(bb1)Imerf ⋅−+(bb 11)Reerf ⋅−−bb { b1 > 0 }. 2222aa
∞ π 2.z sin α+β+γ22 z z exp ( βz) dz =(1 −i) ( i β−β×) ∫ ( ) 113 0 82α
(β + iβ)2 1+ i β + iβ × expi γ − 1 A + erf ⋅ 1 − (1+ i )(β + iβ )× 2 1 1 4iα 2 2α 2 (β1 − iβ) 1 − i β1 − iβ cos γ × exp − i γ A + erf ⋅ + 2 2 2 4iα 2 2α 2α
2 2 { lim [Re (β1 z ) + Im (α z + β z ) ] = −∞; z→∞
Ak = ±1 if lim Re(ααz) +( 3 − 2 kz) Im( ) = ±∞ }. z→∞
2.9.3.
+∞ 0 n 2 2 n n 2 2 1. ∫ z sin (a z + bz + γ ) exp(− b1z )dz = (−1) ∫ z sin (a z − bz + γ )exp(b1z )dz = 0 −∞
22 ni!1bb b−+ b − b ib (9) = exp11 sin +γ Re erf ⋅1 +1W( nab , , , − b) − n 221 1 22aa 4 2 2a
78 22 ni!1bb b−+ b − b ib (9) − exp11 cos+γ Im erf ⋅1 +1W( nab , , , − b) + n 221 1 22aa 4 2 2a
n! (99) ( ) +sin γ⋅ ReW na ,22 , b ,− b − cos γ⋅ Im W na , , b , − b n 22( 11) ( ) 2
{b1 > 0 if n > 0} .
2 ∞ (ββ− i ) 2. n 22 ni! 1 zsin (α z++ βγ z)exp( β1 z) dz =+ exp −×iγ ∫ 24n 12iα 0
2 1− i β − iβ ( ) (β + iβ) × erf ⋅ 1 + A W 9 (n, α, β, β ) −expi γ − 1 × 1 1 1 2 2 2α 4iα 1+ i β +βi (9) × ⋅1 + αββ + erf AW213 ( n,,, ) 2 2α
(99) ( ) + −γ α22 ββ − γ −α −ββ exp( iW) 22( n, , ,11) exp(iW) ( n , , , )
22 { lim Re(β1z) + Im α zzn + β +( −1) ln z = −∞ ; z→∞ ( )
Ak = ±1 if lim Re( αzk) +( 2 − 3) Im( α z) = ±∞ }. z→∞ 2.9.4. +∞ 0 n n+1 n 1. ∫ z sin(bz + γ )exp(− b1 z )dz = (−1) ∫ z sin(bz − γ )exp(b1 z )dz = 0 −∞ ( ) = (− ) n 9 ( − ) > 1 n!W4 b, b1 {b1 0}. ∞ n β + γ β = − n (9 ) β β 2. ∫ z sin( z )exp( 1 z )dz ( 1) n!W4 ( , 1 ) 0
β22 ≠ −β { 1 , lim Re(β1z) + Im( β z) + nz ln = −∞ }. z→∞
Introduced notations:
E (n / 2 ) n−2l ( ) 2π (1 − i ) (β − iβ) 1) W 9 (n, α, β, β ) = ∑ 1 , 1 1 4α n−l l=0 l!(n − 2l )!(iα 2 )
79 π (1 − i )(− i )n1 ( ) (9 ) α β β = 9 ( + α β β) = W1 (2n1 , , , i ) , W1 2n1 1, , , i 0 ; 2n1 +1 n1 ! 8α
n−E (n / 2 ) l−r n−1−2l+2r ( ) (l −1)! l 4 (β − iβ) 2) W 9 (n, α 2 , β, β ) = ∑ ∑ 1 − 2 1 (2l −1)!(n +1− 2l )! n−l+r l=1 r=1 (r −1)!(iα 2 )
n−1−2l+2r E (n / 2) 1 l r!(β − iβ) − ∑ ∑ 1 , l!(n − 2l )! n−l+r l=1 r=1 (2r )!(iα 2 )
( ) 4 n1 n ! W 9 2n , α 2 , β, iβ = 0 , (9 ) + α 2 β β = 1 ; 2 ( 1 ) W2 (2n1 1, , , i ) 2 n1 +1 (2n1 +1)!(iα )
E (n / 2 ) n−2l ( ) 2π (1+ i ) (β + iβ) 3) W 9 (n, α, β, β ) = ∑ 1 , 3 1 4α n−l l=0 l!(n − 2l )!(− iα 2 )
π (1 + i )i n1 ( ) (9 ) α β − β = 9 ( + α β − β) = W3 (2n1 , , , i ) , W3 2n1 1, , , i 0 ; 2n1 +1 n1 ! 8α
En( /2) l 21l+− nl 2 (9) (n +1!) (−β1) β 4) β β = γ⋅ 1 − W4 ( , 1 ) + cos ∑ 22n 1 2l+− 1! nl 2 ! β +β l=0 ( ) ( ) ( 1 ) − ( ) l 2l n+1−2l n E n / 2 (−1) β β i exp(iγ ) exp(− iγ ) − sin γ ⋅ ∑ 1 = − . (2l )!(n +1 − 2l )! 2 n+1 n+1 l=0 (β1 + iβ) (β1 − iβ)
n 2 2 2 2.10. Integrals of z exp(− α z + βz )sin(α 1 z + β 1 z + γ )
2.10.1.
2 2 +∞ π b − b 1. exp(− a 2 z 2 + bz ) sin(b z + γ )dz = exp 1 × ∫ 1 2a 2 0 4a
bb1 b + ib1 bb1 b + ib1 × sin + γ 1+ Re erf + cos + γ Im erf . 2a 2 2a 2a 2 2a
2 2 +∞ π b − b bb 2. exp(− a 2 z 2 + bz ) sin(b z + γ )dz = exp 1 sin 1 + γ . ∫ 1 2 2 −∞ a 4a 2a
+∞ 2 2 − 2 2 − 2 2 2 π a b a b1 2a1bb1 3. ∫ exp(− a z + bz )sin (a1 z + b1 z + γ ) dz = exp × 2 4 + 2 0 4a 4a1
80 2 2 2 a1b − a1b + 2a bb1 1 b + ib 1 × 1 + γ 1 + + sin Re erf 4a 4 + 4a 2 2 2 2 1 a − ia1 2 a − ia1 a − ia1
2 2 2 a1b − a1b + 2a bb1 1 b + ib 1 + 1 + γ 1 + cos Im erf 4a 4 + 4a 2 2 2 2 1 a − ia1 2 a − ia1 a − ia1
{a > 0 or [a = 0, a1 ≠ 0, b ≤ 0]}. +∞ 22 2 2π 4. ∫ exp(−a z + bz)sin( a11 z+ b z +γ) dz = × 42+ −∞ 2 aa1
a 2b 2 − a 2b 2 − 2a bb a b 2 − a b 2 + 2a 2bb × 1 1 1 2 + 4 + 2 1 1 1 1 + γ + exp a a a1 sin 4 + 2 4 + 2 4a 4a1 4a 4a1 2 2 2 a1b − a1b + 2a bb1 + 4 + 2 − 2 1 + γ a a1 a cos 4 + 2 4a 4a1
{a > 0 or [a = b = 0, a1 > 0]}.
∞ π i ()β−i β 2 5. exp −α22z +β zsin( β z + γ) dz = exp 1 −i γ × ∫ ( ) 1 4α 2 0 4α
2 β − iβ1 (β + iβ1 ) β + iβ1 × erf ±1 − exp + iγ erf ±1 2 2α 4α 2α
22 { lim Re αzz −β−Im( β1 z) + ln z =+∞ , lim Re( α=±∞z) }. zz→∞ ( ) →∞
∞ 22 2 π i 1 6. ∫ exp(−αz +β z)sin ( α11 z +β z +γ) dz = × 4 2 0 α +αi 1
(β − β )2 β − β i 1 i 1 1 × exp − iγ erf + A1 − × 4α 2 + 4iα 2 2 1 2 α + iα1 α − iα1 2 (β+i β1 ) β+i β ×exp +γiAerf 1 + 2 2 44α−i α α2 −α 1 2 i 1
22 2 { lim Re αzz −β−Im α11 z +β z +ln z =+∞ ; z→∞ ( ) ( )
2 Ak = ±1 if lim Reα +( 3 − 2ki) α1 z = ±∞ }. z→∞
81 ∞ exp(iγ) 2πi 7. exp −α22z +β zsin ±α i22 z +β+γ z dz = × ∫ ( ) ( 1 ) β ±β α 0 2( 1 i ) 8
(β ± iβ )2 β ± iβ × exp 1 ± iγ erf 1 + A 2 8α 2 2α
22 { lim Re 2αz − β ziz β1 +ln z = +∞ , lim Re[ (β iz β1 ) ] = −∞ ; z→∞ ( ) z→∞
A = 1 if lim Re(αz ) = +∞ , A = −1 if lim Re(αz) = −∞ }. z→∞ z→∞
2.10.2.
2 2 +∞ π b − b 1. z exp(− a 2 z 2 + bz )sin(b z + γ )dz = exp 1 × ∫ 1 3 2 0 4a 4a
bb1 b + ib1 b + ib1 bb1 × sin + γ b Reerf − b1 Imerf + b + cos + γ × 2a 2 2a 2a 2a 2
b++ ib11 b ib sin γ ×b11Reerf + bbImerf ++ . 22aa 2a2
2 2 +∞ π b − b 2. z exp(− a 2 z 2 + bz )sin(b z + γ )dz = exp 1 × ∫ 1 3 2 −∞ 2a 4a
bb1 bb1 × bsin + γ + b1 cos + γ . 2a 2 2a 2
+∞ 2 2 2 2 a sin γ + a1 cos γ 3. ∫ z exp(− a z + bz )sin (a1z + b1z + γ )dz = + 4 + 2 0 2a 2a1
π a22 b− a 22 b −22 abb ab2−+ ab 2 a 2 bb + exp 1 1 1sin 1 11 1+γ × 4 42 42 44aa++11 44aa
b + ib b + ib b + ib × Re 1 erf 1 + 1 + 2 2 2 2 2 (a − ia1 ) a − ia1 2 a − ia1 (a − ia1 ) a − ia1
b+ ib b ++ ib b ib +Im 1erf 11+× 22 2 22 (a−− ia11) a ia2 a − ia 1( a −− ia 11) a ia ab2−+ ab 222 a bb ×cos1 11 1+γ {a > 0 or [a = 0, a ≠ 0, b < 0]}. 42 1 44aa+ 1
82 +∞ π ab22−− ab 22 2 abb 4. zexp − a22 z + bzsin a z2 ++ b zγ dz = exp 1 11× ∫ ( ) ( 11) 42 −∞ 2 44aa+ 1
2 2 2 a1b − a1b + 2a bb1 b + ib × sin 1 + γ Re 1 + 4a 4 + 4a 2 2 2 1 (a − ia1 ) a − ia1
2 2 2 a1b − a1b + 2a bb1 b + ib + cos 1 + γ Im 1 {a > 0}. 4a 4 + 4a 2 2 2 1 (a − ia1 ) a − ia1
∞ 2 π (β−i β ) 22 i 1 5. ∫ zexp(−α z +β z)sin( β11 z + γ) dz =( β−i β )exp −i γ × αα32 0 84
2 β−ii β (β+i β ) β+ β γ 111 sin ×erf± 1 −( β+ii β1 ) exp + γ erf± 1 + 22αααα22 42
22 { lim Re αzz −β−Im( β1 z) =+∞, lim Re( α=±∞z) }. zz→∞ ( ) →∞
∞ 2 2 2 2 α sin γ + α1 cos γ πi 6. ∫ z exp(− α z + βz )sin(α1z + β1z + γ )dz = + × α 4 + α 2 8 0 2 2 1
β − β (β − β )2 β − β i 1 i 1 i 1 × exp − iγ erf + A1 − 2 2 4α 2 + 4iα 2 (α + iα1 ) α + iα1 1 2 α + iα1
β + β (β + β )2 β + β i 1 i 1 i 1 − exp + iγ erf + A2 2 2 4α 2 − 4iα 2 (α − iα1 ) α − iα1 1 2 α − iα1
22 2 { lim Re αzz − β −Im α11 z + β z = +∞ ; Ak = ±1 if z→∞ ( ) ( )
2 lim Reα +( 3 − 2ki) α1 z = ±∞ }. z→∞
∞ 2π (β iβ) 7. z exp(− α 2 z 2 + βz )sin (± iα 2 z 2 + β z + γ ) dz = 1 × ∫ 1 3 0 32α 2 (β±i β1 ) β±i β1 iexp(± ii γ) exp( i γ) ×exp ±γiAerf + ± 222α 22 88αα2(ββ i 1 )
22 { lim Re( 2αz − β ziz β1 ) = +∞ , lim [Re(βz iβ1z ) + ln z ] = −∞ ; z→∞ z→∞ A = 1 if lim Re(αz ) = +∞ , A = −1 if lim Re(αz) = −∞ }. z→∞ z→∞
83
2.10.3.
+∞ 2 2 − 2 2 − n 2 2 2 n! π a b a b1 2a1bb1 1. ∫ z exp(− a z + bz )sin (a1 z + b1 z + γ ) dz = exp × n+1 4 + 2 0 2 4a 4a1
2 22 (10) ab−+ ab2 a bb b+ ib W( naa,,11 ,, bb) ×sin 1 11 1+γ Re erf 1 +1 1 + 42 44aa+ 22−− 1 2 a ia11a ia
2 2 2 2 2 2 2 n! π a b − a b − 2a1bb1 a1b − a1b + 2a bb1 + exp 1 cos 1 + γ × n+1 4 + 2 4 + 2 2 4a 4a1 4a 4a1
(10 ) + W (n, a, a , b, b ) b ib1 1 1 1 × Im erf +1 + 2 2 2 a − ia1 a − ia1
n! ( ) (10) + 10 ( ) γ +γ [ReW2 n, a, a1 , b, b1 sin ImW2 ( naa , ,11 , bb ,) cos 2 n
{a > 0 or [a = 0, a1 ≠ 0, b < 0] for n > 0} .
+∞ 2 2 − 2 2 − n 2 2 2 n! π a b a b1 2a1bb1 2. ∫ z exp(− a z + bz )sin (a1z + b1z + γ ) dz = exp × n 4 + 2 −∞ 2 4a 4a1
2 22 (10) ab1−+ ab 12 a bb 1 W( n,, a a11 ,, b b) × sin11+γ Re + 44aa42+ 2 1 a− ia1
2 22 (10) ab−+ ab2 a bb W( naa,,11 ,, bb) +cos1 11 1+γ Im 1 42 44aa+ 2 1 a− ia1
{a > 0 for n > 0} .
2 ∞ π (β−i β ) n 22 2 ni! 1 3. ∫ zexp(−α z +β z)sin( α11 z +β z +γ) dz = exp −i γ × n+22α+ α 0 2 44i 1
(10 ) β − β W (n, α, − α , β, − β ) (β + β )2 i 1 1 1 1 i 1 × erf + A1 − exp + i γ × 2 2 4α 2 − 4iα 2 α + iα1 α + iα1 1
(10 ) β + β W (n, α, α , β, β ) i 1 1 1 1 n!i × erf + A2 + × 2 2 2 n+1 2 α − iα1 α − iα1
84 × − γ (10 ) α − α β − β − γ (10 ) α α β β [exp( i )W2 (n, , 1 , , 1 ) exp(i )W2 (n, , 1 , , 1 )]
22 2 { lim Re αzz −β−Im α11 z +β zn −−( 1) ln z =+∞ ; z→∞ ( ) ( )
2 Ak = ±1 if lim Reα +( 3 − 2ki) α1 z = ±∞ }. z→∞
∞ (−1)n n!i exp( iγ ) 4. z n exp(− α 2 z 2 + βz )sin(± iα 2 z 2 + β z + γ )dz = ∫ 1 n+1 0 2(β iβ1 ) 2 n!i 2π (β ± iβ1 ) β ± iβ1 exp ± iγ erf + A × n+1 2 2 4α 8α 2 2α (10) 22(10) ×W12( ni, ααβ±β+ , ,,11) exp( ±γiWni) ( , ααβ±β , ,, )
{ lim Re(βziz β1 ) + nln z = −∞ , z→∞
22 lim Re( 2αz− ββ ziz 1 ) −( n −1) ln z = +∞ ; A = 1 if lim Re(αz ) = +∞ , z→∞ z→∞ A = −1 if lim Re(αz) = −∞ }. z→∞
Introduced notations:
En( /2) nl−2 (10) (β+i β ) 1) Wn( ,αα , ,, ββ) = 1 , 1 11∑ nl− l=0 2 ln!( − 2! l) ( α −α i1 )
(10) 1 (10 ) Wn(2 ,αα , ,, β i β=) , W (2n1 +1, α, α1 , β, iβ) = 0 ; 1 11 n 1 2 1 ni11! (α −α)
−− + n− En( /2) l lr− n12 lr 2 (10) (li−1!) 4 (β+ β ) 2) Wn( ,αα , ,, ββ) = 1 − 2 11 ∑∑nlr−+ lr=11(2ln− 1!) ( +− 1 2 l) ! = 2 (ri−1!) ( α −α1 )
En( /2) l n−−12 lr + 2 1 ri!(β+ β1 ) (10 ) − , W (2n1 , α, α1 , β, iβ) = 0 , ∑∑ nlr−+ 2 lr=11ln!( − 2! l) = 2 (2!ri) (α −α1 )
85 4n1 n ! (10 ) + α α β β = 1 . W2 (2n1 1, , 1 , , i ) 2 n1 +1 (2n1 +1)!(α − iα1 )
n 2.11. Integrals of z erf (αz + β)exp(β1 z )sin(β 2 z + γ )
2.11.1. +∞ 0 1. ∫ erf (az )exp(− b1z )sin(b2 z + γ )dz = ∫ erf (az )exp(b1z )sin(b2 z − γ )dz = 0 −∞
2 − 2 1 b1 b2 b1b2 = exp sin − γ × 2 + 2 2 2 b1 b2 4a 2a
b12−− ib b12 ib b12 b ×bbb1Re erf − 21Im erf − −cos −γ × 2a 22 aa
b12−− ib b12 ib ×b2Re erf +− bb 12Imerf {a > 0, b1 > 0} . 22aa
∞ 1 (β + iβ )2 2. erf (αz )exp(β z )sin(β z + γ )dz = exp 1 2 + iγ × ∫ 1 2 (β − β ) 2 0 2 2 i 1 4α
β + iβ 1 (β − iβ )2 β − iβ × erf 1 2 ±1 + exp 1 2 − iγ erf 1 2 ±1 2 2α 2(β 2 + iβ1 ) 4α 2α
2 2 22 {β21 ≠ −β , lim Re αzz − β1 −Im( β 2 z) + 2ln z = +∞ , z→∞ ( )
lim Re(β12zz) + Im( β) = −∞, lim Re( α z) = ±∞ }. zz→∞ →∞
2.11.2. +∞ 0 1. ∫ erf (az + b)exp(− b1z )sin(b2 z + γ )dz = ∫ erf (az − b)exp(b1z )sin(b2 z − γ )dz = 0 −∞
2 − 2 + erf (b) 1 b1 b2 4abb1 b1b2 + 2abb2 = (b1 sin γ + b2 cos γ ) + exp sin γ − × 2 + 2 2 + 2 2 2 b1 b2 b1 b2 4a 2a
b12− ib b12 −+ ib b12 b2 abb 2 ×bb12 +Imerf b + − b1Re erf b + −cos γ− × 22aa2a2
b1 − ib2 b1 − ib2 × b2 Reerf b + + b1 Imerf b + − b2 {a > 0, b1 > 0} . 2a 2a
86 ∞ erf (β) 2. ∫ erf (αz + β)exp(β1z )sin(β 2 z + γ )dz = (β 2 cos γ − β1 sin γ ) + β 2 + β 2 0 1 2
i 1 (β1 − iβ 2 )(β1 − 4αβ − iβ 2 ) β1 − iβ 2 + exp − iγ erf β − 1 − 2 2 β1 − iβ 2 4α 2α
1 (β1 + iβ 2 )(β1 − 4αβ + iβ 2 ) β1 + iβ 2 − exp + iγ erf β − 1 2 β1 + iβ 2 4α 2α 22 { lim Reαz +αβ−β 2 zz12 −Im( β z) + 2ln z =+∞ , z→∞ ( )
22 lim [Re(β1z ) + Im(β 2 z ) ] = −∞, β21 ≠ −β, lim Re( αz) = ±∞ }. z→∞ z→∞
2.11.3. +∞ 0 n n n 1. ∫ z erf (az )exp(− b1z )sin(b2 z + γ )dz = (−1) ∫ z erf (az )exp(b1z )sin(b2 z − γ )dz = 0 −∞
2 − 2 b1 b2 b1b2 b1 − ib2 (11) = exp sin − γ Re erf −1 W (a, 0, − b , b ) − 2 2 1 1 2 4a 2a 2a
b22−− b b b b ib −1 2 12−γ 1 2 −(11) −+ exp22 cos Im erf 1W1 ( a , 0, bb12 , ) 42aa 2 a
(11) (11) +sin γ⋅ ReW22( a , 0,− bb12 ,) + cos γ⋅ Im W( a , 0,− bb12 , ) {ab>>0,1 0} .
∞ 1 β22 −β β β 2. zn erf(α z) exp( β z) sin ( β z +γ) dz = exp1 2 exp i 12+γ × ∫ 12 22 0 2i 42αα
β1 +i β 2 (11) ββ12 ×erf ±1Wi( α , 0, β12 , β) − exp − +γ × 2α 1 α2 2
β −βi (11) i (11) × 12± αβ−β+ −γ αβ−β− erf 1W12( , 0,12 , ) exp( iW) ( , 0,12 , ) 22α
(11) 22 −exp(iW γ) 2 ( α, 0, ββ12 , ) { lim Re αz − β12 z −Im( β zn) −( −2) ln z = +∞ , z→∞ ( )
22 lim [Re(β1z ) + Im(β 2 z ) + n ln z ] = −∞, β21 ≠ −β, lim Re( αz) = ±∞ }. z→∞ z→∞
2.11.4.
+∞ n 1. ∫ zerf( az+ b) exp( − b12 z) sin ( b z +γ) dz = 0
87 0 n n =( −1) ∫ z erf( az− b) exp( b12 z) sin ( b z−γ) dz = −∞
b22−+ b42 abb b b+ abb =exp1 2 1 sin γ− 12 2 × 22 42aa
22 b− ib (11) b−+ b4 abb ×−+Re 1 erf b12 W( ab, ,− b , b) + exp12 1× 1 12 2 2a 4a
b b + 2abb b − ib × γ − 1 2 2 − + 1 2 (11) − + cos Im1 erf b W1 (a, b, b1 , b2 ) 2a 2 2a
(11) (11) +sin γ⋅ ReW22( ab , ,− b12 , b) + cos γ⋅ Im W( ab , ,− b12 , b) {a > 0, b1 > 0} .
∞ β 2 − β 2 − αββ n i 1 2 4 1 2. z erf (αz + β)exp(β1z )sin(β 2 z + γ )dz = exp × ∫ 2 2 0 4α
β12 β −2 αββ 2 β1 +i β 2 (11) ×exp iW+γerf β− 1(α , β , β , β) − 2 1 12 2α 2α 2αββ − β β β − iβ − 2 1 2 − γ β − 1 2 (11) α β β − β + expi erf 1 W1 ( , , 1 , 2 ) 2α 2 2α 1 ( ) + [exp(iγ )W 11 (α, β, β , β ) − exp(− iγ )W (11) (α, β, β , − β )] 2i 2 1 2 2 1 2
22 { lim Reαz +αβ−β− 2 zz12Im( β zn) −−( 2) ln z =+∞ , z→∞ ( )
22 lim [Re(β1z ) + Im(β 2 z ) + n ln z ] = −∞, β21 ≠ −β, lim Re( αz) = ±∞ }. z→∞ z→∞
Introduced notations:
n+1−k E (k / 2 ) k−2l ( ) n n! 1 (β − 2αβ + iβ ) 1) W 11 (α, β, β , β ) = − 1 2 , 1 1 2 ∑ k ∑ 2k−2l k=02 β1 + iβ2 l=0 l!(k − 2l )!α
E (n / 2 ) ( ) n! 1 W 11 (α, β, β , iβ − 2iαβ) = ; 1 1 1 n+1 ∑ n+1−2k (2α) k=0 k!(− β)
n+1 ( ) 1 1 n n! 2) W 11 (α, β, β , β ) = n!erf (β) − + exp(− β 2 ) × 2 1 2 ∑ k −1 β1 + iβ 2 π k =12
88 n+1−k − ( ) l−r k−1−2l+2r 1 k E k / 2 (l −1)! l 4 (β − 2αβ + iβ ) × − 1 2 − ∑ ∑ 2k−1−2l+2r β1 + iβ 2 l=1 (2l −1)!(k +1− 2l )!r=1 (r −1)!α
( ) k−1−2l+2r E k / 2 1 l r!(β − 2αβ + iβ ) − 1 2 , ∑ ∑ 2k−1−2l+2r l=1 l!(k − 2l )!r=1 (2r )!α
n+1 n (11) 1 (−1!) n 2 W2 (α, β , β11 , ii β − 2 αβ) = n ! erf ( β) − + exp(−β) × 2αβ παn+1
− ( ) n E n / 2 (k −1)! × . ∑ n+2−2k k =1 (2k −1)!(2β)
2n+1 2 2 2.12. Integrals of z erf (αz + β)exp(− α 1 z ) sin(α 2 z + γ )
2.12.1. +∞ 2 2 a1 sin γ + a2 cos γ 1. ∫ z erf (az + b)exp(− a1z )sin(a2 z + γ )dz = erf (b) + 2 + 2 0 2a1 2a2
2 2 2 2 2 a a a1 + a + a a a b + exp − b 2 1 2 sin γ + 2 × 2 + 2 4 + 2 + 2 + 2 4 + 2 + 2 + 2 2a1 2a2 a a1 a2 2a a1 a a1 a2 2a a1
a+ ia ab a ×Re121 − erf +× 22 22+− +− 22aa+ a a12 ia a a12 ia 12
aa2++ a 22 a aab22 ×exp −b2 11 2 cos γ+ 2 × 422 2 422 2 aaa+++1222 aa 1 aaa+++12 aa 1
+ a1 ia2 ab × Im 1− erf { a1 > 0 }. 2 2 a + a1 − ia2 a + a1 − ia2
+∞ 22 2. ∫ zerfi( az+ b) exp( − a12 z) sin ( a z+γ) dz = 0
0 22 =∫ zerfi( az − b) exp( − a12 z) sin ( a z+γ) dz = −∞ a sin γ + a cos γ a = 1 2 erfi (b) + × 2a 2 + 2a 2 2a 2 + 2a 2 1 2 1 2
89 2 2 2 2 2 a + a − a a1 a a b × exp b 2 1 2 sin γ + 2 × 4 + 2 + 2 − 2 4 + 2 + 2 − 2 a a1 a2 2a a1 a a1 a2 2a a1
a + ia ab a × 1 2 + + × Re 1 erf 2 2 2a 2 + 2a 2 a1 − a − ia2 a1 − a − ia2 1 2
a222+− a aa aab22 ×expb2 12 1cos γ+ 2 × 422 2 422 2 aaa++−1222 aa 1 aaa++−12 aa 1
a + ia ab × Im 1 2 1+ erf 2 2 a1 − a − ia2 a1 − a − ia2
> 2 =>≠≤2 { aa1 or a12 a0, a 0, ab 0 }.
+∞ 22 2 3. ∫ zerfi( az+ b) exp( − a z) sin ( a1 z+γ) dz = 0
0 22 2 =∫ zerfi( az − b) exp( − a z) sin ( a1 z+γ) dz = −∞
aa2 sinγ+ cos γ a =1 erfi (bb) +×exp( 2 ) 42+ 42 22aa1 (aa+ 11) 8 a
2 2 a b 2 1+ i 2 × sin γ + (a − a1 )Re erf ab − (a + a1 )× a1 2a1
++22 11i 22ab i ×Im erf ab+ a − a11 +cos γ+(a + a )Re erf ab + a 22aa111
1+ i +22 − ++ (a a11)Im erf ab a a {aa>0,1 >≤ 0, b 0}. 2a1
+∞ a 4. zerf( az+ b) exp − a z22 sin a z+γ dz = × ∫ ( 12) ( ) 22 −∞ aa12+
2 2 2 2 2 a a1 + a + a a a b × exp − b 2 1 2 sin γ + 2 × 4 + 2 + 2 + 2 4 + 2 + 2 + 2 a a1 a2 2a a1 a a1 a2 2a a1
90 a + ia a 2a b 2 a + ia × 1 2 + γ + 2 1 2 Re cos Im 2 a 4 + a 2 + a 2 + 2a 2a 2 a + a1 − ia2 1 2 1 a + a1 − ia2
{a1 > 0}.
+∞ a 5. zerfi( az+ b) exp − a z22 sin a z+γ dz = × ∫ ( 12) ( ) 22 −∞ aa12+
2 2 2 a + a − a a a 2 a b 2 × 2 1 2 1 γ + 2 × exp b sin a 4 + a 2 + a 2 − 2a 2 a a 4 + a 2 + a 2 − 2a 2 a 1 2 1 1 2 1
a + ia a 2 a b 2 a + ia × 1 2 + γ + 2 1 2 Re cos Im 2 a 4 + a 2 + a 2 − 2a 2 a 2 a1 − a − ia2 1 2 1 a1 − a − ia2
> 2 =>2 ≠= { aa1 or aa120, a 0, b 0 }.
+∞ a 6. zerfi( az) exp− a22 z sin a z 2+γ dz = × ∫ ( ) ( 1 ) 42 −∞ (aa+ 11) 2 a
×aa22 −sin γ+ aa +cos γ {a > 0}. ( 11) ( ) 1
∞ αsin γ+α cos γ 7. zerf(α z +β) exp −αz22 sin α z + γ dz = 12× ∫ ( 12) ( ) 22 0 22α+α12 α 1 α + iα × (β) + − β2 1 2 − γ × erf exp i 4 2 α 2 + α + iα (α 2 − iα1 ) α + α1 + iα 2 1 2 αβ 1 × A1 − erf + × 2 2 α + α1 + iα 2 (α 2 + iα1 ) α + α1 − iα 2 α − iα αβ × − β2 1 2 + γ − exp i A2 erf α 2 + α − iα 2 1 2 α + α1 − iα 2
22 2 2 { lim Re αzz + α12 +2 αβ z − Im α z + ln z = z→∞ ( ) ( )
22 =lim Re α12zz − Im α = +∞ ; z→∞ ( ) ( )
2 Ak = ±1 if lim Re α + α12 +(3 − 2ki) α z = ±∞ }. z→∞
91 ∞ (1 i)α 8. zerf(α z +β) exp α22 ±i α zsin α z 2 +γ dz = × ∫ ( 11) ( ) 2 0 82α11( α±i α)
2 2 α ± 2iα 1± i αβ erf (β)exp(± iγ ) × exp± i β 1 + γ erf ⋅ − A + 2α 2 2 1 α1 8α1 4iα
2 γ exp −β iiexp( ) ( ) 2 1 erf (β+) { lim Re(αβz ) ± Im(α1z )+ ln z = 4α2 πβ z→∞ 2
22 2 22 =lim Re( αβz) = +∞ , lim Reα zz 2Imα1 = lim Re α z = −∞ ; z→∞ zz→∞ ( ) ( ) →∞ ( )
A = 1 if lim Re( 1 iz) α1 = +∞ , A = −1 if lim Re( 1 iz) α1 = −∞ }. z→∞ z→∞
2.12.2. +∞ na! 1. z21n+ erf( az+ b) exp − a z2 sin a z 2+γ dz = × ∫ ( 12) ( ) 2 0
2 2 2 2 2 a a1 + a + a a a b × exp − b 2 1 2 sin γ + 2 × 4 + 2 + 2 + 2 4 + 2 + 2 + 2 a a1 a2 2a a1 a a1 a2 2a a1
(12) ab W( aa,,12 a , b) n! a ×Re 1 − erf 1 +× 222 a+− a12 ia a +− a12 ia
2 22 (12) aa1++ a a ab W( aa,,12 a , b) ×−expb2 12 Im 1− erf 1 × aaa422+++2 aa 2 22 12 1 a+− aia1 2 a+− aia12
22 aab nb!erf ( ) 1 ×cos γ+ 2 + Re sin γ+ aaa422+++2 aa 2 2 n+1 12 1 (a12− ia )
1 +Im cos γ+ n+1 (a12− ia )
na! (12) (12) + ReW22( aaab ,12 , ,) ⋅ sin γ+ Im W( aaab ,12 , ,) ⋅ cos γ {a1 > 0}. 2 π
+∞ 21n+ 2 2 2. ∫ zerfi( az+ b) exp( − a12 z) sin ( a z+γ) dz = 0
92 0 na! a222+− a aa = z21n+ erfi( az− b) exp − a z2 sin a z 2+γ dz = exp b2 12 1 × ∫ ( 12) ( ) 422 2 −∞ 2 aaa++−122 aa 1
22 (12) aab ab W( ia,, a12 a , ib) ×sin γ+ 21Re erf +1 + aaa422++−2 aa 2 22 12 1 aa1−− ia 2 aa 12−− ia
na! a222+− a aa aab22 + expb2 12 1cos γ+ 2 × 422 2 422 2 2 aaa++−1222 aa 1 aaa++−12 aa 1 (12) ab W( ia,, a12 a , ib) ×+Im 1 erf 1 + −−2 −−2 a12 a ia a12 a ia
n!erfi(b) 1 1 n!a + Re sin γ+ Im cos γ + × 2 n+1 n+1 (a1 − ia2 ) (a1 − ia2 ) 2 π
(12) (12) × ReW( ia , a , a , ib) ⋅ sin γ+ Im W( ia , a , a , ib) ⋅ cos γ 2212 12
> 2 =>≠<2 > } { aa1 or a12 a0, a 0, ab 0 for n 0 .
+∞ 2n+ 1 22 2 3. ∫ zerfi( az+ b) exp( − a z) sin ( a1 z+γ) dz = 0
0 na! a22 b = 2n+ 1 − − 22 2+γ = 2 γ+ × ∫ zerfi( az b) exp( a z) sin ( a1 z) dz exp(b ) sin −∞ 2a1 a1 (12 ) 2 1+ i W1 (ia, a , a1 , ib) n!a 2 × Re1+ erf ab + exp(b )× 1− i 2a1 2a1
(12) 2 ab22 1+ i W1 ( ia, a ,, a1 ib) ×cos γ+ Im 1+ erf ab + ai 1− 1 2a1
nb!erfi( ) 11 + γ+ γ + Re ++sin Im cos 2 22nn11 a−− ia a ia ( 11) ( )
na! (12) 2 + ReW2 ( ia , a , a1 , ib) ⋅ sin γ+ 2 π
+ImW(12) ia , a2 , a , ib ⋅γ cos { > > < – for > }. 2 ( 1 ) a 0, a1 0, b 0 n 0
93 +∞ 2n+1 2 2 4. ∫ z erf (az + b)exp (− a1z ) sin (a2 z + γ ) dz = −∞
2++ 22 22 aa1 a12 a aab2 =na! exp − b2 sin γ+ × 422+++ 2422+++ 2 aaa1222 aa11 aaa12 aa
(12 ) W (a, a , a , b) a 2 a b 2 × 1 1 2 + γ + 2 × Re cos 2 a 4 + a 2 + a 2 + 2a 2 a a + a1 − ia2 1 2 1 ( ) W 12 (a, a , a , b) 1 1 2 × Im {a1 > 0}. 2 a + a1 − ia2 +∞ 2n+1 2 2 5. ∫ z erfi (az + b)exp(− a1 z )sin(a2 z + γ )dz = −∞
a222+− a aa aab22 =na! exp b2 12 1sin γ+ 2 × 422 2 422 2 aaa++−1222 aa 1 aaa++−12 aa 1 (12 ) 2 2 W (ia, a1 , a2 , ib) a a b × Re 1 + cos γ + 2 × 4 2 2 2 − 2 − a + a + a − 2a a a1 a ia2 1 2 1 ( ) W 12 (ia, a , a , ib) 1 1 2 2 × Im { aa1 > for n > 0 }. 2 a1 − a − ia 2
∞ ni! α 6. z21n+ erf(α z +β) exp −αz2 sin α z 2 + γ dz = × ∫ ( 12) ( ) 4 0
2 (12 ) β (α − iα ) αβ W (α, α , α , β) × − 1 2 + γ − 1 1 2 + exp i erf A1 α 2 + α − iα 2 2 1 2 α + α1 − iα 2 α + α1 − iα 2 2 (12) β( α12 +αi ) αβ W (,α α12 , −α ,) β +exp − −γiA −erf 1 + 2 2 α +α +i α α22 +α + α α +α + α 12 12ii12
n+1 α n!1 (12) +exp(iW γ) erf ( β)+( αα,,,12 α β) − 4ii α −α π 2 12
n+1 1 α (12 ) − exp(− iγ )erf (β) + W (α, α , − α , β) α + α 2 1 2 1 i 2 π
94 22 2 2 { lim Re αzz + α12 +2 αβ z − Im α z −( 2 n − 1) ln z = z→∞ ( ) ( )
22 =lim Re α12z − Im α z − 2 nz ln = +∞ ; z→∞ ( ) ( )
2 Ak = ±1 if lim Re α + α12 +(2k − 3) iz α = ±∞ }. z→∞
∞ 2n+1 2 2 2 n!(1 i )α 7. ∫ z erf (αz + β)exp[(α ± iα1 )z ]sin (α1 z + γ ) dz = × 0 8 α1
2 2 2 α β 1± i αβ (12 ) 2 × exp− β ± i + γ A − erf ⋅ W (α, − α iα , ± α , β)± α 1 1 1 2 1 2 α1 + n 1 n! α (12 ) 2 1 ± exp(± iγ ) W (α, − α iα1 , ± α1 , β)+ erf (β) − ± 4i π 2 α 2 ± α 2i 1
2 ni! erf (β) exp(−β ) n (2k ) ! ±γ + exp(i )+ ∑ +− 4 22n112 k nk −α πβ k=0 k!2αβ −α ( ) ( ) ( )
2 { lim Re( 2αβz) ± 2Im α1 z −( 2 n − 1) ln z = lim Re( αβznz) − ln = +∞ , zz→∞ ( ) →∞
22 2 22 lim Reαz 2Im α+=α+=−∞1 z 2 nz ln lim Rez 2 nz ln ; zz→∞ ( ) ( ) →∞ ( ) A = 1 if lim Re[(1 i ) α1 z ]= +∞ , A = −1 if lim Re( 1 iz) α1 = −∞ }. z→∞ z→∞
Introduced notations:
n+1−k ( ) n (2k )! 1 1) W 12 (α, α , α , β) = ∑ × 1 1 2 k! α − iα k =0 1 2 k (αβ)2k −2l × , ∑ 2k −l l=0 l 2 4 l!(2k − 2l )!(α + α1 − iα 2 )
n+1−k ( ) n (2k )! 1 W 12 (α, α , α , 0) = ; 1 1 2 ∑ k k =0 k 2 2 α1 − iα 2 4 (k!) (α + α1 − iα 2 )
+− n (2!k ) nk1 2) (12) 2 1 W2 (α, α12 , α , β) = exp( −β ) ∑ × k=1 ki! α12 −α
95 2k−− 12 lr + 2 kl1 rl!(αβ) k( −1!) × −× ∑∑ −+ ∑ lk!2− 2 l ! lr− 2 2 klr 2l− 1!2 k +− 1 2 l ! lr=11( ) = 4 2!riα +α − α l=1( ) ( ) ( ) ( 12)
2k−− 12 lr + 2 l (αβ) ( ) × , W 12 (α, α , α , 0) = 0 . ∑ 2 klr−+ 2 1 2 r=1 ri−1! α2 +α − α ( ) ( 12)
n 2 2 2.13. Integrals of z erf (αz + β)exp(− α1z + β1z )sin(α 2 z + β 2 z + γ ),
n 2 n 2 z erf (αz )exp(− α1z )sin(βz ) , z erf (αz )exp(− α1z )cos(βz )
2.13.1.
+∞ 0 2 2 2 2 1. ∫ erf (a z )exp(− a1z ) sin (a2 z + γ ) dz = − ∫ erf (az ) exp (− a1z )sin (a2 z + γ ) dz = 0 −∞
11a++ ia ia 1a++ ia ia = Re ln 12 cosγ+ Im ln 12 sin γ 2 πa12 + ia a 12 +− ia ia a12+ ia a 12 +− ia ia
{a1 > 0 or [a1 = 0, a2 ≠ 0]}.
+∞ π b22− b bb 2. erf(az) exp − a22 z + bzsin ( b z +γ) dz = exp11 sin +γ × ∫ ( ) 1 22 0 4a 42aa
2 22− b+ ib1 π bb11bb ×Re erf+ 1 + exp cos+γ × 4a 22 22a 42aa 2 b + ib1 × Imerf +1 {a > 0}. 2 2a
+∞ 22 1 3. ∫ erfi(a z) exp(− a12 z) sin ( a z+γ) dz = × 0 2 π + + + + 1 a1 ia2 a 1 a1 ia2 a × Re ln sin γ −Im ln cos γ a1 + ia2 a1 + ia2 − a a1 + ia2 a1 + ia2 − a 2 { aa1 > }. +∞ +∞ 1 4. erf (a z )exp (− a z 2 )sin(b z )dz = erf (az )exp(− a z 2 ) sin (bz )dz = ∫ 1 2 ∫ 1 0 −∞
96 π b2 ab = − > exp erfi { a1 0 }. 44aa1122 2 a11+ aa
+∞ +∞ 1 5. erfi (az )exp(− a z 2 )sin(bz )dz = erfi (az )exp(− a z 2 )sin(bz )dz = ∫ 1 2 ∫ 1 0 −∞ π b2 ab = − > 2 exp erf {}aa1 . 44aa1122 2 a11− aa
+∞ 2 2 6. ∫ erf (az + b)exp(− a1z + b1z )sin(a2 z + b2 z + γ )dz = −∞
2 − 2 − + 2 − 2 a1b1 a1b2 2a2 b1b2 2a1b1b2 a2 b1 a2 b2 = π exp sin + γ × 2 + 2 2 + 2 4a1 4a2 4a1 4a 2
2 2 1 2a b + ab + 2ia b − iab 2a b b + a b − a b × 1 1 2 2 − 1 1 2 2 1 2 2 + γ × Re erf cos a + ia 2 4a 2 + 4a 2 1 2 2 a1 + ia2 a + a1 + ia2 1 2 1 2a b + ab + 2ia b − iab × Im erf 1 1 2 2 a + ia 2 1 2 2 a1 + ia2 a + a1 + ia2
{a1 > 0 or [ab11= = 0, a 2 ≠ 0]}.
2 2 +∞ π b − b 7. erf (az )exp − a 2 z 2 + bz sin(b z + γ )dz = exp 1 × ∫ ( ) 1 −∞ a 2 2a
bb1 b + ib1 bb1 b + ib1 × sin + γ Re erf + cos + γ Imerf {a > 0} . 2a 2 2 2a 2a 2 2 2a +∞ 2 2 8. ∫ erfi (az + b)exp (− a1z + b1z )sin (a2 z + b2 z + γ )dz = −∞
2 − 2 − + 2 − 2 a1b1 a1b2 2a2 b1b2 2a1b1b2 a2 b1 a 2 b2 = π exp sin + γ × 2 + 2 2 + 2 4a1 4a2 4a1 4a 2
1 22a b++ ab ia b − iab ×−Re erfi 112 2 a+ ia 2 12 2 a1+ ia 21 a −+ a ia 2
2 2 2a b b + a b − a b 1 2a b + ab + 2ia b − iab − 1 1 2 2 1 2 2 + γ 1 1 2 2 cos Im erfi 4a 2 + 4a 2 a + ia 2 1 2 1 2 2 a1 + ia2 a1 − a + ia2 2 { aa1 > }.
97 ∞ 1 exp(iγ ) 9. α− α22 αγ += × ∫ erf( z) exp( 12 z) sin ( z) dz 0 4 π αα12− i
αα−−ii αexp(−iγ ) αα++ii α ×+ln 12 ln 12 αα12−+ii α αα 12 + i αα 12 +− ii α
22 { lim Reα−α12z Im zz + ln = z→∞ ( ) ( )
22 2 2 =lim Re αzz + α12 −Im α z + 2ln z = +∞ , z→∞ ( ) ( )
2 22 2 2 α2 ≠−α 12, α ≠−( α +α 1) }.
∞ π (β + iβ )2 10. erf (αz )exp(− α 2 z 2 + βz ) sin(β z + γ ) dz = exp 1 + iγ × ∫ 1 8iα 2 0 4α 222 β+ii β11π (β−i β1 ) β− β ×erf± 1 − exp −γi erf± 1 αα8iα 2 22 4α 22
22 { lim Re(αzz −β) −Im( β1 z) + ln z = z→∞ 22 =lim Re 2 αzz − β −Im( β1 z) + 2ln z = +∞ , lim Re(αz) = ±∞ }. z→∞ ( ) z→∞
∞ π β2 11. erf (αz )exp(− α z 2 )sin(βz ) dz = ± exp − × ∫ 1 4α 0 2 α1 1 αβ 2 ×erfi { lim Re(α1 z) − Im( β+ zz) ln = 2 z→∞ 2 α1 α + α1
22 2 2 =lim Re( αzz + α1 ) −Im( β z) + 2ln z = +∞ , lim Re α + α1 z = ±∞ }. z→∞ z→∞ ( )
∞(T ) 2 πi 12. erf(α+βz ) exp −αz22 +β zsin α z +β+γ z dz = × ∫ ( 11) ( 2 2) 2 ∞(T1 )
2 A (β − iβ ) α(β − iβ ) + 2β(α + iα ) × 1 exp 1 2 − iγerf 1 2 1 2 − α + iα 4α1 + 4iα 2 2 1 2 2 α1 + iα 2 α + α1 + iα 2
2 A (β + iβ ) α(β + iβ ) + 2β(α − iα ) − 2 exp 1 2 + iγerf 1 2 1 2 α − iα 4α1 − 4iα 2 2 1 2 2 α1 − iα 2 α + α1 − iα 2
98 α22 −β − α +β + = { lim Re( 11zz) Im( 2 z 2 z) ln z zT→∞( 1 )
= α+α+αβ−β−α+β+22 2 2 = lim Re( z1 z2 zz 1) Im( 22 z z) 2ln z zT→∞( 1 )
= α22 −β − α +β + = lim Re( 11zz) Im( 2 z 2 z) ln z zT→∞( 2 )
= α+α+αβ−β−α+β+22 2 2 =+∞ lim Re( z1 z2 zz 1) Im( 22 z z) 2ln z , zT→∞( 2 )
α 2 ≠ −α 2 ; = ± if α 2 + α + ( − ) α = 2 1 Ak 1 lim Re 1 3 2k i 2 z z→∞(T2 )
= − α 2 + α + ( − ) α = ±∞ , = if lim Re 1 3 2k i 2 z Ak 0 z→∞(T1 )
α22 +α + − α ⋅ α +α + − α =+∞}. lim Re 12(3 2ki) z lim Re 12(3 2ki) z zT→∞( 12) zT→∞( )
ν 2 2 13. ∫ erf (αz )exp(− α1z ) sin (α 2 z + γ ) dz = 0 . −ν
ν 2 14. ∫ erf (αz )exp(− α1z ) cos (βz ) dz = 0 . −ν
2.13.2.
+∞ 1 bb22− 1. zerf( az) exp − a22 z + bzsin ( b z +γ) dz = exp1 × ∫ ( ) 1 22 0 44aa
bb1 bb1 π+(b ib1 ) × sin + γ Re θ + cos + γ Imθ { θ= × 2a 2 2a 2 2a
2 2 b+ ib1 (b+ ib1 ) b+ ib1 ×+1 erf +2 exp −+1 erf ,a > 0 }. 2 22a 8a 22a
+∞ +∞ 221 2. ∫∫zerf( az) exp(−= a11 z) cos( bz) dz zerf( az) exp( −= a z) cos( bz) dz 0 2 −∞
99 1 2a b 2 πb b 2 ab = exp − − exp − erfi 2 4a1 2 + 4a + 4a a1 4a1 2 + 2 a a1 1 2 a a1 a1
{a1 > 0}.
+∞ +∞ 221 3. ∫∫zerfi( az) exp(− a11 z) cos( bz) dz = zerf i( az) exp( −= a z) cos( bz) dz 0 2 −∞
12ab22π b bab = exp−−exp erf 2 44aa11−−2 44aa− a1 22 a11 a 1 2 a1 aa
2 { aa1 > }.
+∞ 22 4. ∫ zerf( az+ b) exp( − a11 z + b z)sin ( a 2 z+ b 2 z +γ) dz = −∞
2 − 2 − 2 − 2 + 1 a1 (b1 b2 ) 2a 2 b1b2 a 2 (b1 b2 ) 2a1b1b2 = exp sin + γ Reθ − 2 2 + 2 2 + 2 4a1 4a 2 4a1 4a 2
22 a b−+ b2 abb π− 2( 1 2) 112 (b12 ib ) −cos +γIm θ { θ= × 22 44aa+ (a++ ia) a ia 12 1212 22a b++ ab ia b − iab 2a ×erf 112 2 +× 2 a+ ia a22 ++ a ia( a + ia) a ++ a ia 12 12 12 12
2 (22a112 b++ ab ia b − iab 2) ×−exp ,a > 0 }. 2 1 4 a+ ia a ++ a ia ( 12)( 12)
+∞ 22 5. ∫ zerfi( az+ b) exp( − a11 z + b z)sin ( a 2 z+ b 2 z +γ) dz = −∞
2 − 2 − 2 − 2 + 1 a1 (b1 b2 ) 2a 2 b1b2 a 2 (b1 b2 ) 2a1b1b2 = exp sin + γ Imθ + 2 2 + 2 2 + 2 4a1 4a 2 4a1 4a 2
22−+ a2( b 1 b 2) 2 abb 112 π−(b12 ib ) ++cos γθRe { θ= × 44aa22+ a++ ia a ia 12 ( 1212)
100 ab − 2a b + 2ia b + iab 2ia × 2 2 1 1 + × erf + − 2 + ( + ) − 2 + 2 a1 ia2 a1 a ia2 a1 ia2 a1 a ia2
++ − 2 (22a112 b ab ia b iab 2) 2 ×>exp , aa1 }. + −+2 4(a1 ia 21)( a a ia 2)
∞ 2 1 (β+i β ) 6. zerf(α z) exp −α22 z +β zsin ( β z + γ) dz = exp 1 +i γ × ∫ ( ) 1 22 0 84iαα
2 2 π β+ii β (β+i β ) β+ β 111 ×( β+i β1 )erf± 1 + 2 exp − erf± 1 − 2α ααα2 22 8 22
2 2 (β−i β ) π β−i β 1 1 1 − exp −ii γ ( β− β1 )erf± 1 + 222α α 84iαα 22 (β − iβ )2 β − iβ + 2 exp− 1 erf 1 ±1 2 8α 2 2α
22 { lim Re αzz −β −Im( β1 z) = z→∞ ( )
22 =lim Re( 2 αzz − β) −Im( β1 z) + ln z = +∞ , lim Re(αz) = ±∞ }. z→∞ z→∞
∞ 12α β2 7. zerf(α z) exp −α z2 cos( β z) dz = ± exp− − ∫ ( 1 ) 2 4α1 2 44α+α 0 α +α1 1
πβ β2 αβ − − { lim Reαzz2 − Im β= exp erfi ( 1 ) ( ) α 4α1 2 z→∞ 1 2 α1 α + α1
22 2 2 =lim Re α+αzz11 −Im( β+ zz) ln =+∞ , lim Re α+αz =±∞ }. zz→∞ ( ) →∞ ( )
∞(T2 ) 22 8. ∫ zerf(α z +β) exp( −α11z +β z)sin ( α 2 z +β 2 z +γ) dz = ∞(T1 )
2 ii(β −βi ) (13) =A exp 12−γiW (1, αα , , α , ββ , , β) −A × 14 44α+i α 1 1 2 12 24 12 (β + iβ )2 × 1 2 + γ (13) α α − α β β − β exp i W1 (1, , 1 , 2 , , 1 , 2 ) 4α1 − 4iα 2
101 α22 −β − α +β = { lim Re( 11zz) Im( 2 z 2 z) zT→∞( 1 )
= α+α+αβ−β−α+β+22 2 2 = lim Re( z1 z2 zz 1) Im( 22 z z) ln z zT→∞( 1 )
22 = lim [Re(α11zz −β) −Im( α 2 z +β 2 z) = zT→∞( 2 )
= α+α+αβ−β−α+β+22 2 2 =+∞ lim Re( zz12 zz 1) Im( 22 zzz) ln , zT→∞( 2 )
α 2 ≠ −α 2 ; = ± if α2 +α + − α = 2 1 Ak 1 lim Re 12(3 2ki) z zT→∞( 2 )
= − α2 + α + − α = ±∞ , = if lim Re 12(3 2ki) z Ak 0 zT→∞( 1 )
α22 +α + − α ⋅ α +α + − α =+∞ lim Re 12(3 2ki) z lim Re 12(3 2ki) z }. zT→∞( 12) zT→∞( )
ν 2 9. ∫ zerf(α z) exp( −α1 z) sin( β z) dz = 0 . −ν
2.13.3. +∞ (2n)! 2n − 2 2 + γ = γ ⋅ (13) − 1. ∫ z erf (az )exp( a1z )sin (a2 z ) dz [sin ReW2 (a, a1 , a2 ) 0 n! π
− γ ⋅ (13) cos ImW2 (a, a1 , a2 )] {a1 > 0 for n > 0} .
+∞ bb22− 2. zn erf( az) exp − a22 z + b zsin ( b z +γ) dz =exp 1 × ∫ ( ) 1 2 0 4a
bb (13) bb (13) ×11 +γ + +γ sin ReW33( abb , ,11 ,1) cos ImW( abb , , ,1) 22aa22
{a > 0} .
+∞ +∞ 22nn1 2 2 3. ∫∫zerf( az) exp(−= a11 z) sin ( b z) dz zerf( az) exp( −= a z) sin ( b z) dz 0 2 −∞
2 (2n)! b (13) = − exp − ImW (2n, a, a , b) {a > 0}. 2n+1 4 1 1 2 4a1
102 +∞ +∞ 21nn++2 1 21 2 4. ∫∫zerf( az) exp(−= a11 z) cos( bz) dz zerf( az) exp( −= a z) cos( bz) dz 0 2 −∞
2 (2n + 1!) b (13) =−+exp ReW( 2 n 1, aa , , b) {a > 0}. n+1 4 1 1 4 4a1
+∞ (2n)! 2n − 2 2 + γ = γ ⋅ (13) + 5. ∫ z erfi (az )exp( a1z )sin (a2 z )dz [sin ImW2 (ia, a1 , a2 ) 0 n! π
(13) +cos γ⋅ ReW( ia , a , a ) { aa> 2 }. 2 12 1
+∞ +∞ 1 6. z 2nerfi (az )exp(− a z 2 )sin(bz )dz = z 2nerfi (az )exp(− a z 2 )sin(bz )dz = ∫ 1 2 ∫ 1 0 −∞ 2 (2n)! b (13) = exp − ReW (2n, ia, a , b) { aa> 2 }. 2n+1 4 1 1 2 4a1
+∞ +∞ 1 7. z 2n+1erfi (az )exp(− a z 2 )cos(bz )dz = z 2n+1erfi (az )exp(− a z 2 )cos(bz )dz = ∫ 1 2 ∫ 1 0 −∞ 2 (2n +1)! b (13) = exp − ImW (2n +1, ia, a , b) { aa> 2 }. n+1 4 1 1 4 4a1
+∞ n 22 8. ∫ zerf( az+ b) exp( − a11 z + b z)sin ( a 2 z+ b 2 z +γ) dz = −∞
2 n! (b12− ib ) (13) =sin γ⋅ Re exp W( naa,,,,,, a bb b) − n 44a+ ia 1 12 12 2 12
2 n! (b12− ib ) (13) −cos γ⋅ Im exp W( naa,,,,,, a bb b) n 44a+ ia 1 12 12 2 12
{ a1 > 0 for n > 0 }. +∞ n 2 2 9. ∫ z erfi (az + b)exp(− a1 z + b1 z )sin (a2 z + b2 z + γ )dz = −∞
2 n! (b12− ib ) (13) =cos γ⋅ Re exp W( niaa,,,,,, a ibbb) + n 44a+ ia 1 12 12 2 12
2 n! (b12− ib ) (13) +sin γ⋅ Im exp W( niaa,,,,,, a ibbb) { aa> 2 }. n 44a+ ia 1 12 12 1 2 12
103
∞ 2n 2 2 (2n)!i 10. ∫ z erf (αz )exp(− α1z )sin (α 2 z + γ ) dz = × 0 2 πn!
× − γ (13) α α α − γ (13) α α − α [exp( i )W2 ( , 1 , 2 ) exp(i )W2 ( , 1 , 2 )]
22 { lim Reα12z −α Im z −−( 2 nz 1) ln = z→∞ ( ) ( )
22 2 2 =lim Re α+α−αzz12Im z −−( 2 n 2) ln z =+∞ , z→∞ ( ) ( )
2 22 2 2 α2 ≠−α 12,( α ≠− α +α 1 )}. ∞ n 2 2 11. ∫ z erf (αz )exp(− α z + βz )sin(β1z + γ )dz = 0
2 2 1 (β + iβ ) ( ) 1 (β − iβ ) = exp 1 + iγW 13 (α, β, β , ±1) − exp 1 − iγ × 2 3 1 2 2i 4α 2i 4α
× (13) (α β − β ± ) 22 W3 , , 1 , 1 { lim Re(αz −β z) −Im( β1 zn) −( −1) ln z = z→∞ 22 =lim Re( 2 α−β−β−−z z) Im( 1 zn) ( 2) ln z =+∞ , lim Re(αz) = ±∞ }. z→∞ z→∞
∞ 2 (2n)!i β ( ) 12. z 2nerf (α z )exp(− α z 2 )sin(βz )dz = ± exp − W 13 (2n, α, α , β) ∫ 1 2n+1 4α 4 1 0 2 1
2 lim [Re(α1z )− Im(βz ) − (2n −1)ln z ]= z→∞
22 2 2 =lim Re( α+α−β−−zz1 ) Im( zn) ( 2 2) ln z =+∞ , lim Re α + α1 z = ±∞ . z→∞ z→∞ ( ) }
∞ (2n + 1!) β2 13. z21n+ erf(α z) exp −α z2 cos( β z) dz = ± exp− × ∫ ( 1 ) n+1 α 0 4 4 1
× (13) + α α β 2 W4 (2n 1, , 1 , ) { lim Re(α1 z) − Im( β− z) 2 nz ln = z→∞ 22 2 2 =lim Re( α+α−β−−zz1 ) Im( zn) ( 2 1) ln z =+∞ , lim Re α + α1 z = ±∞ }. z→∞ z→∞ ( )
∞(T2 ) n 2 2 14. ∫ z erf (αz + β)exp(− α1z + β1z )sin(α 2 z + β2 z + γ ) dz = ∞(T1 )
104 2 ni! (ββ12− i ) (13) =−−Aexp iWγ( n,,,,,, αα α ββ β) 2n+1 1 44αα+ i 1 12 12 12 (β + iβ )2 − 1 2 + γ (13) α α − α β β − β A2 exp i W1 (n, , 1 , 2 , , 1 , 2 ) 4α1 − 4iα 2
α22 −β − α +β − − = { lim Re( 11z z) Im( 2 z 2 zn) ( 1) ln z zT→∞( 1 )
= α22 −β − α +β − − = lim Re( 11z z) Im( 2 z 2 zn) ( 1) ln z zT→∞( 2 )
= α+α+αβ−β−α+β−−22 2 2 = lim Re( z1 z2 zz 1) Im( 22 z z) ( n2) ln z zT→∞( 1 )
= α+α+αβ−β−α+β−−22 2 2 =+∞ lim Re( z1 z2 zz 1) Im( 22 z z) ( n2) ln z , zT→∞( 2 )
α 2 ≠ −α 2 ; A = ±1 if α2 +α + − α = 2 1 k lim Re 12(3 2ki) z zT→∞( 2 )
= − α2 + α + − α = ±∞ = lim Re 12(3 2ki) z , Ak 0 if zT→∞( 1 )
α22 +α + − α ⋅ α +α + − α =+∞ . lim Re 12(3 2ki) z lim Re 12(3 2ki) z } zT→∞( 12) zT→∞( )
ν 2n 2 2 15. ∫ z erf (αz )exp(− α1z )sin (α 2 z + γ )dz = 0 . −ν
ν 2n 2 16. ∫ z erf (αz )exp(− α1z )cos(βz )dz = 0 . −ν
ν 2n+1 2 17. ∫ z erf (αz )exp(− α1z )sin(βz )dz = 0 . −ν
Introduced notations:
(13) π 1) Wn1 ( ,αα ,1 , α 2 ,, ββ 12 , β) = × α+α12i
nl−2 2βα+( ii α) +αβ−β( ) En( 2) (β−β i) ×+erf 1 2 1 2 12 ∑ nl− 2 l=0 − α+α 2 α+α12ii α +α+α 12 ln!( 2! l) ( 12 i )
105 2 1 2 β( α1 +ii α 2) +α( β 12 − β) +−exp × 2 4(α+αi ) α2 +α +i α α +α12 +i α 12( 12)
+− n− En( 2) nl12 l 2lr+− 12 (li−1!) ( β−β12) 2( 2r− 2!) ×× ∑∑ 2ln− 1! +− 1 2 l ! r−1! lr=11( ) ( ) = ( )
−− 2rq−− 22 r−1 2rq 12 α 2 β( α1 +ii α 2) +α( β 12 − β) ×−∑ −+ − 22rq−− q=0 nlr2 q 2 qr!2( − 2 − 2 q) !( α+α12 i ) ( α +α+α12i )
nl−2 En( 2) (β−βi ) l −∑∑12 (r −×1!) lr=11ln!( − 2! l) =
− 2 rq 2rq−− 12 r−1 (α) 2 β( α1 +ii α 2) +α( β 12 − β) × ∑ , nlr−+ −2 q 21rq−− q=0 qr!2−− 1 2 q ! α+α i α2 +α+αi ( ) ( 12) ( 1 2)
(13) (13) Wn11(21 ,αα , 12 , α ,0,0,0) =Wn( 21 , αα , 12 , α ,0, i β 22 , β) = 0,
(13) (13) Wn11(21+ 1, αα , 12 , α ,0,0,0) =Wn( 21 + 1, αα , 12 , α ,0, i β 22 , β) =
+− n ! α n1 22nl1 12 ( 2!l) = 1 ∑ , +− l 2ni+ 1! α2 +α + α l=0 21nl1 2 ( 1) 12(li!) (α+α12) ( α +α+α12i)
(13) (13) W1 (2n1,α,α1,α2 , β,0,0) = W1 (2n1,α,α1,α2 , β,iβ2 , β2 ) =
π β α+αi 1 = erf 12−× α+α α+α n1 22 ni11212! ( i) α +α1 +in α 2 1! α +α 12 + i α
+− 2 lr1 2lr+− 12 2 n1 −1 l αβ(2 ) β( α+α12i ) l! ( ) ×−exp , 2 ∑∑nl− 21l+− r α +α +i α lr=00(α+αi ) 1 = 2 12 12 rl!2( + 1 − 2 r) !( α +α12 + i α )
(13) (13) W1 (2n1 +1,α,α1,α2 , β,0,0) = W1 (2n1 +1,α,α1,α2 , β,iβ2 , β2 ) =
n ! β2 ( α+αi ) =1 exp −×12 2 α2 +α +i α (2ni1+ 1!) α +α 12 + α 12
106 lr− 2nr1 +− 12 2lr+− 12 2 n1 (2!l) l 2 αβ( ) × ; ∑∑n+−12 l lr− lr=00li! (α+α) 1 = 2 12 rlr!2( − 2) !( α +α12 + i α )
(13) i α+12ii α −α 2) W2 (αα,,12 α) = ln + 21n+ α+ii α +α (2 α+α12i ) 12
2 n−1 (l!) +α ; ∑ nl− l+1 l=0 2 (2li+ 1!) 4( α+α12) ( α +α+α 12 i)
2 E (n / 2 ) n−2k ( ) πn! β + iβ (β + iβ ) 3) W 13 (α, β, β , A) = erf 1 + A 1 + 3 1 n+1 ∑ k n−2k 4α 2 2α k =0 4 k!(n − 2k )!(2α)
nk−2 En( /2) (β+i β ) k−1 nl!! 1 (13) +2 ∑∑Wl(2+ 1, αββ , ,1 , A) + n+12 n−− k kl 5 α+ kl=10kn!( −α 2 k) !2( ) = 4( 2l 1!)
nk+−12 n− En( /2) ki!(β+ β ) k−1 1 1 (13) + ∑∑Wl(2,αββ , ,1 , A) ; nk+−12 l! 5 kl=10(2!kn) ( +− 12!2 k) ( α) =
En( /2) k nk−2 (13) π iαβ (−β1) 4) Wn( ,,,αα β=) erf + 4 1 n+−2 ∑ nk i α 2 k=0 kn!( −α 2! k) 112 α11 α +α
En( /2) α αβ22 βnk−2 k + exp ∑∑(l −×1) ! + 2 n 124α α +α kl=11kn!( − 2! k) = i α1 +α 11( )
−− klr+− 2 2lr 12 l−1 (−1) ( αβ) n− En( /2) (k −β1!) nk+−12 × −× ∑∑21lr−− rk=01nkl− +−22 r = (2kn− 1!) ( +− 1 2 k) ! rl!2( − 1 − 2 r) ! α11( α +α )
2lr−− 22 +− klr+− 2 kl22kl 12 ( 2l − 2!) −1 (−1) ( αβ) × ; ∑∑ −− l −1! nkl− +−22 r 22lr lr=10( ) = rl!2− 2 − 2 r ! α α +α ( ) 11( )
2 ni!2 (β+i β ) β+ β (13) 111 5) Wn5 ( 11,,,,αββ A) = exp − erf +A × n1 +22 (−2) 8α 22α
107 En( /2) nr−22 1 (β+ii β ) 1 n ! (β+ β ) ×∑ 11+1 exp −× r nr11−+21 n 2 r=0 2rn !( 1 −α 2 r) !2( ) ( −2) π4α
( ) E n1 / 2 (β + iβ )n1 −2r r−1 q!(β + iβ )2q+1 × ∑ 1 ∑ 1 + n1 −2r r−q 2q+1 r=1 r!(n1 − 2r )!(2α) q=0 2 (2q +1)!(2α)
− ( ) n1 E n1 / 2 r!(β + iβ )n1 +1−2r r−1 2r−q (β + iβ )2q + ∑ 1 ∑ 1 . n1 +1−2r 2q r=1 (2r )!(n1 +1− 2r )!(2α) q=0 q!(2α)
108 2.14. Integrals of z n [± 1 − erf (αz + β)],
n z [erf (α 1 z + β 1 ) erf (α 2 z + β 2 )] 2.14.1. +∞ 0 1 β 1. [1− erf (az + β)] dz = − [−1− erf (az − β)] dz = exp(− β2 )+ [erf (β) −1] ∫ ∫ a 0 −∞ πa {a > 0}.
+∞ β β 2. [erf (az + β ) − erf (az + β )]dz = 1 [1− erf (β )]+ 2 [erf (β ) −1]+ ∫ 1 2 a 1 a 2 0 1 + [exp(− β2 )− exp(− β2 )] {a > 0}. π a 2 1
+∞ ββ12 3. ∫ erf (a11 z +β−) erf ( a 2 z +β=− 2) dz 1 erf ( β+1) erf( β−+2) 1 0 aa12
1122 +exp( −β21) −exp( −β ) {a1 > 0, a2 > 0}. ππaa21
+∞ 2 4. ∫[erf (az + β1 ) − erf (az + β2 )] dz = (β1 − β2 ) {a > 0}. −∞ a
+∞ ββ12 5. ∫ erf (a11 z +β) −erf ( a 2 z +β 2) dz =2 − {a1 > 0, a2 > 0}. −∞ aa12
∞ 1 β 6. ±1 − erf ( αz +β) dz =exp −β2 + erf( β) 1 ∫ ( ) α 0 πα
{ lim Reα22z +αβ+ 2 zz 2ln =+∞ , lim Re( α=±∞z) }. zz→∞ ( ) →∞
∞ 1 α + β − α + β = − β2 − − β2 + 7. ∫[erf ( z 1 ) erf ( z 2 )] dz [exp( 2 ) exp( 1 )] 0 πα ββ 21 22 +βerf( 21) 1 −β erf( ) 1 { lim Re(αz + 2 αβ1 zz) + 2ln = αα z→∞
22 =lim Re α+αβ+z 22 zz 2ln =+∞ , lim Re( α=±∞z) }. zz→∞ ( ) →∞
∞ β β 8. [erf (α z + β ) erf (α z + β )] dz = 1 [ A − erf (β )] 2 [ A − erf (β )]− ∫ 1 1 2 2 α 1 1 α 2 2 0 1 2
109 11 − −β22 ± −β exp( 12) exp( ) πα12πα
22 22 { lim Reα1z + 2 αβ 11 zz + 2ln = lim Re(α2z+ 2 αβ 22 zz) + 2ln = +∞ ; z→∞ ( ) z→∞
Ak = 1 if lim Re(αk z) = +∞ , z→∞
Ak = −1 if lim Re(αk z) = −∞ ; lim Re(α12zz) ⋅ lim Re( α) = ±∞ }. z→∞ zz→∞ →∞
2.14.2.
+∞ 0 21β+2 1. z−1 erf ( az +β) dz = z −−1 erf ( az −β) dz = × ∫∫ 2 0 −∞ 4a β ×[1− erf (β)]− exp(− β2 ) {a > 0}. 2 πa 2
+∞ 1 2. zerf( az +β−) erf ( az +β=) dz βexp −β−β22 exp −β+ ∫ 1 2 2 1( 12) ( 2) 0 2 πa
β2 + β2 + 2 1 1 2 2 1 + [erf (β1 ) −1]+ [1− erf (β2 )] {a > 0}. 4a 2 4a 2
+∞ 1 β1 2 β2 2 3. ∫ z[erf (a1z + β1 ) − erf (a2 z + β2 )] dz = exp(− β )− exp(− β ) + 2 π 2 1 2 2 0 a1 a2 21ββ22++ 21 + 12ββ−+ − {a > 0, a > 0}. 22erf( 12) 1 1 erf ( ) 1 2 44aa12
+∞ β2 − β2 4. z[erf (az + β ) − erf (az + β )]dz = 2 1 {a > 0}. ∫ 1 2 2 −∞ a
+∞ 2121β+22 β+ 5. zerf ( a z +β) −erf ( a z +β) dz =21 − {a > 0, a > 0}. ∫ 11 2 2 221 2 −∞ 22aa21
∞ 2β2 +1 β 6. z[±1− erf (αz + β)] dz = [±1− erf (β)]− exp − β2 ∫ 2 2 ( ) 0 4α 2 πα
{ lim Reα22z +αβ+ 2 zz ln =+∞ , lim Re( α=±∞z) }. zz→∞ ( ) →∞
110 ∞ 1 7. z[erf (αz + β ) − erf (αz + β )] dz = β exp − β2 − β exp − β2 + ∫ 1 2 2 [ 1 ( 1 ) 2 ( 2 )] 0 2 πα
β2 + β2 + 2 1 1 2 2 1 + [erf (β1 ) 1]− [erf (β2 ) 1] 4α 2 4α 2
22 { lim Reαz + 2 αβ1 zz + ln = z→∞ ( )
22 =lim Re αz +αβ+ 22 zz ln =+∞ , lim Re( α=±∞z) }. zz→∞ ( ) →∞
∞ 1 ββ 8. zerf αβ z +erf αβz += dz 12exp −ββ22exp −+ ∫ ( 11) ( 2 2) 22( 12) ( ) 0 2 π αα12
β+22β+ 211221 + erf ( β11) −AA ± 2 −erf ( β 2) αα22 4412
22 22 { lim Reα1z + 2 αβ 11 zz + ln = lim Re(α2z+ 2 αβ 22 zz) + ln = +∞ ; z→∞ ( ) z→∞
Ak = 1 if lim Re(αk z) = +∞ , z→∞
Ak = −1 if lim Re(αk z) = −∞ ; lim Re(α12zz) ⋅ lim Re( α) = ±∞ }. z→∞ zz→∞ →∞
2.14.3.
+∞ 0 1. ∫ z 2 [1− erf (az + β)] dz = − ∫ z 2 [−1− erf (az − β)] dz = 0 −∞
β2 +1 2β3 + 3β = exp(− β2 )+ [erf (β) −1] {a > 0}. 3 πa3 6a3
+∞ 1 2. z2erf( az +β−) erf ( az +β=) dz β+21 exp −β−β+× 221 ∫ 123 ( 2) ( 21) ( ) 0 3 πa
2β3 + 3β 2β3 + 3β × − β2 + 1 1 − β + 2 2 β − { > } exp( 1 ) ] [1 erf ( 1 )] [erf ( 2 ) 1] a 0 . 6a3 6a3
+∞ 3 2β + 3β1 3. z 2 [erf (a z + β ) − erf (a z + β )] dz = 1 [1− erf (β )]− ∫ 1 1 2 2 3 1 0 6a1
23β+β3β+ 221 β+1 − 221− erf ( β ) − 1 exp −β22 + 2exp −β {a > 0, a > 0}. 3[ 2] 33( 12) ( ) 1 2 6a2 33ππ aa 12
111 +∞ 232β+β−β33 −β 3 4. z2 erf( az +β) − erf ( az +β) dz = 1122 {a > 0}. ∫ 12 3 −∞ 3a
+∞ 33 2323β+β12 β+β 5. z2 erf ( a z +β) −erf ( a z +β) dz =12 − ∫ 11 2 2 33 −∞ 33aa12
{a1 > 0, a2 > 0}.
∞ β+231 23β+β 6. z22±1 − erf ( αz +β) dz = exp −β + erf( β) 1 ∫ 33( ) 0 36πα α
{ lim Reα22zz +αβ=+∞ 2 , lim Re( α=±∞ z) }. zz→∞ ( ) →∞
∞ 3 3 2β + 3β2 2β + 3β1 7. z 2 [erf (αz + β ) − erf (αz + β )] dz = 2 [erf (β ) 1]− 1 × ∫ 1 2 3 2 3 0 6α 6α 1 × β + β2 + − β2 − β2 + − β2 [erf ( 1 ) 1] [( 2 1)exp( 2 ) ( 1 1)exp( 1 )] 3 πα3
22 22 { lim Reαzz +αβ= 212 lim Re α z +αβ 2 z =+∞ , lim Re( α=±∞ z) }. zz→∞ ( ) →∞ ( ) z→∞
∞ β3 + β 2 2 1 3 1 8. ∫ z [erf (α1z + β1 ) erf (α 2 z + β2 )]dz = [ A1 − erf (β1 )] α3 0 6 1
23β+β3 1 β+221 β+ 1 22A −erf ( β) − 1exp −β222exp −β 322 33( 1) ( 2) 6α2 3 π αα12
22 { lim Reα1zz + 2 αβ 11 = z→∞ ( )
= α 2 2 + α β = +∞ = lim Re( 2 z 2 2 2 z ) ; Ak 1 if lim Re(α k z ) = +∞ , z→∞ z→∞
Ak = −1 if lim Re(α k z ) = −∞ ; lim Re(α12zz) ⋅ lim Re( α) = ±∞ }. z→∞ zz→∞ →∞
2.14.4.
+∞ 0 n − + β = − n+1 n − − − β = (14 ) β 1. ∫ z [1 erf (az )]dz ( 1) ∫ z [ 1 erf (az )]dz W1 (n, a, , 1) 0 −∞ {a > 0}.
+∞ n + β − + β = (14 ) β − (14 ) β 2. ∫ z [erf (az 1 ) erf (az 2 )] dz W1 (n, a, 2 , 1) W1 (n, a, 1 , 1) 0 {a > 0}.
112 +∞ n + β − + β = (14 ) β − 3. ∫ z [erf (a1z 1 ) erf (a2 z 2 )] dz W1 (n, a2 , 2 , 1) 0
− (14 ) β > > W1 (n, a1 , 1 , 1) {a1 0, a2 0}.
+∞ n + β − + β = (14 ) β − (14 ) β 4. ∫ z [erf (az 1 ) erf (az 2 )]dz n![W2 (n, a, 1 ) W2 (n, a, 2 )] −∞ {a > 0}.
+∞ n 5. ∫ zerf ( a11 z +β) −erf (a 2 z +β 2) dz = −∞
(14) (14) =n! W( na ,,β−) W( na , , β) {a > 0, a > 0}. 2211 2 2 1 2
∞ n ± − α + β = (14 ) α β ± 6. ∫ z [ 1 erf ( z )] dz W1 (n, , , 1) 0
22 { lim Re(αz + 2 αβ zn) −( − 2) ln z = +∞ ; lim Re(αz) = ±∞ }. z→∞ z→∞
∞ n α + β − α + β = (14 ) α β ± − (14 ) α β ± 7. ∫ z [erf ( z 1 ) erf ( z 2 )] dz W1 (n, , 2 , 1) W1 (n, , 1 , 1) 0
22 { lim Reαz + 2 αβ1 zn −( −2) ln z = z→∞ ( )
22 =lim Re( αz + 2 αβ2 zn) −( −2) ln z = +∞ ; lim Re(αz) = ±∞ }. z→∞ z→∞
∞ n 8. ∫ zerf ( α11 z +β) erf ( α 2z +β 2) dz = 0 (14) (14) =±Wn11( , α2 , β 2 , AWn 2) −( , αβ111 ,, A)
22 { lim Reα1z + 2 αβ 11 zn −( −2) ln z = z→∞ ( )
22 =lim Re α2z + 2 αβ 22 zn −( −2) ln z= +∞ ; z→∞ ( )
Ak = 1 if lim Re(α k z ) = +∞ , z→∞
Ak = −1 if lim Re(αk z) = −∞ ; lim Re(α12zz) ⋅ lim Re( α) = ±∞ }. z→∞ zz→∞ →∞
113 Introduced notations:
2 n− En( /2) nk+−12 exp −β (14) n! β ( ) 1) Wn( ,,,αβ A) = A −erf ( β) + × 1 nk+1 ∑ (−α) k=0 4kn !( +− 12 k) ! π
− ( ) ( ) n E n / 2 1 k l!βn−2k +2l E n / 2 k! k βn−2k +2l × − , ∑ ∑ k −l ∑ ∑ k =1 k!(n +1− 2k )!l=1 4 (2l )! k =0 (2k +1)!(n − 2k )!l=0 l!
(14) n ! α= 1 , Wn1 (21 , , 0, A) 21n1 + (21n1 +) πα
(14 ) (2n1 +1)! W (2n1 +1, α, 0, A) = A ; 1 2n1 +2 (n1 +1)!(2α)
n n−E (n / 2 ) n+1−2k ( ) (−1) β 2) W 14 (n, α, β) = , 2 n+1 ∑ 2k −1 α k =0 2 k!(n +1− 2k )!
( ) 1 14 α = (14 ) W2 (2n1 , , 0) 0 , W (2n1 +1, α, 0) = − . 2 2n1 +1 2n1 +2 2 (n1 +1)!α
2.15. Integrals of z n [± 1 − erf (αz + β)]2 , z n [1 − erf 2 (αz + β)],
n 2 z [erf (α 1 z + β 1 ) erf (α 2 z + β 2 )],
n 2 2 z [erf (α 1 z + β 1 ) − erf (α 2 z + β 2 )]
2.15.1.
+∞ 0 222 1. −+β =−−−= ββ −− ∫∫1 erf (az) dz 1 erf (az) dz erf( 2) 1 0 −∞ 2π a
2 2 β 2 −exp( −β) erf( β−−) 1 erf( β−) 1 {a > 0}. πa a
+∞ 2 β −2 +β= β −β+22 β−− 2. ∫ 1 erf (az) dz erf( ) exp( ) erf( ) 1 0 π a a 2 − erf( 2β−) 1 {a > 0}. 2π a
+∞ 0 3. +β −22 +β =− −β + −β = ∫∫erf (a11 z ) erf (a 2 z 2) dz erf (a11 z ) erf (a 2 z 2) dz 0 −∞
114 β 1 1 2 = 1 − β − − β2 + β − β2 + [1 erf ( 1 )] exp( 1 ) erf ( 2 )exp( 2 ) a1 πa1 a2 π
2 2 + [1− erf ( 2β2 )]− β2 [1− erf (β2 )] {a1 > 0, a2 > 0}. 2π +∞ ββ 22+β− +β =12 −2 β− − 2 β+ 4. ∫ erf (a11 z ) erf (a 2 z 2) dz 1 erf ( 1) 1 erf ( 2) 0 aa12
2 erf (β ) erf (β ) + 2 − β2 − 1 − β2 + exp( 2 ) exp( 1 ) π a2 a1 1− erf 2 β erf 2 β− 1 2 ( 21) ( ) ++ {a1 > 0, a2 > 0}. 2π aa 21
+∞ 4 5. ∫ [1− erf 2 (az + β)] dz = {a > 0}. −∞ 2πa
+∞ 2 2 4 1 1 6. ∫ [erf (a1z + β1 ) − erf (a2 z + β2 )] dz = − {a1 > 0, a2 > 0}. −∞ 2π a2 a1
∞ 2 2 7. ∫[±1− erf (αz + β)]2 dz = [erf ( 2β)1]− exp(− β2 )× 0 2πα πα β ×[erf (β) 1]− [erf (β) 1]2 α
{ lim 2Reα22z +αβ+ 2 zz 3ln =+∞ , lim Re( α=±∞z) }. zz→∞ ( ) →∞
∞ 2 β 8. [1− erf 2 (αz + β)] dz = erf (β)exp(− β2 )+ [erf 2 (β) −1]− ∫ α 0 πα 2 − [erf ( 2β)1] 2πα
{ lim 2Reα22z +αβ+ 2 zz 3ln =+∞ , lim Re( α=±∞z) }. zz→∞ ( ) →∞
∞ β 1 9. [erf (α z + β ) erf 2 (α z + β )] dz = 1 [±1− erf (β )]− exp(− β2 )± ∫ 1 1 2 2 α 1 1 0 1 πα1 1 2 2 ± β − β2 + − β + β 2 β − erf ( 2 )exp( 2 ) [A erf ( 2 2 )] 2 [erf ( 2 ) 1] α 2 π 2π
115 α22 + αβ + = { lim Re1 z 211 zz 2ln z→∞ ( )
= α22 + α β + = +∞ = lim 2Re2 z 222 zz 3ln ; A 1 if z→∞ ( )
lim Re(α2 z) = +∞ , A = −1 if lim Re(α2 z) = −∞ ; lim Re(α1 z) = ±∞ }. z→∞ z→∞ z→∞
∞ ββ 10. erf 22(α+β−z ) erf ( α+βz) dz =12 1 − erf 2( β−) 1 − erf 2( β+) ∫ 11 2 2αα 1 2 0 12 2 erf (β ) erf (β ) + 2 − β2 − 1 − β2 + exp( 2 ) exp( 1 ) π α 2 α1
2 A − erf ( 2β2 ) erf ( 2β1 )1 + + 2π α 2 α1
α22 + αβ + = { lim 2Re1 z 211 zz 3ln z→∞ ( )
= α22 + α β + = +∞ = lim 2Re2 z 222 zz 3ln ; A 1 if z→∞ ( )
lim Re(α2 z) = +∞ , A = −1 if lim Re(α2 z) = −∞ ; lim Re(α1z) = ±∞ }. z→∞ z→∞ z→∞ 2.15.2. +∞ 0 2β2 +1 1. z[1− erf (az + β)]2 dz = − z[−1− erf (az − β)]2 dz = [erf (β) −1]2 + ∫ ∫ 2 0 −∞ 4a
ββ222 1 +exp( −β) erf( β−−) 1 erf( 2β−) 1 − exp( −β 2 ) {a > 0}. πa2 22 ππ aa22
+∞ 2β2 +1 2β 2. z 1− erf 2 (az + β) dz = 1− erf 2 (β) + [erf ( 2β)−1]+ ∫ [ ] 2 [ ] 2 0 4a 2πa 1 β +exp( − 2 β22) −erf( β) exp( −β ) {a > 0}. 2ππaa22
+∞ 0 3. +β −22 +β = −β + −β = ∫∫zerf ( a11 z ) erf (a 2 z 2) dz zerf ( a11 z ) erf (a 2 z 2) dz 0 −∞
β2 + β2 + 2 1 1 β1 2 2 2 1 2 = [erf (β1 ) −1]+ exp(− β )+ [1− erf (β2 )]+ 2 π 2 1 2 4a1 2 a1 4a2
1 2ββ + −β+2222β−− β −β exp( 2 22) erf( 222) 1 erf( ) exp( ) ππ22 π 2 22aa22 a 2
116 {a1 > 0, a2 > 0}.
+∞ 2β +1 4. z erf 2 (a z + β ) − erf 2 (a z + β ) dz = 2 1− erf 2 (β ) + ∫ [ 1 1 2 2 ] 2 [ 2 ] 0 4a2 2ββ1 + 22erf 2β−+ 1 exp −β− 2 22erf( β) exp −β− 2( 2) 22( 2) 22( ) 22πa2 ππ aa 22 2β1 +1 2 2β1 1 2 − [1− erf (β1 )]− [erf ( 2β1 )−1]− exp(− 2β )+ 2 π 2 π 2 1 4a1 2 a1 2 a1
β1 +erf( β) exp −β2 {a > 0, a > 0}. 2 11( ) 1 2 πa1
+∞ 4β 5. z 1− erf 2 (az + β) dz = − {a > 0}. ∫ [ ] 2 −∞ 2πa +∞ 2 2 4 β1 β2 6. ∫ z[erf (a1z + β1 ) − erf (a2 z + β2 )] dz = − {a1 > 0, a2 > 0}. 2π 2 2 −∞ a1 a2
∞ 2β 2 +1 β 7. z[±1− erf (αz + β)]2 dz = [erf (β) 1]2 + exp(− β 2 )× ∫ 2 2 0 4α πα 2β 1 ×[erf (β) 1]− [erf ( 2β)1]− exp(− 2β2 ) 2πα 2 2πα2
{ lim Reα22z +αβ+ 2 zz ln =+∞ , lim Re( α=±∞z) }. zz→∞ ( ) →∞
∞ 2β2 +1 2β 8. z 1− erf 2 (αz + β) dz = 1− erf 2 (β) + [erf ( 2β)1]+ ∫ [ ] 2 [ ] 2 0 4α 2πα 1 β +exp( − 2 β22) −erf( β) exp( −β ) 2πα22πα
{ lim Reα22z +αβ+ 2 zz ln =+∞ , lim Re( α=±∞z) }. zz→∞ ( ) →∞
∞ β2 + 2 2 1 1 9. ∫ z[erf (α1z + β1 ) erf (α 2 z + β2 )] dz = [erf (β1 ) 1]+ α 2 0 4 1
β 21β+2 1 2β + 12exp −β22 ±1 − erf ( β) ±exp − 2 β 22× 22( 1) 22 2( ) 2 24πα12α 22πα 2πα 2
β2 2 × [A − erf ( 2β2 )] erf (β2 )exp(− β ) πα 2 2 2
117 22 22 { lim Reα1z + 2 αβ 11 zz + ln =lim Re( α2z + 2 α 22 β zz) + ln = +∞ ; z→∞ ( ) z→∞
A = 1 if lim Re(α2 z) = +∞ , A = −1 if lim Re(α2 z) = −∞ ; lim Re(α1 z) = ±∞ }. z→∞ z→∞ z→∞
∞ β2 + 2 2 2 2 1 2 10. ∫ z[erf (α1z + β1 ) − erf (α 2 z + β2 )] dz = [1− erf (β2 )]+ α 2 0 4 2
2ββ1 + 22erf 2β−A +exp −β− 2 22erf( β) exp −β− 2( 2) 22( 2) 22( ) 22πα2πα 22πα
β2 + 2 1 1 2 2β1 − [1− erf (β1 )]− [erf ( 2β1 )1]− α 2 πα 2 4 1 2 1 1 β − − β22 +1 β −β exp( 2 11) erf( 1 ) exp( ) πα22πα 2 11
22 22 {lim[Re(α1z++ 2 αβ 11 zz ) ln ] = lim Reα2z+ 2 αβ 22 zz + ln = +∞ ; z→∞ z→∞ ( )
A = 1 if lim Re(α2 z) = +∞ , A = −1 if lim Re(α21zz) = −∞ ; lim Re( α) = ±∞ }. z→∞ zz→∞ →∞
2.15.3.
+∞ 0 1. ∫ z n [1− erf (az + β)]2 dz = (−1)n ∫ z n [−1− erf (az − β)]2 dz = 0 −∞ n! 2 2 = β − 2 (15) β − (15) β − × [erf ( ) 1] W1 (n, ) W2 (n, ) (− a )n+1 π 2π
(15) 1 (15) × erf ( 2β)−1]W (n, β) − W (n, β) {a > 0}. 3 π 4
+∞ n! (15) 2. zn 1− erf 22( az +β) dz = 1− erf ( β) W( n, β) + ∫ n+1 1 0 (−a)
21(15) (15) > +erf( 2ββ) −+ 1Wn34( , ) Wn( , β) {a 0}. 2ππ
118 +∞ n 2 n+1 3. ∫ z [erf (a1z + β1 ) − erf (a2 z + β2 )] dz = (−1) × 0
0 n! × zn erf ( a z −β) +erf 2 (a z −β) dz = × ∫ 11 2 2 n+1 −∞ (−a1 ) (15) 1!(15) n 2 ×erf( β−11) 1Wn( , β+) Wn( , β 1) +1 − erf ( β2) × 12π n+1 (−a2 )
(15) 21(15) (15) ×Wn1( , β+2) erf( 2 β− 22) 1Wn34( , β+) Wn( , β 2) 2ππ
{a1 > 0, a2 > 0}.
+∞ n! 4. zn erf 22( a z +β) −erf (a z +β) dz = 1− erf 2( β) × ∫ 11 2 2n+1 2 0 (−a2 )
(15) 21(15) (15) × β+ β− β+ β − Wn1( , 2) erf( 2 22) 1Wn34( , ) Wn( , 2) 2ππ
n!2 (15) −1 − erf 2 ( β) Wn( , β+) × n+1 111 (−a ) 2π 1
(15) 1 (15) × β− β+ β erf( 211) 1Wn34( , ) Wn( , 1) {a1 > 0, a2 > 0}. π
+∞ 4 ( ) 5. z n 1− erf 2 (az + β) dz = (−1)n n! W 15 (n, β) {a > 0} . ∫ [ ] n+1 3 −∞ 2πa +∞ n 2 2 n 4 1 (15) 6. ∫ z [erf (a1z + β1 ) − erf (a2 z + β2 )] dz = (−1) n! W (n, β2 ) − 2π n+1 3 −∞ a2
1 (15) −βWn( , ) {a > 0, a > 0}. n+1 3 1 1 2 a1
∞ 22n!2 (15) (15) 7. zn ±1 − erf ( α z +β) dz = erf( β) 1 Wn( , β) Wn( , β) − ∫ n+1 12 0 (−α) π
2 1 − β (15) β − (15) β [erf ( 2 ) 1]W3 (n, ) W4 (n, ) 2π π { lim 2Reα22z + 2 αβ z +( 3 − nz) ln = +∞ ; lim Re(αz) = ±∞ }. z→∞ ( ) z→∞
119 ∞ n! 8. z n 1− erf 2 (αz + β) dz = × ∫ [ ] n+1 0 (− α)
(15) 2 (15) 1 (15) × [1− erf 2 (β)] W (n, β) + [erf ( 2β)1] W (n, β)+ W (n, β) 1 2π 3 π 4
{ lim Reα22z + 2 αβ zn −( − 2) ln z = +∞ for n > 0 ; lim Re(αz) = ±∞ }. z→∞ ( ) z→∞
∞ n! 9. z n erf (α z + β ) erf 2 (α z + β ) dz = × ∫ [ 1 1 2 2 ] n+1 0 (− α1 )
(15) 1!n ×erf( β) 1 Wn( , β) + W(15) (,) n β ± × 11 12 1n+1 π ()−α2
2 (15) 2 ×[1 − erf ( β22 )]Wn1 ( , β ) + [erf ( 2β− )A ] × 2π
(15) 1 (15) ×W34( n,, β+22) Wn( β) π
22 { lim Reα1z + 2 αβ 11 zn −( −2) ln z = z→∞ ( )
22 = lim Reα2z+ 2 αβ 22 zn −( −2) ln z = +∞ for n > 0 ; A = 1 if z→∞ ( )
lim Re(α 2 z ) = +∞ , A = −1 if lim Re(α 2 z ) = −∞ ; lim Re(α1 z) = ±∞ }. z→∞ z→∞ z→∞
∞ n n 2 2 (−1) n! 10. ∫ z [erf (α1z + β1 ) − erf (α 2 z + β 2 )]dz = × α n+1 0 1
(15) 2 (15) × −2 β β+ β β+ 1 erf ( 11) Wn13( , ) erf( 2 1) 1Wn( , 1) 2π
1!(15) n 2 (15) 2 +Wn( , β1) + 1− erf ( β22) Wn( , β+) × ππ41n+1 (−α2 ) 2
(15) 1 (15) × [erf ( 2β 2 )− A]W (n, β 2 ) + W (n, β 2 ) 3 π 4
22 { lim Reα1z + 2 αβ 11 zn −( −2) ln z = z→∞ ( )
22 =lim Re α2z + 2 α 22 β zn −( −2) ln z = +∞ for n > 0 ; A = 1 if z→∞ ( )
lim Re(α 2 z ) = +∞ , A = −1 if lim Re(α2 z) = −∞ ; lim Re(α1 z) = ±∞ }. z→∞ z→∞ z→∞
120 Introduced notations:
n− En( /2) nk+−12 (15) β (15) 1) Wn, β= , W (2n , 0) = 0 , 1 ( ) ∑ k 1 1 k=0 4kn !( +− 12! k)
(15) 1 Wn(21 += 1, 0 ) ; 1 n1+1 4(n1 + 1!)
En( /2) k nkl−+22 (15) k! β 2) Wn( ,β) = exp −β2 ∑∑− 2 ( ) +− kl=00(2k 1!) ( nk 2) != l !
− ( ) n E n / 2 1 k l!βn−2k +2l − , ∑ ∑ k −l k =1 k!(n +1− 2k )! l=1 4 (2l )!
(15) n ! ( ) = 1 15 ( + ) = Wn2 (21 ,0) , W2 2n1 1, 0 0 ; (2n1 + 1!)
En( /2) nk−2 k (15) k!β (2!l) 3) Wn( , β=) ∑∑, 3 +− l 2 kl=00(2k 1!) ( nk 2) != 8!(l )
n (15) n ! 1 (2!k ) (15) Wn(2 ,0) = 1 , W (2n + 1, 0) = 0 ; 3 1 ∑ k 2 3 1 (2n1 + 1!) k=0 8!(k )
n− En( /2) nk+−12 k 21l− (15) β (l −β1!) 4) Wn( , β) = ∑∑erf(β) exp −β2 + 4 +− ( ) kl− kl=11kn!( 12! k) = 4( 2l − 1!)
2 2 − exp(−β 2 ) kl−+11(lk!) β22r− En( /2) ( +β1!) nk2 +∑∑ −× ∑ 2l+ 1! 2 klr−− 2k+− 2! nk 2 ! π lr=01( ) = 2(r − 1!) k=0( ) ( )
2 − kβ2l exp(−β 2 ) kl(2lr) !( −β 1!) 21r ×4erf( β) exp −β2 + , ( )∑ ∑∑2 lr− l=0l! π lr=11 = 8( 2r − 1!) (l!) ( ) 15 ( ) = W4 2n1 , 0 0 ,
n 2 ( ) 1 1 (k!) W 15 (2n +1, 0) = . 4 1 ∑ 2n +1−k (n1 +1)! π k =0 2 1 (2k +1)!
121 n 2.16. Integrals of z exp(βz )[± 1 − erf (αz + β1 )],
n z exp(βz )[erf (α1z + β1 ) erf (α 2 z + β 2 )]
2.16.1.
+∞ 0 1. ∫ exp(βz )[1− erf (az + β1 )] dz = − ∫ exp(− βz )[−1− erf (az − β1 )] dz = 0 −∞
1 1 β2 − 4aββ β − 2aβ = [erf (β ) −1] + exp 1 erf 1 +1 {a > 0}. 1 2 β β 4a 2a
+∞ 1 β2 − 4a ββ 2. exp(βz )[erf (a z + β ) − erf (a z + β )]dz = exp 1 1 × ∫ 1 1 2 2 β 2 0 4a1
2 2a11β −β 1 β −42aa2 ββ 2 2β 2 −β ×erf −−1 exp erf −+1 22aaβ 2 124a2 1 +erf( β) − erf ( β) {a > 0, a > 0}. β 211 2
+∞ 3. ∫ exp(βz )[erf (a1z + β1 ) − erf (a2 z + β2 )] dz = −∞ 2 2 2 β − 4a2ββ 2 β − 4a1ββ1 = exp − exp {a1 > 0, a2 > 0}. β 2 2 4a2 4a1
∞ 11β2 −4 αββ 4. exp(βz) ± 1 − erf ( αz +β) dz = erf( β) 1 − exp1 × ∫ 11 ββ 2 0 4α
2αβ1 − β × erf 1 2α
22 { lim Reαz +αβ−β+ 2 1 zz2ln z =+∞ , lim Re( α=±∞z) }. zz→∞ ( ) →∞
∞ 2 1 β − 4α1ββ1 5. ∫ exp(βz )[erf (α1z + β1 ) erf (α 2 z + β2 )] dz = exp × β α 2 0 4 1
2 2α1β1 − β 1 β − 4α 2ββ 2 2α 2 β2 − β × erf − A1 exp erf − A2 − 2α β α 2 2α 1 4 2 2
1 22 − [erf (β1 ) erf (β2 )] { lim Re(α1z + 2 α 11 β zz −β) +2ln z = β z→∞
122 22 =lim Re( α2z +αβ−β+ 2 22 zz) 2ln z =+∞ ; Ak = 1 if lim Re(α k z ) = +∞ , z→∞ z→∞
Ak = −1 if lim Re(α k z ) = −∞ ; lim Re(α12zz) ⋅ lim Re( α) = ±∞ }. z→∞ zz→∞ →∞ 2.16.2.
+∞ 0 1. ∫ z exp(βz )[1− erf (az + β1 )] dz = ∫ z exp (− βz )[−1− erf (az − β1 )] dz = 0 −∞
β2 − 2a 2 − 2aββ β2 − 4aββ β − 2aβ = 1 exp 1 erf 1 +1 + 2 2 2 2a β 4a 2a
1 1 2 + [1− erf (β1 )]+ exp (− β ) {a > 0}. β2 πaβ 1
+∞ 1 1 2. z exp(βz )[erf (a z + β ) − erf (a z + β )]dz = [erf (β ) − erf (β )]+ × ∫ 1 1 2 2 2 1 2 0 β πβ
β2 − 2 − ββ 1 2 1 2 2a1 2a1 1 × exp(− β )− exp(− β ) + × a 2 a 1 2β2 2 1 2a1
β2 −4a ββ 2 aβ −β β22 −22aa − ββ ×exp1 1 erf 11 −+1 2 2 2× 2 22 42aa122a1 β
2 β − 4a 2 ββ 2 2a 2 β 2 − β × exp 1− erf {a1 > 0, a2 > 0}. 2 2a 4a2 2
+∞ 2 + ββ − β2 2a1 2a1 1 3. ∫ z exp(β z )[erf (a1z + β1 ) − erf (a2 z + β2 )] dz = × 2β2 −∞ a1
2 2 + ββ − β2 2 β − 4a1ββ1 2a2 2a2 2 β − 4a2ββ 2 × exp − exp {a1 > 0, a2 > 0}. 2 2β2 2 4a1 a2 4a2 ∞ 1 β2 − 2α 2 − 2αββ 4. z exp(βz )[±1− erf (αz + β )] dz = exp − β2 + 1 × ∫ 1 ( 1 ) 2 2 0 παβ 2α β
β2 − 4αββ β − 2αβ 1 × exp 1 erf 1 ±1 − [erf (β ) 1] 2 2 1 4α 2α β
22 { lim Reαz +αβ−β+ 2 1 zzln z =+∞ , lim Re( α=±∞z) }. zz→∞ ( ) →∞
∞ β2 − α 2 − α ββ 2 1 2 1 1 5. ∫ z exp(βz )[erf (α1z + β1 ) erf (α 2 z + β2 )] dz = × α 2β2 0 2 1
123 2 22 β−αββ41 1 2 αβ−β11 β−α−αββ222 2 2 ×experf −×A 2 α 1 22 42α12 2 1 αβ
2 β−αββ422 2 αβ−β22 1 ×experf −A + erf( β) erf ( β) − 22α 2 12 4α2 2 2 β
11 1 − exp −β22exp −β { lim Reα22z + 2 α β zz −β +ln z = ( 12) ( ) ( 1 11 ) πβ αα12z→∞
22 =lim Re( α2z +αβ−β+ 2 22 zz) ln z =+∞ ; Ak = 1 if lim Re(αk z) = +∞ , z→∞ z→∞
Ak = −1 if lim Re(αk z) = −∞ ; lim Re(α12zz) ⋅ lim Re( α) = ±∞ }. z→∞ zz→∞ →∞ 2.16.3.
+∞ 0 n − n+1 n −β − − −β = 1 ∫ zexp(β z) 1 − erf ( az +β1 ) dz = ( 1) ∫ z exp( z) 1 erf ( a z) dz 0 −∞
n+1 1 (16) =( −1) n ! 1 − erf ( β) +Wa( , β , β ,1) {a > 0}. n+1 111 β
+∞ n 2. ∫ z exp(βz )[erf (a1z + β1 ) − erf (a2 z + β 2 )]dz = 0
n+1 1 =( −1) n ! erf( β) − erf ( β) − n+1 12 β
(16) (16) > > −Wa11( 11, ββ , ,1) + Wa( 2, ββ , 2 ,1) {a1 0, a2 0}.
+∞ n 3. ∫ z exp(βz )[erf (a1z + β1 ) − erf (a2 z + β2 )]dz = −∞ n+1 (16) (16) =( −1) nWa !( ,,ββ) − Wa( ,, ββ ) {β ≠ 0, a > 0, a > 0}. 2211 2 2 1 2
∞ n 4. ∫ z exp(βz )[± 1 − erf (αz + β1 )]dz = 0
n+1 1 (16) =( −1) nW ![± 1 − erf ( β )] +( αββ , , , ± 1) n+1 111 β
22 { lim Reαz +αβ−β−− 2 1 zz( n2) ln z =+∞ , lim Re( α=±∞z) }. zz→∞ ( ) →∞
124 ∞ n 5. ∫ z exp(βz )[erf (α1z + β1 ) erf (α 2 z + β2 )]dz = 0
n+1 1 =( −1) n ! erf( β) erf ( β) − n+1 12 β
(16 ) (16 ) −W (α1 , β, β1 , A1 ) ± W (α 2 , β, β 2 , A2 ) 1 1
22 { lim Reα1z + 2 α 11 β zz −β −( n −2) ln z = z→∞ ( )
22 =lim Re α2z +αβ−β−− 2 22 zz( n2) ln z =+∞ ; z→∞ ( )
Ak = 1 if lim Re(αk z) = +∞ , Ak = −1 if lim Re(α k z ) = −∞ ; z→∞ z→∞
lim Re(α12zz) ⋅ lim Re( α) = ±∞ }. zz→∞ →∞
Introduced notations:
2 n (16) β −42 αββ αβ − β 1 1) WA(αββ, , ,) = exp11erf −A × 1 1 21∑ kn+− k 42αβ2α k=0
( ) − E k / 2 (2αβ − β)k 2l 1 n 1 × 1 + exp(− β 2 ) × ∑ 2k −2l 1 ∑ k n+1−k l =0 l!(k − 2l )!α π k =12 β
− ( ) + − k −1−2l+2r k E k / 2 l! l 4l 1 r (2αβ − β) × ∑ ∑ 1 − + − 2k −1−2l+2r l=1 (2l )!(k 1 2l )!r=1 (r −1)!α
Ek( /2) k−−12 lr + 2 1 l (r −1!) ( 2 αβ − β) − ∑∑1 , lk!− 2! l 2k−− 12 lr + 2 lr=11( ) = (2r −α 1!)
2 (16) ββ WA(αβ , , , ) = exp − × 1 2 2α 4α
n− En( /2) En( /2) 2!k 1 ×−∑∑A ; n+−22 kk 2 − 1 nk+−12 2k π kk=10(2!k ) βα = k!2βα( )
− 2 n Ek( /2) kl2 (16) β −4 αββ 1 (2αβ − β) 2) W (αββ, ,) = 2exp1 1 , 2 1 2 ∑∑kn+−1 k 22k− l 4αkl=00 2 β= lk!( −α 2! l)
125 2 En( /2) (16) ββ 1 W (αβ , , ) = 2exp − . 2 22∑ nk+−12 k 2α 4α k=0 k!2βα( )
n 2 2 2.17. Integrals of z exp(− α z + βz ) [± 1 − erf (α 1 z + β 1 )],
n 22 zexp(−α zz +β) erf ( α11 z +β) erf ( α 22 z +β )
2.17.1.
+∞ 0 22 22 1. ∫∫exp(−a z) − 1 erf ( az +β) dz =−exp( −a z ) −− 1 erf (az −β) dz = 0 −∞
2 π β = 1 − erf {a > 0}. 4a 2
+∞ 0 22 22 2. ∫∫exp(−a z) − 1 erf ( a11 z) =− dz exp( −a z ) −−1 erf (a z) = dz 0 −∞ 1 a = arctan {a1 > 0}. πa a1
+∞ 0 2 2 2 2 3. ∫ exp(a z ) [1− erf (a1 z )]dz = − ∫ exp(a z ) [−1− erf (a1 z )]dz = 0 −∞
1 a1 + a = ln {a1 > a} . 2 πa a1 − a
+∞ 2 2 4. ∫ exp(− a z ) [erf (az + β) − erf (a1 z )]dz = 0
2 1 a πβ =arctan −− 1 erf {a > 0, a1 > 0} . π a aa1 4 2
+∞ 22 1 aa 5. ∫ exp(−a z) erf( a12 z) − erf ( a z) = dz arctan− arctan 0 πa aa21
{a1 > 0, a2 > 0}. +∞ 1 a − a a − a 6. exp(a 2 z 2 ) [erf (a z ) − erf (a z )]dz = ln 1 − ln 2 ∫ 1 2 a + a a + a 0 2 π a 1 2
{a1 > a, a2 > a}.
126 +∞ 2 2 7. ∫ exp(− a z ) [erf (az + β1 ) − erf (az + β 2 )]dz = 0 2 2 π β 2 β1 = 1 − erf − 1 − erf {a > 0} 4a 2 2
+∞ π β 2 8. exp(− a 2 z 2 + βz ) [erf (a z + β ) − erf (a z + β )]dz = exp × ∫ 1 1 2 2 2 −∞ a 4a
a β + 2a 2β a β + 2a 2β × erf 1 1 − erf 2 2 {a > 0, a > 0}. 1 2 2 + 2 2 + 2 2a a a1 2a a a 2
+∞ π β2 9. exp a22 z+β zerf ( a z +β−) erf ( a z +β=) dz exp − × ∫ ( ) 11 2 2 2 −∞ a 4a
2a 2β − a β 2a 2β − a β × erfi 1 1 − erfi 2 2 {a > a, a > a}. 1 2 2 − 2 2 − 2 2a a1 a 2a a 2 a
∞ 2 π β 10. exp(− α 2 z 2 ) [±1− erf (αz + β)]dz = 1 erf ∫ 4α 0 2
{ lim Re α22z +αβ+ zzln =+∞ , lim Re( α=±∞z) }. zz→∞ ( ) →∞ ∞ 1 α 11. −αα22 ±− = { α22 ≠ −α , ∫ exp( z ) 1 erf ( 1z) dz arctan 1 0 πα α1
22 22 lim Re αzz + α11 +2ln z = +∞ ; ± Re( α z) > 0 if z → ∞}. z→∞ ( )
∞ 2 22 π β 12. exp−αz erf( α+β z ) erf ( α=−z) dz A −erf ± ∫ ( ) 1 α 0 4 2 1 α ± arctan { lim Re α22z + αβ zz +ln = →∞ ( ) πα α1 z
= α 2 2 + α 2 2 + = +∞ 2 2 lim [Re( z 1 z ) 2ln z ] , α ≠ −α ; z→∞ 1 A = 1 if lim Re(α z ) = +∞ , A = −1 if lim Re(αz ) = −∞ ; z→∞ z→∞
±Re( αzz +β) ⋅ Re( α1 ) > 0 if z → ∞}.
∞ 22 1 αα 13. exp−αz erf( α z) erf ( α z) dz = − arctan arctan ∫ ( ) 12 αα 0 πα 12
127 22 22 22 22 { lim Re αzz +α12 +2ln z = lim Re α zz +α +2ln z =+∞, zz→∞ ( ) →∞ ( )
α 2 ≠ −α 2 α 2 ≠ −α 2 ± α⋅ α > → ∞ 1 , 2 ; Re( 12zz) Re( ) 0 if z }.
∞ 22 14. ∫ exp(−αz) erf ( α+β− z 12) erf ( α+βz) dz = 0
β 2 β 2 π 2 π 1 22 = 1 erf − 1 erf { lim Re(αz + αβ1 zz) +ln = 4α 2 4α 2 z→∞ 22 =lim Re αz +αβ+2 zzln =+∞ , lim Re( α=±∞z) }. zz→∞ ( ) →∞
∞(T2 ) 22 15. ∫ exp(−αz +β z) erf ( α11 z +β) erf ( α 2z +β 2) dz = ∞(T1 )
π β2 αβ+αβ2222 αβ+αβ = expAA erf 11 erf 2 2 2 12 α 4α 22 22 22α α +α12α α +α
{ lim Re α+α+αβ−β+22z 22 z2 zz2ln z = ( 1 11 ) zT→∞( 1)
2 2 2 2 = lim [Re(α z + α z + 2α1β1 z − βz )+ 2ln z ]= →∞( ) 1 z T2 = α 2 2 + α 2 2 + α β − β + = lim [Re( z 2 z 2 2 2 z z ) 2ln z ] z→∞(T1 )
= α 2 2 + α 2 2 + α β − β + = +∞ ; lim [Re( z 2 z 2 2 2 z z ) 2ln z ] z→∞(T2 )
= if α 2 + α 2 = − α 2 + α 2 = +∞ , Ak 1 lim Re k z lim Re k z z→∞(T2 ) z→∞(T1 )
= − if α 2 + α 2 = − α 2 + α 2 = −∞ , Ak 1 lim Re k z lim Re k z z→∞(T2 ) z→∞(T1 )
= if α 2 + α 2 ⋅ α 2 + α 2 = +∞ ; Ak 0 lim Re k z lim Re k z z→∞(T1 ) z→∞(T2 )
±Re( α11zz +β) ⋅ Re( α 2 +β 2) >0 if z → ∞(T1 ) and z → ∞(T2 )} .
2.17.2.
+∞ 0 2 2 2 2 1. ∫ z exp(− a z ) [1− erf (a1 z + β)]dz = ∫ z exp(− a z ) [−1− erf (a1 z − β)]dz = 0 −∞
128 22 1 aa11a β β = 1 − erf ( β) + exp− erf −1 2a2 222 aa22+ 22 2aaa++111 aa
{a1 > 0}.
+∞ 0 2 2 2 2 2. ∫ z exp(a z ) [1− erf (a1 z + β)]dz = ∫ z exp (a z ) [−1− erf (a1 z − β)]dz = 0 −∞
2 2 1 a α β a β = [erf (β) −1]+ 1 exp 1− erf 1 2 2 2 2a 2 2 − 2 a − a 2 − 2 2a a1 a 1 a1 a
{a1 > a} .
+∞ 0 22 22 3. ∫∫zexp( a z) − 1 erf ( az += b) dz zexp( a z ) −− 1 erf (a z −= b) dz 0 −∞ 1 1 = [erf (b) −1]+ exp(− b 2 ) {a > 0, b > 0}. 2a 2 2 πa 2 b +∞ 1 4. z exp(− a 2 z 2 ) [erf (a z + β ) − erf (a z + β )]dz = [erf (β ) − erf (β )]+ ∫ 1 1 2 2 2 1 2 0 2a
22 aa β aaβ +1 exp −−1 1 erf 11 −2 × 22 222+aa+ 22++ 222 22aaa11 aa12 aaa
2 2 a β a β × exp − 2 1− erf 2 2 {a > 0, a > 0}. 2 2 1 2 a + a 2 + 2 2 a a 2 +∞ 2 2 a1 5. ∫ z exp(a z ) [erf (a1 z + β1 ) − erf (a 2 z + β 2 )]dz = × 2 2 2 0 − 2a a1 a a22ββ a aa22β ×exp1 erf 11 −−1 2 exp2 × aa22−−22 222 aa22 12aa12−−2 aaa
a22β 1 ×erf −1 + erf( β21) − erf ( β) {a1 > a, a2 > a}. 22 2a2 aa2 −
+∞ 22 6. ∫ zexp( a z) erf( a1 z +β) − erf ( az + b) dz = 0
129 0 22 22 a1 a β = ∫ zexp( a z) erf( a1 z −β) − erf ( a z − b) dz = exp× 22 2 aa22− −∞ 2aa1 − a 1
2 exp −b −β a1β ( ) erf(b) erf ( ) ×erf −+1 + {a>0, a1 >> ab , 0} . 22 22πab22 a aa1 −
+∞ 2 2 7. ∫ z exp(a z ) [erf (az + b1 ) − erf (az + b2 )]dz = 0
0 22 =∫ zexp( a z) erf( a z − b12) − erf ( az − b) = dz −∞
1 11 1 = erf(bb) − erf ( ) + exp−b22 − exp − b 2221 ( 2) ( 1) 22aaπ bb21
{a > 0, b1 > 0, b2 > 0}.
+∞ πβ β 2 8. z exp (− a 2 z 2 + βz )[erf (a z + β ) − erf (a z + β )]dz = exp × ∫ 1 1 2 2 3 2 −∞ 2a 4a
2a 2β + a β 2a 2β + a β a × erf 1 1 − erf 2 2 + 1 × 2 + 2 2 + 2 2 2 + 2 2a a a1 2a a a2 a a a1 2 22 2 22 β−44aa β−11 ββ a β−44aa β−2 ββ 2 ×−exp12 exp 2 22 22 44aa++ 22+ 2 44aa 12 aa a2
{a1 > 0, a2 > 0}.
+∞ π β β 2 9. z exp(a 2 z 2 + βz )[erf (a z + β ) − erf (a z + β )]dz = exp − × ∫ 1 1 2 2 3 2 −∞ 2a 4a
a β − 2a 2β a β − 2a 2β a × erfi 1 1 − erfi 2 2 − 1 × 2 − 2 2 − 2 2 2 − 2 2a a1 a 2a a2 a a a1 a
2 2 2 2 2 2 β + 4a β − 4a1ββ1 a β + 4a β − 4a2ββ 2 × exp 1 + 2 exp 2 2 2 2 2 4a − 4a 2 2 − 2 4a − 4a 1 a a2 a 2
{a1 > a, a2 > a}. ∞ 1 α 10. z exp(− α 2 z 2 )[±1 − erf (α z + β)]dz = [±1 − erf (β)] + 1 × ∫ 1 2 2α α 2 α 2 + α 2 0 2 1
130 α 2β 2 α β × exp − erf 1 − A 2 2 α + α α 2 + α 2 1 1
22 22 { lim Re α+α+αβ+z11 z2 zz ln =+∞; z→∞ ( )
22 22 A = 1 if lim Re α + α1 z = +∞ , A = −1 if lim Re α + α1 z = −∞ ; z→∞ ( ) z→∞ ( )
±Re( α1 z +β) > 0 if z → ∞}.
∞ 11 11. zexpα22 z ± 1 − erf ( αz +β) dz = erf( β) 1 + exp −β2 ∫ ( ) 22 ( ) 0 22α πα β
{ lim Re(αβz) = +∞ , ±Re( αz +β) > 0 if z → ∞}. z→∞
∞ 1 12. zexp−α22 z erf ( α+β z ) erf ( α+β=z) dz erf( β) erf ( β−) ∫ ( ) 11 2 22 1 2 0 2α
α α22 β αβ α 1 1 11 2 − exp− erf −±A1 × ααα222+αα22+ αα22 ++ ααα222 2211 12
αβ22 αβ 2 22 ×−experf −A2 αα22+ αα22+ 2 2
22 22 { lim Re α+α+αβ+z1 z2 11 zz ln = z→∞ ( )
= α 2 2 + α 2 2 + α β + = +∞ lim [Re( z 2 z 2 2 2 z ) ln z ] ; z→∞
22 Ak = 1 if lim Re α + αk z = +∞ , Ak = −1 if z→∞ ( )
22 lim Re α + αk z = −∞ ; ±Re( α11zz +β) ⋅ Re( α 2 +β 2) >0 if z → ∞}. z→∞ ( )
∞ 2 2 α1 13. ∫ z exp(α z )[erf (α1 z + β1 ) erf (αz + β 2 )]dz = × α 2 α 2 − α 2 0 2 1
22 2 α β αβ erf(ββ) erf ( ) exp(−β2 ) ×exp1 erf 11 −−A 12± 2222 2 2 α12 − α α −α 22α πα β 1 22 22 { lim Re α−α+αβ+1z z211 zz ln = lim Re( αβ=+∞2 z) ; zz→∞ ( ) →∞
131 = α 2 − α 2 = +∞ = − α 2 − α 2 = −∞ A 1 if lim Re 1 z , A 1 if lim Re 1 z ; z→∞ z→∞
±Re( αzz +β2) ⋅ Re( α 11 +β) >0 if z → ∞}.
∞ 1 14. zexpα22 z erf( α+β− z ) erf ( α+β=z) dz erf( β−) erf ( β+) ∫ ( ) 1 22 21 0 2α
11 1 + exp−β22 − exp −β { lim [Re(αβzz) = lim Re( αβ) = +∞ }. 2 ( 21) ( ) 12 2 πα ββ21zz→∞ →∞
∞(T ) 2 πβ β2 15. zexp −α22 z +β z erf α z +βerf αz +β dz = exp× ∫ ( ) ( 11) ( 2 2) 24αα32 ∞(T1 )
2α 2β + α β 2α 2β + α β × A erf 1 1 A erf 2 2 + 1 2 α α 2 + α 2 α α 2 + α 2 2 1 2 2
β 2 − α 2β 2 − α ββ 1 A1α1 4 1 4 1 1 + exp 2 2 2 α α 2 + α 2 4α + 4α 1 1
β 2 − α 2β 2 − α ββ A2 α 2 4 2 4 2 2 exp 2 2 α 2 + α 2 4α + 4α 2 2
α 2 2 + α 2 2 + α β − β + = { lim [Re( z 1 z 2 1 1 z z ) ln z ] z→∞(T1 )
= α 2 2 + α 2 2 + α β − β + = lim [Re( z 1 z 2 1 1 z z ) ln z ] z→∞(T2 )
= lim Re α2z +α 22 z +2 α β zz −β +ln z = ( 2 22 ) zT→∞( 1 )
= α 2 2 + α 2 2 + α β − β + = +∞ lim [Re( z 2 z 2 2 2 z z ) ln z ] ; z→∞(T2 )
= if α 2 + α 2 = − α 2 + α 2 = +∞ , Ak 1 lim Re k z lim Re k z z→∞(T2 ) z→∞(T1 )
= − if α 2 + α 2 = − α 2 + α 2 = −∞ , Ak 1 lim Re k z lim Re k z z→∞(T2 ) z→∞(T1 )
= if α 2 + α 2 ⋅ α 2 + α 2 = +∞ ; Ak 0 lim Re k z lim Re k z z→∞(T1 ) z→∞(T2 )
±Re( α11zz +β) ⋅ Re( α 2 +β 2) >0 if z → ∞(T1 ) and z → ∞(T2 )} .
132 2.17.3. +∞ 1. ∫ z 2m exp(− a 2 z 2 ) [1− erf (az + β)]dz = 0
0 (17) = −2m − 22 − − −β = β { > } . ∫ zexp( a z ) 1 erf (az) dz W1 ( a, ,1) a 0 −∞
+∞ 0 2m 2 2 2m 2 2 2. ∫ z exp(− a z ) [1− erf (a1z )]dz = − ∫ z exp(− a z ) [−1− erf (a1z )]dz = 0 −∞
2 (2!mk) 1 a m−1 ( !) = − {a > 0}. ++arctg a1 ∑ 1 mm21 a 22mk− 2 2k1 m! π 4 a 1 k=0 2k++ 1 !( 2 a ) aa ( ) ( 1 )
+∞ 0 2m 2 2 2m 2 2 3. ∫ z exp(a z ) [1− erf (a1z )]dz = − ∫ z exp(a z ) [−1− erf (a1z )]dz = 0 −∞
m 2 (2m )!(−1!) aa+ m−1 (k ) = 1 − {a > a} . +ln a1 ∑ −+1 21m aa− 2mk 22k1 m! π (2a) 1 k=0 2k+− 1! 4 a aa − ( ) ( ) ( 1 )
+∞ 2m 2 2 (2m)! 4. ∫ z exp(− a z ) [erf (az + β) − erf (a1z )]dz = × 0 m! π m−1 2 1 a (k!) (17) × − −β ++arctg a1 ∑ Wa1 ( , ,1) mm21 a 22mk− 22k1 4 a 1 k=0 2k++ 1!2 a aa ( ) ( ) ( 1 )
{a > 0, a1 > 0} .
+∞ (2m)! 5. z 2m exp(− a 2 z 2 ) [erf (a z ) − erf (a z )] dz = × ∫ 1 2 m 2m+1 0 4 m! πa
2 aa(2!mk) m−1 ( !) aa ×−+ 12− arctg arctg ∑ −+ + aa 2mk 22k11 22k 21m! π k=0 2+ 1!4 ++ ( k) ( a) ( aa12) ( aa)
{a1 > 0, a2 > 0}.
+∞ (−1)m (2m)! 6. z 2m exp(a 2 z 2 ) [erf (a z ) − erf (a z )]dz = × ∫ 1 2 2m+1 0 m! π (2a )
a − a a − a (2m)! m−1 (k!)2 × ln 1 − ln 2 + ∑ × a + a a + a m−k 1 2 m! π k =0 (2k +1)!(− 4a 2 )
133 a a × 1 − 2 {a > a, a > a}. k +1 k +1 1 2 2 − 2 2 − 2 (a1 a ) (a2 a ) +∞ 2m 2 2 7. ∫ z exp(− a z ) [erf (az + β1 ) − erf (az + β 2 )]dz = 0
= (17 ) β − (17 ) β W1 (a, 2 ,1) W1 (a, 1 ,1) {a > 0} . +∞ 2m+1 2 2 8. ∫ z exp(− a z ) [1− erf (a1z + β)]dz = 0 0 2m+1 2 2 = ∫ z exp(− a z ) [−1− erf (a1z − β)]dz = −∞
m! (17) = 1 − erf ( β) −W aa2 , ,β ,1 {a > 0}. m+1 2 ( 1 ) 1 2(a2 ) 9.
+∞ 0 2m+1 2 2 2m+ 1 22 ∫ z exp(a z ) [1− erf (a1z + β)]dz = ∫ zexp( a z ) − 1 − erf (az1 −β) dz = 0 −∞ m! ( ) = [1− erf (β)]−W 17 (− a 2 , a , β,1) {a > a} . m+1 2 1 1 2(− a 2 )
+∞ 0 10. ∫ z 2m+1 exp(a 2 z 2 ) [1− erf (az + b)]dz = ∫ z 2m+1 exp(a 2 z 2 ) [−1− erf (az − b)]dz = 0 −∞
m mm!!2 (2!k ) = 1 − erf(bb) + exp( − ) ∑ m+−1 mk 21k+ 2(−−a22) πa k=0 k!2( a) ( ab)
{a > 0, b > 0}.
+∞ 2m+1 2 2 11. ∫ z exp(− a z ) [erf (a1z + β1 ) − erf (a2 z + β 2 )] dz = 0 m! = [erf (β1 ) − erf (β 2 )]+ 2(a 2 ) m+1 ( ) ( ) + 17 2 β − 17 2 β > > W2 (a , a1 , 1 , 1) W2 (a , a2 , 2 , 1) {a1 0, a2 0}. +∞ 2m+1 2 2 12. ∫ z exp(a z ) [erf (a1z + β1 ) − erf (a2 z + β 2 )]dz = 0
134 m! (17) (17) = β−β+erf( ) erf ( ) W − aa22, , β− ,1 W − aa, , β ,1 m+1 1 2 22( 11) ( 2 2) 2(−a2 )
{a1 > a, a2 > a}. +∞ 2m+1 2 2 13. ∫ z exp(a z ) [erf (a1z + β) − erf (az + b)]dz = 0
0 2m+ 1 22 = ∫ zexp( a z) erf( a1 z −β) − erf ( az − b) dz = −∞
m! (17) = erf( β) − erf (b) + W − aa2 , , β ,1 + m+1 2 ( 1 ) 2(−a2 )
− m! m (−1)m k (2k )! + exp(− b 2 ) {a>0, a >> ab , 0} . 2m+2 ∑ 2k+1 1 πa k=0 k!(2b)
+∞ 2m+1 2 2 14. ∫ z exp(a z ) [erf (az + b1 ) − erf (az + b2 )]dz = 0
0 2m+1 2 2 = ∫ z exp(a z ) [erf (az − b1 ) − erf (az − b2 )] dz = −∞ mm!!m (2!k ) = erf(bb) − erf ( ) + × m+−1 12 ∑ mk 2!(−−a22) πa k=0 ka( )
1 1 × exp(− b 2 )− exp(− b 2 ) {a > 0, b > 0, b > 0}. 2k +1 2 2k +1 1 1 2 (2ab2 ) (2ab1 )
+∞ n − 2 2 + β ( + β ) − ( + β ) = (17 ) 2 β β − 15. ∫ z exp ( a z z ) [erf a1 z 1 erf a 2 z 2 ] dz W3 (n, a , a1 , , 1 ) −∞
− (17 ) 2 β β + (17 ) β β − (17 ) β β W3 (n, a , a2 , , 2 ) W4 (n, a, a1 , , 1 ) W4 (n, a, a2 , , 2 )
{a > 0, a1 > 0, a2 > 0}. +∞ n 2 2 + β + β − + β = (17 ) − 2 β β − 16. ∫ z exp(a z z ) [erf (a1z 1 ) erf (a2 z 2 )] dz W3 (n, a , a1 , , 1 ) −∞
− (17 ) − 2 β β + (17 ) β β − (17 ) β β W3 (n, a , a2 , , 2 ) W5 (n, a, a1 , , 1 ) W5 (n, a, a2 , , 2 )
{a > 0, a1 > a, a2 > a} . ∞ 2m − α 2 2 ± − α + β = (17 ) α β ± 17. ∫ z exp( z ) [ 1 erf ( z )]dz W1 ( , , 1) 0
135 { lim Re α22z +αβ− zm( −1) ln z =+∞ , lim Re( α=±∞z) }. zz→∞ ( ) →∞
∞ (2m)! 1 α 18. z 2m exp(− α 2 z 2 ) [±1− erf (α z )] dz = arctg − ∫ 1 m 2m+1 α 0 m! π 4 α 1 m−1 (k!)2 − α 1 ∑ m−k k+1 k=0 + α 2 α 2 + α 2 (2k 1)!(4 ) ( 1 )
α22 + α 22 − − = +∞ > { lim Re zzm1 (2 2) ln z (for m 0 ); z→∞ ( )
± Re(α1z ) > 0 if z → ∞}. ∞ (2m)! 1 α 19. z 2m exp(− α 2 z 2 ) [erf (αz + β) erf (α z )]dz = ± arctg − ∫ 1 m 2m+1 α 0 m! π 4 α 1
− 2 m 1 (k!) ( ) − α −W 17 (α, β, A) 1 ∑ m−k k+1 1 k=0 + α 2 α 2 + α 2 (2k 1)!(4 ) ( 1 )
{ lim Re α22z + αβ zm −( −1) ln z = z→∞ ( )
= α 2 2 + α 2 2 − − = +∞ > lim [Re( z 1 z ) (2m 2)ln z ] (for m 0 ); z→∞ A = 1 if lim Re(αz ) = +∞ , A = −1 if lim Re(αz ) = −∞ ; z→∞ z→∞
±Re( αzz +β) ⋅ Re( α1 ) > 0 if z → ∞}.
∞ (2!m) 20. z2m exp−α 22z erf( α z) erf ( α z) dz = − × ∫ ( ) 12 mm21+ 0 4!m πα
2 αα(2!mk) m−1 ( !) ×−arctg arctg × αα ∑ mk− 12m! π k=0 (2k +α 1!4) ( 2 )
αα × 12 kk++11 α22 +α α 22 +α ( 12) ( )
α 2 2 + α 2 2 − − = { lim [Re( z 1 z ) (2m 2)ln z ] z→∞
= α22 + α 22 − − = +∞ > lim Re zzm2 (2 2) ln z (for m 0 ); z→∞ ( )
± Re(α1z )⋅ Re(α 2 z ) > 0 if z → ∞}.
136 ∞ 2m 2 2 21. ∫ z exp (− α z ) [erf (αz + β1 ) − erf (αz + β 2 )]dz = 0
(17) (17) = αβ ± − αβ ± WW11( ,21 ,1) ( , ,1)
22 { lim Re αz + αβ1 zm −( −1) ln z = z→∞ ( )
22 =lim Re α+αβ−−z2 zm( 1) ln z =+∞ , lim Re( α=±∞z) }. zz→∞ ( ) →∞
∞ 2m+1 2 2 22. ∫ z exp(− α z ) [±1− erf (α1z + β)]dz = 0 m! = ± − β − (17 ) α 2 α β [ 1 erf ( )] W2 ( , 1 , , A) 2(α 2 ) m+1
22 22 { lim Re α+α+αβ−−z11 z2 zm( 2 1) ln z =+∞; z→∞ ( )
22 A = 1 if lim Reα + α1 z = +∞ , z→∞
22 A = −1 if lim Reα + α1 z = −∞ ; ±Re( α1z +β) > 0 if z → ∞}. z→∞
m ∞ (−1!) m 23. z2m+ 1expα 22z ± 1 − erf αz +β dz = erfβ 1 + ∫ ( ) ( ) 22m+ ( ) 0 2α
1 m (2!k ) +exp −β2 ∑ π ( ) k 21k+ k=0 (−β4!) k
{ lim Re( αβzmz) − ln = +∞ , ± Re( αz + β) > 0 if z → ∞}. z→∞
∞ 2m+1 2 2 24. ∫ z exp(− α z ) [erf (α1z + β1 ) erf (α 2 z + β 2 )]dz = 0
m! (17 ) (17 ) = [erf (β ) erf (β )] + W (α 2 , α , β , A ) W (α 2 , α , β , A ) m+1 1 2 2 1 1 1 2 2 2 2 2(α 2 )
22 22 { lim Re α+α+αβ−−z1 z2 11 zm( 2 1) ln z = z→∞ ( )
22 22 =lim Re α+α+αβ−z2 z2 22 zm( 2 − 1) ln z =+∞, As = 1 if z→∞ ( ) 22 22 lim Reα + αs z = +∞ , As = −1 if lim Reα + αs z = −∞ ; z→∞ z→∞
137 ±Re( α11zz +β) ⋅ Re( α 2 +β 2) >0 if z → ∞}. ∞ 2m+ 1 22 25. ∫ z exp(αz) erf ( α11 z +β) erf ( αz +β 2) dz = 0 m! (17) = erf β erf β +WA −α2,,, α β m+1 ( 1) ( 2) 2 ( 11 ) 2(−α2 ) m! m (2!k ) exp −β2 mk+1 ( 2 ) ∑ 21k+ 2 π( −α2 ) k=0 (−β4!) k 2 22 22 { lim Re α−α+αβ−−1z z211 z( 2 m 1) ln z = lim Re( αβ−2zmz) ln =+∞; zz→∞ ( ) →∞ 22 22 A = 1 if lim Reα1 − αz = +∞ , A = −1 if lim Reα1 − αz = −∞ ; z→∞ z→∞ ±Re( αzz +β2) ⋅ Re( α 11 +β) >0 if z → ∞}.
∞ m (−1!) m 26. z2m+ 1expα 22z erf α+β− z erf α+βz dz = erfβ − erf β + ∫ ( ) ( ) ( 11) 22m+ ( ) ( ) 0 2α
1 m (2k )! 1 1 + − β 2 − − β 2 ∑ exp ( 1 ) exp( ) π − k β 2k+1 β 2k+1 k=0( 4) k! 1
{ lim Re(αβzmz) − ln = lim Re( αβ1zmz) − ln = +∞ }. zz→∞ →∞
∞(T2 ) n 22 27. ∫ zexp(−α z +β z) erf ( α11 z +β) erf ( α 2z +β 2) dz = ∞(T1)
= (17 ) α 2 α β β + (17 ) α α β β A1 [W3 (n, , 1 , , 1 ) W4 (n, , 1 , , 1 )]
(17) (17) AW n,,,,α2 α ββ +W( n ,,,, αα ββ ) 234( 22) 22
{ lim Re α+α+αβ−β−−22z 22 z2 zz( n2) ln z = ( 1 11 ) zT→∞( 1 )
= α 2 2 + α 2 2 + α β − β − − = lim [Re( z 1 z 2 1 1z z ) (n 2)ln z ] z→∞(T2 )
= α 2 2 + α 2 2 + α β − β − − = lim [Re( z 2 z 2 2 2 z z ) (n 2)ln z ] z→∞(T1 )
= α 2 2 + α 2 2 + α β − β − − = +∞ lim [Re( z 2 z 2 2 2 z z ) (n 2)ln z ] ; z→∞(T2 )
22 22 As = 1 if lim Reα +αsszz =− lim Re α +α =+∞, zT→∞( 21) zT→∞( )
138 22 22 As = −1 if lim Reα +αsszz =− lim Re α +α =−∞, zT→∞( 21) zT→∞( )
22 22 As = 0 if lim Reα +αsszz ⋅ lim Re α +α =+∞; zT→∞( 12) zT→∞( )
±Re( α11zz +β) ⋅ Re( α 2 +β 2) >0 if z → ∞(T1 ) and z → ∞(T2 )} .
Introduced notations:
2 − ( ) (2m)! π β (2m)! m 1 1) W 17 (α, β, A) = A − erf + k! × 1 m+1 2m+1 2m+1 ∑ 4 m!α 2 m!(2α) k=0
2 1 β2 ββk 2kl+− 12 exp(−β ) ×exp −−A erf + × ∑ l 222 l=0 2lk !2( +− 1 2 l) ! π
k1! lkrlββ2klr−+ 22 !2 llrklr−2 −+ 22 ×−∑ ∑∑ ∑ , lk!2+− 1 2 l ! lr− 2l+ 1!2 k − 2 l ! r ! l=1( ) rl= 102( 2!r) = ( ) ( ) r= 0 m−1 2 (17) (2!mk) π 1 ( !) WA(α=, 0, ) −∑ ; 1 21m+ m + 1π 21mk+− mk!α+ 4 k=0 2( 2 1!)
22 (17) m!α αβ αβ 2) WAα2 ,,, αβ = 11exp−A −erf × 2 ( 1 ) 22 α22 +α α +α α22 +α 2 111
22kl− mk(2!k ) (αβ) m!α × 1 +1 exp −β2 × ∑∑m+−12 k kl− ( ) kl=002 = l 22 2 π k!(α ) 4lk !2( − 2 l) !( α +α1 )
2k−− 12 lr + 2 m(2!kr) kl1 !(αβ) ×−1 ∑m+−12 k ∑∑ klr−+ k=12 lr= 11lk!2( − 2 l) != lr− 22 kr!(α ) 4( 2!) (α +α1 )
kl2k−− 12 lr + 2 (l −1!) (αβ1 ) −∑∑ , (2l− 1!2) ( kl +− 1 2) ! 2klr−+ lr=11= (r −1!) α22 +α ( 1 )
m (17) m!α (2!k ) Wαα2, , 0, AA = 1 ; 2 ( 1 ) ∑ mk+−1 k 2 α22 +α k=0 k 2 2 22 1 4!(k ) (α) ( α +α1 )
139 2 22 (17) n! β−αβ−αββ44 3) Wn,,,,α2 α ββ = exp1 11× 3 ( 11) n 22 2 22 44α+α (2α) α +α1 1
n− En( /2) +− kl+−1 k!βnk12 k 4( 2l − 2!) ×× ∑∑ +− − kl=11(2!kn) ( 12! k) = ( l 1!)
k −2−l+2r 2l−2r l 2 2l+1−2r 2 E (n / 2 ) n−2k (α ) α (2α β1 + α1β) β × ∑ 1 − ∑ × 2l−1−r ( − ) r=1 − − α 2 + α 2 k =1 k! n 2k ! (r 1)!(2l 2r )!( 1 )
−+ −− 2kl2 r 22lr− 2 2l 12 r kl−1 (α) α1(2 α β 11 +αβ) ×−l 1! , ∑∑( ) 21lr−− lr=10 = rl!2( − 1 − 2 r) ! α22 +α ( 1 )
2 (17) αβn! α β Wn(,,,,)α2 α β−11 = exp× 3 1 22n 24ααα2 α 22 +α (2 ) 1
− ( ) + − − k −1+l n E n / 2 k!β n 1 2k k−14 k l (2l )!(α 2 ) × ∑ ∑ , ( ) ( + − ) l k=1 2k ! n 1 2k ! l=0 2 α 2 + α 2 (l!) ( 1 ) 2 2 ( ) (2m)! α β W 17 (2m, α 2 , α , 0, β )= − exp − 1 × 3 1 1 2 2 m α 2 + α 2 α + α 4 m! 1 1
− 2kl−− 12 mk(k −1!) −1 αβ22kl(2 ) × 1 1 , ∑∑mk+−1 21kl−− kl=10α2 = − − α22 +α ( ) lk!2( 1 2 l) !( 1 )
2 2 ( ) m! α β W 17 (2m +1, α 2 , α , 0, β )= exp − 1 × 3 1 1 2 2 α 2 + α 2 α + α 1 1
mk(2!k ) αβ2k+− 12 lkl 2 − 2 × 11 ; ∑∑mk+−1 21k− kl=002 = l 22 k!(α ) 4lk !2( − 2 l) !( α +α1 )
2 2 ( ) n! π β 2α β + α β 4) W 17 (n, α, α , β, β ) = exp erf 1 1 × 4 1 1 n 2 2 α 4α α α 2 + α 2 2 1
E (n / 2 ) β n−2k × , ∑ 2n−2k k =0 k!(n − 2k )!α
140 ( ) (2m)! π αβ W 17 (2m, α, α , 0, β ) = erf 1 , 4 1 1 m 2m+1 4 m!α α 2 + α 2 1
(17 ) + α α β = W4 (2m 1, , 1 , 0, 1 ) 0 ;
2 2 ( ) n! π β 2α β − α β 5) W 17 (n, α, α , β, β ) = exp − erfi 1 1 × 5 1 1 n 2 2 α 4α α α 2 − α 2 2 1 ( ) − − E n / 2 (−1)n k β n 2k × , ∑ 2n−2k k =0 k!(n − 2k )!α
( ) (2m)! π αβ W 17 (2m, α, α , 0, β ) = erfi 1 , 5 1 1 m 2m+1 (− 4) m!α α 2 − α 2 1
(17) Wm5 (2+ 1, αα ,11 , 0, β) = 0 .
n 2 2.18. Integrals of z erf (αz + β 1 )exp[− (αz + β 2 ) ],
2 zzn ±1 − erf( α +β) exp −( α z +β ) , 12
n 2 n 2 z exp(βz )erf (αz + β1 ), z exp(βz ) [1 − erf (αz + β1 )] ,
n 2 2 z exp(βz )[erf (α1 z + β1 ) − erf (α 2 z + β 2 )]
2.18.1. +∞ 2 2 π β1 − β 2 1. ∫ erf (az + β1 )exp[− (az + β 2 ) ] dz = 1+ erf {a > 0}. 4a 2 −β1 a
+∞ 2 2 2 π π β 2. ∫ erf (az + β)exp(− a z )dz = − 1− erf {a > 0}. 0 2a 4a 2 +∞ 2 2 π β 3. ∫ erf (az )exp[− (az + β) ] dz = 1− erf {a > 0}. 0 4a 2 +∞ 2 4. ∫ [1− erf (az + β1 )]exp[− (az + β 2 ) ] dz = −β1 a
141 β1 a 2 =− −−1 erf(az −β) exp −(az −β) dz = ∫ 12 −∞ 2 π π β1 − β 2 = [1+ erf (β1 − β 2 )]− 1+ erf {a > 0}. 2a 4a 2 +∞ 2 5. 1 − erf(az +β) exp −(az +β) dz = ∫ 0
0 2 π 2 =−−−1 erf(az −β) exp −(az −β) dz = −1 erf ( β) {a > 0}. ∫ −∞ 4a
+∞ 0 22 6. −1 erf(az) exp −+β=−−−( az) dz 1 erf(az) exp −−β=( az) dz ∫∫ 0 −∞ 2 π π β = [1− erf (β)]− 1− erf {a > 0}. 2a 4a 2 +∞ 2 π β1 − β 2 7. ∫ erf (az + β1 )exp[− (az + β 2 ) ] dz = erf {a > 0}. −∞ a 2
∞ 2 2 π β12 −β 8. ∫ erf(αz +β12) exp −( αz +β) dz = erf ±1 4α 2 −β1 α
2 2 { lim [Re(α z + αβ1 z + αβ 2 z )+ ln z ]= z→∞
22 =lim Re α+αβ+z 22 zz ln =+∞ , lim Re( α=±∞z) }. zz→∞ ( ) →∞
∞ 2 22 ππβ 9. erf(αz +β) exp −αz dz = − erf 1 ∫ ( ) 24αα 0 2
{ lim [Re(α 2 z 2 )+ ln z ]= lim [Re(α 2 z 2 + αβz )+ ln z ]= +∞, z→∞ z→∞
lim Re(αz ) = ±∞ }. z→∞
∞ 2 π β 10. erf (αz )exp [− (αz + β)2 ] dz = erf 1 ∫ α 0 4 2
{ lim [Re(α 2 z 2 + αβz )+ ln z ]= lim [Re(α 2 z 2 + 2αβz )+ ln z ]= +∞, z→∞ z→∞
lim Re(αz ) = ±∞ }. z→∞
∞ 2 11. ±−1 erf( α+βz ) exp −α+β( z) dz = ∫ 12 −β1 α
142 2 ππβ12 −β = 1 ± erf ( β12 −β) −1 ± erf 24αα2
22 { lim Re αz +αβ+αβ12 z zz +ln =+∞ , lim Re( α=±∞z) }. zz→∞ ( ) →∞
∞ π 12. [±1− erf (αz + β)]exp[− (αz + β)2 ] dz = [erf (β) 1]2 ∫ α 0 4
{ lim Reα22z +αβ+ 2 zz ln =+∞ , lim Re( α=±∞z) }. zz→∞ ( ) →∞
∞ 2 π π β 13. [±1− erf (αz )]exp[− (αz + β)2 ] dz = [1 erf (β)]− 1 erf ∫ α α 0 2 4 2
{ lim Re α22z +αβ+ zzln =+∞ , lim Re( α=±∞z) }. zz→∞ ( ) →∞
−β2 α 2 π 2 β1 − β 2 14. ∫ erf (αz + β1 )exp[− (αz + β 2 ) ] dz = erf . 2α 2 −β1 α
−(β1 +β2 ) (2α) 2 15. ∫ erf (αz + β1 )exp[− (αz + β 2 ) ] dz = −β1 α
π 22β12 −β β 12 −β = erf − erf . 42α 2 2.18.2.
+∞ β a 1. ∫ exp(− bz )erf 2 (az + β)dz = ∫ exp(bz )erf 2 (az − β)dz = −β a −∞
2 1 b 2 + 4abβ 2b = − { > > } exp 1 erf a 0, b 0 . b 4a 2 4a
+∞ 2 2 1 β− 4a ββ1 2. exp(ββz) 1−+= erf ( az1 ) dz exp× ∫ β 4a2 −β1 a
2 β ββ 2 1 1 × 1+ erf − exp − {a > 0}. 4a β a
+∞ 2 2 2 2 1 β 2β 3. exp(βz ) [erf (a1z ) − erf (a2 z )] dz = exp 1+ erf − ∫ β 2 4a 0 4a2 2
143 2 2 1 β 2β − exp 1+ erf {a1 > 0, a2 > 0}. β 2 4a 4a1 1 +∞ 4 β 2 − 4aββ 2β 4. exp(βz ) [1− erf 2 (az + β )] dz = exp 1 erf {a > 0}. ∫ 1 β 2 −∞ 4a 4a
+∞ 4 β 2 − 4a ββ 5. exp(βz ) [erf 2 (a z + β ) − erf 2 (a z + β )] dz = exp 2 2 × ∫ 1 1 2 2 β 2 −∞ 4a2
2 2β 4 β − 4a1ββ1 2β × erf − exp erf {a1 > 0, a2 > 0}. 4a β 2 4a 2 4a1 1
∞ 2 1 β2 −4 αββ 2β β2 α +β =− 1 ± 6. ∫ exp( z) erf ( z1 ) dz exp erf 1 βα4α2 4 −β1 α
2 2 { lim [Re(2α z + 4αβ1z − βz )+ 3ln z ]= +∞, z→∞ lim Re(βαzz) = −∞ , lim Re( ) = ±∞ }. zz→∞ →∞
∞ 2 2 1 β − 4αββ1 7. ∫ exp(βz ) [1− erf (αz + β1 )] dz = exp × β α 2 −β1 α 4
2 β ββ 2 1 1 2 2 × erf ±1 − exp − { lim [Re(α z + 2αβ1z − βz )+ 2ln z ]= 4α β α z→∞
22 =lim Re 2 αz +αβ−β+ 4 1 zz3ln z =+∞ , lim Re(αz ) = ±∞ }. z→∞ ( ) z→∞
2 ∞ β 2 β 2 2 1 2 8. ∫ exp(βz )[erf (α1z ) − erf (α 2 z )] dz = exp erf ±1 − β α 2 4α 0 4 2 2 2 β 2 β 1 2 − exp erf + A β α 2 4α 4 1 1
22 22 { lim Re α12zz −β +2ln z = lim Re α zz −β +2ln z = zz→∞ ( ) →∞ ( )
= α 2 2 − β + = α 2 2 − β + = +∞ lim [Re(2 1 z z ) 3ln z ] lim [Re(2 2 z z ) 3ln z ] ; z→∞ z→∞
A = 1 if lim Re(α1z ) = +∞ , A = −1 if lim Re(α1z ) = −∞ ; z→∞ z→∞
lim Re(α2z) = ±∞ }. z→∞
144 2 (β−2αβ1 ) (2α ) 2 2 1 β − 4αββ1 9. ∫ exp(βz )erf (αz + β1 )dz = exp × β α 2 −β1 α 4
ββ2 2 β ×−exp erf22 2erf . 2 42αα 4 α
2 (β−4αβ1 ) (4α ) 2 2 1 β − 4αββ1 10. ∫ exp(βz )erf (αz + β1 )dz = exp × β α 2 −β1 α 4
β β 2 2 2 × 2erf − erf . 4α 4α
2.18.3.
+∞ β −β 2 2 2 ( 12) 1. ∫ zerf( az +β12) exp −(az +β) dz =exp − × 4a2 2 −β1 a
2 ββ−− πβ ββ ×+ 12− 2 + 12 { > } 1 erf 2 1 erf a 0 . 224a
+∞ 2 2 2 ββ 2. zerf( az) exp −( az +β) dz =exp − 1 − erf − ∫ 2 2 0 4a 2
2 πβ β − 1− erf {a > 0}. 4a 2 2 +∞ 2 3. z1 − erf( az +β) exp −(az +β) dz = ∫ 12 −β1 a
β1 a 2 2 πβ β − β =z −−1 erf( az −β) exp −−β(az) dz =21 + erf 12 − ∫ 122 −∞ 4a 2
2 β −β β − β πβ 2 ( 12) 12 2 −exp − 1+ erf − 1 + erf ( β12 −β) + 222 42aa2
1 2 +exp −( β12 −β ) {a > 0}. 2a2
145 +∞ 0 2 2 4. z−1 erf( az +β) exp −(az +β) dz = z −−1 erf( az −β) exp −(az −β) dz = ∫∫ 0 −∞
1 πβ 2 = erf (− β 2 ) [1− erf (β)]− [1− erf (β)]2 − [1− erf ( 2β)] {a > 0}. 2a 2 4a 2 4a 2
+∞ 0 5. ∫ z[1− erf (az )]exp[− (az + β)2 ] dz = ∫ z[−1− erf (a z )]exp[− (az − β)2 ] dz = 0 −∞ 2 πβ β 2 β 2 β πβ = − − − − − × 1 erf exp 1 erf 4a 2 2 4a 2 2 2 2a 2 1 ×[1− erf (β)]+ exp(− β 2 ) {a > 0}. 2a 2 +∞ 2 πβ β − β 1 6. zerf( az +β) exp −(az +β) dz = 2erf 21+ × ∫ 1222 −∞ aa2 2 2 (β21 −β ) ×−exp {a > 0}. 2
∞ 2 1 β12 −β 7. ∫ zerf(α z +β12) exp −( αz +β) dz = erf ±1 × 4α2 2 −β1 α
2 (β12 −β ) β12 −β ×2 exp − − πβ2 erf ±1 2 2
22 { lim 2Re αz + αβ12 z + αβ zz +ln = z→∞ ( )
22 =lim Re αzz +αβ=+∞ 22 , lim Re( α=±∞ z) }. zz→∞ ( ) →∞
∞ 2 1 β 8. zerf(α z) exp −( α z +β) dz =− erf 1 × ∫ 2 0 4α 2
ββ2 × − + πβ 2 exp erf 1 2 2
{ lim 2Re α22z +αβ+ zzln = lim Re α22 z +αβ=+∞ 2 z , lim Re( α=±∞z) }. z→∞ ( ) zz→∞ ( ) →∞
∞ 2 1 β12 −β 9. ∫ z±−1 erf( α+β z 12) exp −α+β( z) dz = erf ±1 × 4α2 2 −β1 α
146 2 β1 − β 2 (β1 − β 2 ) × πβ 2 erf ±1 − 2 exp− − 2 2
πβ 2 1 2 − [1± erf (β1 − β 2 )]± exp[− (β1 − β 2 ) ] 2α 2 2α 2
22 { lim 2Re αz +αβ+αβ12 z zz +ln =+∞ , lim Re( α=±∞z) }. zz→∞ ( ) →∞
∞ 2 2 10. z±−1 erf( α+β z ) exp −α+β( z) dz = erf 2β 1 − ∫ 2 ( ) 0 4α
πβ 2 1 2 − erf( β) 1 − exp( −β) [ erf ( β ) 1] 42αα22
{ lim 2Reα22z +αβ+ 2 zz ln =+∞ , lim Re( α=±∞z) }. zz→∞ ( ) →∞ ∞ 2 2 1 β β 11. z[±1− erf (αz )]exp[− (αz + β) ] dz = erf 1 2 exp − + ∫ 2 0 4α 2 2
β πβ 1 2 + πβ erf 1 − [1 erf (β)]± exp(− β ) 2 2α 2 2α 2
{ lim 2Re α22z +αβ+ zzln =+∞ , lim Re( α=±∞z) }. zz→∞ ( ) →∞
−β2 α 2 2 β12 −β 12. ∫ zerf(α z +β12) exp −( αz +β) dz = erf × 2α2 2 −β1 α
2 (ββ− ) πβ β− β 1 ×−exp 12 − 2erf 2 12−erf (ββ −) . αα22 12 222 2
−(β1 +β2 ) (2α ) 2 2 β1 − β 2 13. ∫ z erf (αz + β1 )exp[− (αz + β 2 ) ]dz = erf × α 2 2 −β1 α 4
2 (β12 −β ) πβ222 β 12 −β β 12 −β ×−exp + erf −erf − 222 4α 2
1 β − β (β − β )2 − erf 1 2 exp− 1 2 . 2 2α 2 4
147
2.18.4.
+∞ β a 1. ∫ z exp(− bz )erf 2 (az + β)dz = − ∫ z exp(bz )erf 2 (az − β)dz = −β a −∞
2 2a 2 − b 2 − 2abβ b 2 + 4abβ 2b = − + exp 1 erf . 2a 2b 2 4a 2 4a
2 b 2 + 8abβ 2b + − { > > } exp 1 erf a 0, b 0 . 2πab 8a 2 4a
+∞ a + ββ ββ 2 2. β −2 +β = 11− + × ∫ zexp( z) 1 erf ( az1 ) dz exp aβ2 a 2πβa −β1 a 2 2 2 2 β − 8aββ1 2β β − 2a − 2aββ1 β − 4aββ1 × exp erf +1 + exp × 2 2 2 2 8a 4a 2a β 4a
2 β 2 × erf +1 {a > 0}. 4a
+∞ β 2 β 2 2 2 2 3. ∫ z exp(βz )[erf (a1z ) − erf (a2 z )] dz = exp erf +1 + 2πa β 2 4a 0 2 8a2 2
2 β−222a ββ2222 β + 2 exp erf+− 1 exp× 22 2 2 24aa22β 4a2 2πβa1 8a1
2 2 2 β 2a − β β 2 β 2 1 2 × erf +1 + exp erf +1 {a1 > 0, a2 > 0}. 4a 2β 2 2 4a 1 2a1 4a1 1 +∞ 2 β 2 − 8aββ 2 4. z exp(βz ) [1− erf 2 (az + β )] dz = exp 1 + ∫ 1 β 2 −∞ a 8a 2π
β22 −22aa − ββ ββ2 2 +1 exp erf {a > 0}. β 2 aa8a 4
+∞ 2 2 2 2 β − 8a2ββ 2 5. z exp(βz )[erf (a1z + β1 ) − erf (a2 z + β 2 )] dz = exp × ∫ a β 2 −∞ 2 8a2
22 2 2 22β −22aa −22 ββ ββ2 β −8a ββ ×+2 − 11 × exp erf exp 2π αβ 224aaβ 2 88aa2121
148 22 2 2 β −22aa −11 ββ ββ2 ×+1 { > > }. exp erf a1 0, a2 0 2π αβ 2 4a 118a1
∞ 22α2 + αββ − β 22 β −4 αββ 2 11 6. ∫ zexp(β z) erf ( α z +β1 ) dz = exp × 24αβ22 α2 −β1 α
2 2 2β 2 β − 8αββ1 2β × erf ±1 − exp erf ±1 2 4α 2παβ 8α 4α
22 { lim Re 2αz + 4 αβ1 zz − β +2ln z = +∞ , z→∞ ( )
lim Re(βzz) + ln = −∞ , lim Re( αz) = ±∞ }. zz→∞ →∞ ∞ 22β2 −8 αββ β 7. zexp(β z) 1 − erf 2 ( αz +β) dz = exp1 erf± 1 + ∫ 1 2 2παβ 8α 4α −β1 α
2 β 2 − 2α 2 − 2αββ β 2 − 4αββ 2β α + ββ ββ + 1 exp 1 erf ±1 + 1 exp − 1 2 2 2 2 2α β 4α 4α αβ α
2 2 { lim [Re(α z + 2αβ1z − βz )+ ln z ]= z→∞
22 =lim Re( 2 αz +αβ−β+ 4 1 zz) 2ln z =+∞ , lim Re(αz ) = ±∞ }. z→∞ z→∞
∞ 22ββ2 8. zexp(β z) erf22( α z) − erf ( α z) dz = exp erf±+ 1 ∫ 12πα β 2 4α 0 2 2 8α2 2
2 βα22− 2 ββ2222 β + 2 exp erf±− 1 exp× αβ22 α2 α α2 22 44 22 2πα1 β 81
22 2 22β2α −β ββ2 × ++1 + erf AAexp erf α44αα22βα 2 112411
22 22 { lim Re α12zz −β +ln z = lim Re α zz −β +ln z = zz→∞ ( ) →∞ ( )
22 22 =lim Re 2 α−β+12zz2ln z = lim Re 2 α−β+zz2ln z =+∞; zz→∞ ( ) →∞ ( )
A = 1 if lim Re(α1z ) = +∞ , A = −1 if lim Re(α1z ) = −∞ ; z→∞ z→∞
lim Re(α2z) = ±∞ }. z→∞
149 β− αβ α2 ( 221) ( ) 22 2 β−α−αββ221 9. ∫ zexp(β z) erf ( α z +β1) dz = × 2αβ22 −β1 α
β22 −24 αββ ββ2 β − αββ ×+exp11erf 2 exp erf + 22 24αα22ααπ αβ
22α2 + αββ − β 22 β −4 αββ 24β + 11experf 2 −× 22 2 αβ 4α 4α 2π αβ
β 2 − 8αββ 2β × exp 1 erf . 2 8α 4α
2 (β−4αβ1 ) (4α ) 2 2 2 3β − 8α − 8αββ1 10. ∫ z exp(βz )erf (αz + β1 )dz = × α 2β 2 −β1 α 4
β22 −4 αββ ββ2 3β − 16 αββ ×+exp11erf 2 exp erf + 22 4αα44ααπ αβ 16
22α2 + αββ − β 22 β −4 αββ 22β + 11experf 2 −× 22 2 24αβ α 4α 2π αβ
β 2 − 8αββ 2β × exp 1 erf . 2 8α 4α 2.18.5. +∞ 2 (18) 1. znerf( az +β) exp −(az +β) dz = W( n, a , β , β ,1) {a > 0}. ∫ 1 2 1 12 −β1 a
+∞ 2m + β − 2 2 = (18) β − (18) β 2. ∫ z erf (az )exp( a z ) dz W1 (2m, a, , 0,1) W2 (2m, a, , 0) 0 {a > 0}.
+∞ 2 (18) 3. znerf( az) exp −( az +β) dz = W( n, a , 0,β ,1) {a > 0}. ∫ 1 0 +∞ 21n+ 4. zn −1 erf( az +β) exp −+β(az) dz =−( 1) × ∫ 12 −β1 a
β1 a 2 ×zn −−1 erf( az −β) exp −−β(az) dz = ∫ 12 −∞
150 (18) (18) =W31( na, , ββ12 , ,1) − W( na , , ββ12 , ,1) {a > 0}.
+∞ 21n+ 5. zn −1 erf( az +β) exp −+β(az) dz =−( 1) × ∫ 0
0 2 n! (18) ×zn −−1 erf( az −β) exp −−β(az) dz = W( n,β ,1) {a > 0}. ∫ n+1 4 −∞ (−a)
+∞ 21n+ 6. zn 1 − erf( a z) exp −( az +β) dz =( −1) × ∫ 0
0 2 (18) (18) ×zn −1 − erf( a z) exp −( az −β) dz = W( n,,0,,1 aβ) − W( n ,,0,,1 a β ) {a > 0}. ∫ 31 −∞
+∞ 2 (18) 7. znerf( az +β) exp −(az +β) dz = W( n,,, a β β ) {a > 0}. ∫ 1 2 5 12 −∞
∞ n α + β − α + β 2 = (18) α β β ± 8. ∫ z erf ( z 1 )exp[ ( z 2 ) ] dz W1 (n, , 1 , 2 , 1) −β1 α
22 { lim 2Re αz + αβ12 z + αβ zn −( −2) ln z = z→∞ ( )
22 =lim Re α+αβ−−z 2 2 zn( 1) ln z =+∞ , lim Re( α=±∞z) }. zz→∞ ( ) →∞
∞ 2m α + β − α 2 2 = (18) α β ± − (18) α β 9. ∫ z erf ( z )exp( z ) dz W1 (2m, , , 0, 1) W2 (2m, , , 0) 0
{ lim [Re(α 2 z 2 + αβz )− (m −1)ln z ]= z→∞
=lim Re α−−22zmz( 2 1) ln =+∞ , lim Re( α=±∞z) }. zz→∞ ( ) →∞
∞ 2 (18) 10. znerf(α z) exp −( α z +β) dz = W( n, α , 0, β , ± 1) ∫ 1 0
{ lim Reα22z + 2 αβ zn −( − 1) ln z = z→∞ ( )
=lim 2Re α+αβ−−22z zn( 2) ln z =+∞ , lim Re( α=±∞z) }. zz→∞ ( ) →∞
∞ 2 (18) 11. zn ±−1 erf( α+βz ) exp −α+β( z) dz = W( n, αββ±− , , , 1) ∫ 1 2 3 12 −β1/ α
151 − (18) α β β ± W1 (n, , 1 , 2 , 1) 2 2 { lim [2 Re(α z + αβ1z + αβ 2 z )− (n − 2)ln z ]= +∞ , lim Re(αz) = ±∞ }. z→∞ z→∞
∞ 2 n! (18) 12. zn ±−1 erf( α+βz ) exp −α+β( z) dz = W( n,β , ± 1) ∫ n+1 4 0 (−α)
{ lim 2Reα22z +αβ−− 2 zn( 2) ln z =+∞ , lim Re( α=±∞z) }. zz→∞ ( ) →∞
∞ 2 (18) 13. zn ±1 − erf( αz) exp −( α z +β) dz = W( n, α , 0, β , ± 1) − ∫ 3 0
− (18) α β ± W1 (n, , 0, , 1)
{ lim 2Re α22z +αβ−− zn( 2) ln z =+∞ , lim Re( α=±∞z) }. zz→∞ ( ) →∞
−β2 α 14. n α + β − α + β 2 = (18) α β β . ∫ z erf ( z 1 )exp[ ( z 2 ) ] dz W2 (n, , 1 , 2 ) −β1 α
−( β12 +β) (2 α) 2 (18) 15. znerf(α+β z ) exp −α+β( z) dz = W( n,,, αββ ) . ∫ 1 2 6 12 −β1 α 2.18.6. +∞ β a n 1. ∫∫znnexp(− bz) erf 22( az +β=−) dz( 1) z exp( bz) erf ( az−β=) dz −β a −∞
2 n 44a b+β ab n !(18) b = exp W( ma , ,β , +β , 1) 21∑ nm+− 1 π 4!am=0 mb 2a
{a > 0, b > 0}.
+∞ ββ n n! 2. zn exp(β z) 1 − erf 2 ( az +β) dz =exp − 1 × ∫ 1 ∑ nm+−1 a = −β −β1 a m 0 m!( )
m 2 β 4a ββ(18) × −1 − exp W( ma , ,β , β− , 1) {a > 0}. 2 1 11 aaπ 4a 2
+∞ 4 n n! 3. z n exp(βz ) [erf 2 (a z ) − erf 2 (a z )] dz = × ∫ 1 2 ∑ n−m 0 πβ m=0 m!(− β)
2 2 β (18) β β × a2 exp W m, a2 , 0, − ,1 − a1 exp × 2 1 2a 2 4a2 2 4a1
152 (18) β ×− W1 ( ma ,1 , 0, ,1) {a12>>0,а 0} . 2a1
+∞ 4a β 2 − 4aββ n n! 4. z n exp(βz ) [1 − erf 2 (az + β )] dz = exp 1 × ∫ 1 2 ∑ n−m −∞ πβ 4a m=0 m!(− β)
(18) β ×W(,, ma β , β− ) {a > 0}. 5 112a +∞ 4 n n! 5. z n exp(βz ) [erf 2 (a z + β ) − erf 2 (a z + β )] dz = × ∫ 1 1 2 2 ∑ n−m −∞ πβ m=0 m!(− β)
2 β −4a ββ (18) β ×a exp22W(,,,) ma β β− −a × 2 2 5 2222a 1 4a2 2
2 β −4a ββ (18) β ×exp11W(,,,) ma β β− {a > 0, a > 0}. 2 5 111 1 2 4a1 2a1
∞ 4!α β2 −4 αββ n n 6. zn exp(β z) erf 2 ( α z +β) dz = exp1 × ∫ 1 21∑ nm+− π 4α = −β −β1 α m 0 m!( )
(18) β × αβ β − ± 22 Wm1 ( , ,11 , , 1) { lim Re( 2αz +αβ−β− 4 1 zz) ( n −3) ln z =+∞ , 2α z→∞
lim Re(βznz) + ln = −∞ , lim Re( αz) = ±∞ }. zz→∞ →∞
∞ m ββ n n! β 7. znexp(β z) 1 − erf 2 ( αz +β) dz =exp − 11 − − ∫ 1 ∑ nm+−1 αα = −β −β1 α m 0 m!( )
2 4α ββ(18) − expWm( ,αβ , , β − , ± 1) 2 1 11 α π 4α 2
22 { lim Reαz + 2 αβ1 zz − β −( n −2) ln z = z→∞ ( )
22 =lim Re 2 αz +αβ−β−− 4 1 zz( n3) ln z =+∞ , lim Re( α=±∞z) }. zz→∞ ( ) →∞
∞ 4!n n 8. znexp(β z) erf22( α− z) erf ( α z) dz = ∑ × ∫ 12π nm+−1 0 m=0 m!(−β)
22 β(18) ββ × α α − ± −α × 1exp Wm1 (, 12 ,0, ,1) exp αα222α 44121
153 (18) β × α− Wm1 ( ,2 , 0, , A ) 2α2
22 22 { lim Re α−β−−12z zn( 2) ln z = lim Re α−β−−z zn( 2) ln z = zz→∞ ( ) →∞ ( )
= α 2 2 − β − − = α 2 2 − β − − = +∞ lim [Re(2 1 z z ) (n 3)ln z ] lim [Re(2 2 z z ) (n 3)ln z ] ; z→∞ z→∞
A = 1 if lim Re(α2 z) = +∞ ; A = −1 if lim Re(α2z) = −∞ ; z→∞ z→∞
lim Re(α1z) = ±∞ }. z→∞
2 (β−2αβ1 ) (2α ) 2 n 2 β − 4αββ1 9. ∫ z exp(βz )erf (αz + β1 )dz = exp × α 2 −β1 α 4
n m n!4α(18) ββ −2 αβ1 ×∑ Wm(,,αβ , β − ) −× nm+−12 π 2 11 2α m=0 m!(−β) 2α
β 2 β × 2 exp erf . 4α 2 2α
2 (β−4αβ1 ) (4α ) 2 n 2 β − 4αββ1 10. ∫ z exp(βz )erf (αz + β1 )dz = exp × α 2 −β1 α 4
n m n!4α(18) ββ −4 αβ1 2 β ×∑ Wm(,,,)αβ β − − erf . nm+−12π 6 11 24αα m=0 m!(−β) 4α
Introduced notations:
2 E (n / 2 ) n−2k ( ) π n! β − β (− β ) 1) W 18 (n, α, β , β , A) = erf 1 2 + A 2 + 1 1 2 n+1 ∑ k 4α 2 k=0 4 k!(n − 2k )!
n− En( /2) nk+−12 k−1 n!1k!β (18) +2 ∑∑2 Wl(2,ββ , , A) − n+1 +− 7 12 (−α) kl=10(2!kn) ( 12! k) = l !
En( /2) nk−2 k−1 β l! (18) −∑∑2 Wl(2+ 1, ββ , , A) , − kl− 7 12 kl=10kn!( 2! k) = 4( 2l + 1!)
154 E (n / 2) n−2k ( ) πn! (− β) n! W 18 (n, α, β, β, A) = + × 1 n+1 ∑ k n+1 4α k=0 4 k!(n − 2k )! (− α)
− ( ) + − A n E n / 2 k!β n 1 2k k −1 (2l )! × − ∑ ∑ l 2 2 k =1 (2k )!(n +1− 2k )!l=08 (l!)
En( /2) − 2 1 βnk2 k−1 (l!) − ∑∑ , π kn!− 2! k 21kl+− kl=10( ) = 2( 2l + 1!)
2 (18) (2!n1) π β Wn(21 ,αβ , , 0, A) = erf +A + 1 nn11++1 21 4!n1 α 2 n −1 (2!n1) 1 k! (18) +2 ∑ Wk7 (2+β 1, , 0, A) , 21n11+− nk nk1!α+k=0 4( 2 1!)
n (18) n ! 1 1 (18) Wn2+ 1, αβ , , 0, A = 1 WkA2 ,β , 0, ; 1 ( 1 ) 22n + ∑ 7 ( ) α 1 k=0 k!
n− En( /2) nk+−12 (18) n! k!β 2) Wn,,,α β β = erf β −β 2 + 2 ( 12) n+1 ( 2 1) ∑ (−α) k=1 (2!kn) ( +− 12! k)
En( /2) nk−2 nn!!π β −β (−β ) +erf 2 21 2 +×2 n++11∑ kn 2α−2 k=0 4kn !( 2! k) (−α)
En( /2) − βnk2 k−1 l! ×× ∑∑2 − kl− kl=10kn!( 2! k) = 4( 2l + 1!)
n−E (n / 2 ) n+1−2k − ( ) k!β k 11 ( ) ×W 18 (2l +1, β , β ) − ∑ 2 ∑ W 18 (2l, β , β ) , 8 1 2 ( ) ( + − ) 8 1 2 k=1 2k ! n 1 2k ! l=0 l!
(2n )! π β (2n )! (18) α β = 1 2 − 1 × W2 (2n1 , , , 0) erf 2 2n1 +1 2n1 +1 n1 !(2α) 2 n1 !α n −1 1 k! (18) × Wk2+β 1, , 0 , ∑ nk− 8 ( ) k=0 41 ( 2k + 1!)
n (18) n ! erf (β) 1 1 (18) Wn2+ 1, αβ , ,0 =− 1 − Wk2 ,β ,0 ; 28( 1 ) 22n + ∑ ( ) α 1 2!k=0 k
En( /2) nk−2 (18) πn! (−β ) 3) Wn,,,,α β β A = 1 + A erf β −β 2 + 3 ( 12 ) nk+1 ( 1 2) ∑ 2α−k=0 4kn !( 2! k)
155 ( ) n−2k 2l+1 n! E n / 2 β k−1l!(β − β ) + A exp[− (β − β )2 ] ∑ 2 ∑ 1 2 + n+1 1 2 ( − ) k−l (− α) k=1 k! n 2k !l=0 4 (2l +1)!
+ − n−E (n / 2 ) k!β n 1 2k k−1(β − β )2l + ∑ 2 ∑ 1 2 , ( ) ( + − ) k=1 2k ! n 1 2k ! l=0 l!
E (n / 2 ) n−2k ( ) πn! (− β) n! W 18 (n, α, β, β, A) = + A × 3 n+1 ∑ k n+1 2α k=0 4 k!(n − 2k )! (− α)
− ( ) + − n E n / 2 k!β n 1 2k × ∑ , k =1 (2k )!(n +1− 2k )!
π (2n )! (2n )! (18) α β = 1 + β − 1 × W3 (2n1 , , , 0, A) [1 A erf ( )] A 2n1 +1 2n1 +1 n1 !(2α) n1 !α − n1 1 k!β 2k +1` × exp(− β 2 ) , ∑ n −k k =0 4 1 (2k +1)!
2 n 2k ( ) n !exp(− β ) 1 β W 18 (2n +1, α, β, 0, A) = A 1 ; 3 1 2n +2 ∑ 2α 1 k=0 k!
En( /2) k−1 n+−12 kl + 2 (18) 2 1 l! β 4) W( nA,β ,) = exp −β erf ( β) −A ∑∑ − 4 ( ) kn!− 2! k kl− kl=10( ) = 4( 2l + 1!)
− ( ) + − + n E n / 2 k! k−1β n 1 2k 2l − ∑ ∑ − π[erf (β) − A]2 × k=1 (2k )!(n +1− 2k )!l=0 l!
( ) − ( ) − E n / 2 β n 2k exp(− 2β 2 ) E n / 2 β n 2k k−1 (l!)2 × + × ∑ k+1 ∑ k ∑ k=0 4 k!(n − 2k )! 2 π k=1 4 k!(n − 2k )!l=0 (2l +1)!
l2!lr+ ββ2 r n− En( /2) krn+−12 k kl−−11(2!l) ! β2r+ 1 ×−∑ ∑ ∑∑ + r! 2!kn+− 12! k 2 3lr−− 23 r=0 k= 1( ) ( ) lr=10(l!) = 2( 2r + 1!)
n− En( /2) erf( 2β−) A k!βnk+−12 k−1 (2!l) + ∑∑, +− l 2 2 kl=10(2kn) !( 1 2 k )! = 8!(l )
n −1 k 2 (18) π 1 1 2!(k ) W4 (2 nA1 ,0, ) =−+ ∑ , nn11++1 21 + 4nn11 !2 !π k=0 (2k 1!)
156 n (18) n ! 1 (2!k ) W(2 n+=− 1, 0, AA) 1 ; 4 1 ∑ k 2 2 2( 2n1 + 1!) k=0 8!(k )
E (n / 2 ) n−2k ( ) n! π β − β (− β ) n! 2 5) W 18 (n, α, β , β ) = erf 1 2 2 + × 5 1 2 n+1 ∑ k n+1 α 2 k=0 4 k!(n − 2k )! α
2 n+−12 k 2lr− 2 (β −β ) n− En( /2) kl!(−β ) kl−1(2!) (β −β ) ×−exp 12 ∑2 ∑∑12 − 2 (2!kn) ( +− 12! k) lr k=1 lr=004l ! = 2 rl !2( − 2 r) !
En( /2) (−β )nk−2 kl−1 (β − β )2l+− 12 r − ∑2 ∑∑l! 12 , 21kr+ k=12kn !( − 2 k) !lr=00 = 2r !2( l+− 1 2 r) !
(18) (2!nn11) π β (2!2) Wn5 (21 ,αβ , , 0) = erf − × nn1121+21n1 ++ 21 n 1 4!nn11α 2 2 ! α β 2 n1 −1 k β 2k +1−2l × exp − k! , ∑ ∑ l 2 k =0 l=0 2 l!(2k +1− 2l )!
2 n k 22kl− (18) n ! ββ1 (2!k ) Wn(2+ 1, αβ , , 0 ) = 1 exp − ; 5 1 22n+ ∑∑kl 2α 1 2 kl=004k ! = 2 lk !2( − 2 l) !
(18) n! π 22β12 −β β 12 −β 6) Wn6 ( ,,,αβ12 β) = erf −erf × 4αn+1 2 2
( ) n−2k 2 E n / 2 (− β ) n! (β − β ) β − β × 2 + exp− 1 2 erf 1 2 × ∑ k n+1 k =0 4 k!(n − 2k )! α 4 2
E (n / 2) (− β )n−2k k−1l!(β − β )2l+1 × 2 1 2 − ∑ 2k+1 ∑ k=1 2 k!(n − 2k )!l=0 (2l +1)!
n− En( /2) n+−12 k 2l En( /2) nk−2 k!(−β ) k−1 (β − β ) n! (−β ) −∑∑2 12−×2 ∑2 2!k n+− 12! k ln+1 kn!− 2! k kl=10( ) ( ) = 4!l α k= 1( )
k−1 l! (18) × Wl2+ 1, ββ , + ∑ kl− 9 ( 12) l=0 4( 2l + 1!)
n− En( /2) nk+−12 k−1 k!(−β ) 1 (18) + ∑∑2 Wl(2,ββ , ) , (2!kn) ( +− 12! k) l! 9 12 kl=10=
157 (2n )! π β β (18) α β = 1 2 − 2 + W6 (2n1 , , , 0) erf erf n1 +1 2n1 +1 2 4 n1 !α 2
(2n )! β 2 β + 1 exp − erf × 2n1 +1 4 2 n1 !(2α)
nn−−1121k+ 11kk!!β (2!n1) (18) ×∑∑ −2 Wk9 (2+β 1, , 0 ) , + 21n11+− nk kk=00(2k 1!) nk1!α+= 4( 2 1!)
22n k (18) nn!!β ββ1 Wn(2+ 1, αβ , , 0) =− 11exp− erf − × 6 1 22n++∑ kn22 2αα1142k=0 4!k
(18) n1 Wk(2 ,β ,0) × ∑ 9 ; k=0 k!
2 ( ) m ! (β − β ) β − β 7) W 18 (m , β , β , A) = 1 exp− 1 2 erf 1 2 + A × 7 1 1 2 m +1 2 1 2 2 2
− Em( 1/2) mr1 2 (β21 −β ) m1! 2 × ∑ + exp −( β12 −β) × rm1+1 r=0 2!rm( 1 − 2! r) 2 π
( ) − + E m1 / 2 (β − β )m1 2r r−1q!(β − β )2q 1 × ∑ 2 1 ∑ 2 1 + − r−q r=1 r!(m1 2r )! q=0 2 (2q +1)!
m− Em( /2) mr1 +−12 rq− 2q 11r!2(β −β ) r−1 (β −β ) + ∑∑21 21 ; (2!rm) ( +− 12! r) q! rq=101 =
2 ( ) m ! (β − β ) β − β 8) W 18 (m , β , β ) = 1 exp− 2 1 erf 2 1 × 8 1 1 2 m 2 1 2 2 2
− E (m1 / 2) (β − β ) m1 2r × 2 1 − ∑ r r=0 2 r!(m1 − 2r )!
mr−+2 21q Em( 1/2) 1 r−1 2 ββ−−q! ββ m1 ! ( 21) ( 21) −exp −−(ββ21) − ; m1 ∑∑rq 2 π rq=10rm!( 1 −+ 2 r) != 2( 2 q 1!)
m ! ()β −β2 β −β 9) Wm(18) ( ,ββ , ) = 1 exp − 21erf 21× 9 112 m +1 221 2 2
158 − E ( m1 / 2) (β − β ) m1 2r m ! (β − β )2 × ∑ 2 1 + 1 exp− 2 1 × r m1 +1 2 r=0 2 r!(m1 − 2r )! 2 π
+ − m1 −E (m1 / 2) 2 r r!(β − β )m1 1 2r m ! × 2 1 − 1 exp[− (β − β )2 ]× ∑ m +1 2 1 r=1 (2r )!(m1 +1− 2r )! 2 1 π
− + E (m1 / 2) (β − β )m1 2r r−1q!(β − β )2q 1 × ∑ 2 1 ∑ 2 1 + − r−q r=1 r!(m1 2r )! q=0 2 (2q +1)!
m− Em( /2) mr1 +−12 rq− 2q 11r!2(β −β ) r−1 (β −β ) + ∑∑21 21 . (2!rm) ( +− 12! r) q! rq=101 =
±−nnαβ + − αβ +2 2.19. Integrals of ( 1) erf( zz) exp ( ) ,
+ + z 2m 1 [± 1 − erf (αz )]3 , z 2m 1 [± 1 − erf 3 (αz )],
21m+ 3αα 3 2m 2 (α ) − α 2 2 zerf( 12 zz) erf ( ) , z erf z exp( z ),
z 2m [1 − erf 2 (αz )]exp(− α 2 z 2 )
2.19.1.
+∞ π π 1. [1− erf 2 (az + β)]exp [− (a z + β)2 ] dz = [1− erf (β)]− [1− erf 3 (β)] ∫ 2a 6a 0 {a > 0}.
+∞ 2 π 2. ∫ [1− erf 2 (az + β)]exp [− (a z + β)2 ] dz = {a > 0}. −∞ 3a
∞ 2 ππ 3. 1− erf23( α+βz ) exp −α+β( z) dz =[erf( β−) 1] [erf( β) 1] ∫ 62αα 0
{ lim Reα22z +αβ+ 2 zz ln =+∞ , lim Re( α=±∞z) }. zz→∞ ( ) →∞ 2.19.2. +∞ 0 1 1 3 − 3 1. z[1− erf (az )]3 dz = z[−1− erf (az )]3 dz = − {a > 0}. ∫ ∫ 2 π 0 −∞ 2a 2
159 +∞ 0 1 1 3 2. z[1− erf 3 (az )]dz = z[−1− erf 3 (az )]dz = + {a > 0}. ∫ ∫ 2 π 0 −∞ 2a 2 +∞ +∞ 3 3 1 3 3 3. ∫ z [erf (a1z ) − erf (a2 z )] dz = ∫ z [erf (a1z ) − erf (a2 z )]dz = 0 2 −∞ 1 3 1 1 = + − {a1 > 0, a2 > 0} . 4 2π 2 2 a2 a1
∞ 1 1 3 − 3 4. z [±1− erf (αz )]3 dz = ± − ∫ 2 π 0 2α 2
{ lim Reα22zz + ln = +∞ , lim Re( αz) = ±∞ }. zz→∞ ( ) →∞ ∞ 1 1 3 5. z [±1− erf 3 (αz )] dz = ± + ∫ 2 π 0 2α 2
{ lim Reα22zz + ln = +∞ , lim Re( αz) = ±∞ }. zz→∞ ( ) →∞ ∞ 13AA 6. zerf33(α z) erf ( α z) dz =−+12 ∫ 1242π 22 0 αα12
22 22 { lim Reα+12zz ln = lim Re α+ zz ln =+∞; zz→∞ ( ) →∞ ( )
Ak =1 if lim Re(α k z ) = +∞ , Ak = −1 if lim Re(α k z ) = −∞ ; z→∞ z→∞
lim Re(α12zz) ⋅ lim Re( α) = ±∞ }. zz→∞ →∞ 2.19.3. +∞ +∞ 1 11π 1. z2erf 2( az) exp −= a22 z dz z2erf 2( az) exp −= a22 z dz + ∫∫( ) 26( ) 3 0 −∞ 2a 3π {a > 0}.
+∞ +∞ 1 2. z21− erf 2( az) exp −= a22 z dz z21 − erf 2( az) exp −= a22 z dz ∫∫( ) 2 ( ) 0 −∞
π 1 1 = − {a > 0}. 2a3 3 3π
∞ 11π 3. z2erf 2(α z) exp −α22 z dz = ± + ∫ ( ) 3 6 π 0 2α 3
160 { lim Reα22zz − ln = +∞ , lim Re( αz) = ±∞ }. zz→∞ ( ) →∞
∞ 1 π 1 4. z 2 [1− erf 2 (αz )]exp(− α 2 z 2 ) dz = ± − ∫ 3 0 2α 3 3π
{ lim Re α22zz = +∞, lim Re( α) = ±∞ }. zz→∞ ( ) →∞
2.19.4. +∞ +∞ 1 1. z2mmerf 2( az) exp −= a22 z dz z2erf 2( az) exp −= a22 z dz ∫∫( ) 2 ( ) 0 −∞ (2!m) π 1 (19) = + Wm( ) {a > 0}. m! 21mm++ 211 32( a) 3π a
+∞ +∞ 1 2. z2mm1− erf 2(az) exp −= a22 z dz z21 − erf 2(az) exp −= a22 z dz ∫∫( ) 2 ( ) 0 −∞ (2!m) 21π (19) = − Wm( ) {a > 0}. m! 21mm++ 211 32( a) 3π a
+∞ 2m π 3. 1− erf 22m (az +β)exp − (az +β ) dz = {a > 0}. ∫ + −∞ 21ma
∞ (2!m) π 1 (19) 4. z2merf 2(α z) exp −α22 z dz = ± + W( m) ∫ ( ) m! 21mm++ 211 0 32( α) 3πα
{ lim Reα22zmz −( 2 − 1) ln = +∞ , lim Re( αz) = ±∞ }. zz→∞ ( ) →∞
∞ (2!m) 21π (19) 5. z2m 1− erf 2( αz) exp −α22z dz = ± − Wm( ) ∫ ( ) m! 21mm++ 211 0 32( α) 3πα
{ lim Re α22zm −( −1) ln z = +∞ , lim Re( αz) = ±∞ }. zz→∞ ( ) →∞ 2.19.5.
+∞ 0 ( ) 2m+1 − 3 = 2m+1 − − 3 = 19 { > } 1. ∫ z [1 erf (az )] dz ∫ z [ 1 erf (az )] dz W2 (a) a 0 . 0 −∞
161 +∞ 0 2. ∫ z 2m+1 [1 − erf 3 (az )] dz = ∫ z 2m+1 [−1 − erf 3 (az )] dz = 0 −∞
(2m +1)! 1 2 3 (19 ) = + W (m +1) {a > 0}. + + 1 (m +1)! (2a )2m 2 πa 2m 2
+∞ +∞ 1 3. z21mm++erf 3(а z) −= erf 3( а z) dz z21erf 3( а z) −= erf 3( а z) dz ∫∫122 12 0 −∞ (2m +1)! 1 1 2 3 1 1 ( ) = − + − W 19 (m +1) (m +1)! 2m+2 2m+2 π 2m+2 2m+2 1 (2a2 ) (2a1 ) a2 a1
{a1 > 0, a2 > 0} .
+∞ π 4. 1− erfn (az +β) exp − (az +β )2 dz =[ 1 − erf ( β )] − ∫ 2a 0
π + −1 −β erfn 1 ( ) {a > 0}. (2na+ 2)
∞ 2m+1 ± − α 3 = ± (19 ) α 5. ∫ z [ 1 erf ( z )] dz W2 ( ) 0
{ lim 3Reα22zmz −( 2 − 3) ln = +∞ ; lim Re(αz) = ±∞ }. z→∞ ( ) z→∞
∞ (2m + 1!) 1 23 (19) 6. ∫ z21m+ ±−1 erf 3( αz) dz =± + W( m +1) (m +1!) 22mm++ 221 0 (2α) πα
{ lim Reα22zmz −( 2 − 1) ln = +∞ , lim Re( αz) = ±∞ }. zz→∞ ( ) →∞
∞ + (2m +1)! A A 7. z 2m 1 [erf 3 (α z ) erf 3 (α z )] dz = ± 2 1 + ∫ 1 2 (m +1)! 2m+2 2m+2 0 (2α 2 ) (2α1 ) 2 3 A A (19 ) + 2 1 W (m +1) π α 2m+2 α 2m+2 1 2 1
22 22 { lim Reα−−1 zmz( 2 1) ln = lim Re(α2 zm) −( 2 − 1) ln z = +∞ ; z→∞ ( ) z→∞
Ar = 1 if lim Re(α r z ) = +∞ , Ar = −1 if lim Re(α r z ) = −∞ ; z→∞ z→∞
lim Re(α12zz) ⋅ lim Re( α) = ±∞ }. zz→∞ →∞
162 ∞ π 8. (±− 1)nn erf( α+βz ) exp −α+β (z )2 dz =± ( 1)n [ ±− 1 erf ( β− )] ∫ 2α 0 π −( ±− 1)nn++11 erf ( β ) (2n +α 2)
{ lim Reα22z +αβ+ 2 zz ln =+∞ , lim Re( α=±∞z) }. zz→∞ ( ) →∞
Introduced notations:
mk−1 2 (19) (kl!) ( 2!) 1) Wm( )= ; 1 ∑∑2mkll−+ 2 kl=00(2k + 1!) = 2 3!(l )
m 2 k (19 ) (2m +1)! 1 3 (k!) (2l )! 2) W (α)= − 3 − . 2 2m+2 m+1 ∑ 2m−k ∑ l 2 (m +1)!α 4 2π k =0 (2k +1)!2 l=06 (l!)
n 2.20. Integrals of z sin(βz + γ )[± 1 − erf (αz + β1 )] ,
n zzsin(β +γ) erf ( α11 z +β) erf ( α 22 z +β ) ,
n 2 z sin(αz + βz + γ ) [± 1 − erf (α1z + β1 )],
n 2 z sin(αz + βz + γ ) [erf (α1z + β1 ) erf (α 2 z + β 2 )]
2.20.1.
+∞ 0 1. ∫ sin(bz + γ )[1− erf (az + b1 )] dz = ∫ sin(bz − γ )[−1− erf (az − b1 )]dz = 0 −∞ γ 2 + cos 1 b bb1 2ab1 ib = [1− erf (b1 )]− exp − cos − γ 1− Re erf + b b 4a 2 a 2a
bb1 2ab1 + ib + sin − γ Im erf {a > 0}. a 2a
+∞ cos γ 2. sin(bz + γ ) [erf (a z + b ) − erf (a z + b )] dz = [erf (b ) − erf (b )]+ ∫ 1 1 2 2 b 1 2 0
2 1 b bb1 2a11 b+ ib +exp − cos−γ 1 − Re erf + ba2 2 a 4a1 11
163 2 bb1 2a11 b+ ib 1 b +sin −γ Im erf −exp − × a2 ab 2 11 4a2
bb2 22a22 b++ ib bb2 a22 b ib ×cos −γ 1 − Re erf +sin −γ Im erf a2 22 aa 22 a 2
{a1 > 0, a 2 > 0} . +∞ 3. ∫ sin(bz + γ ) [erf (a1 z + b1 ) − erf (a 2 z + b2 )] dz = −∞
2 2 2 b bb b bb = exp − cos 1 − γ − exp − cos 2 − γ {a > 0, a > 0} . 2 2 1 2 b a1 a 2 4a1 4a 2
∞ 2 1 β ββ1 4. sin(βz + γ )[± 1 − erf (αz + β )] dz = exp − exp i γ − × ∫ 1 2β 2 α 0 4α
2αβ1 − iβ ββ1 2αβ1 + iβ × erf 1 + expi − γ erf 1 − 2α α 2α cos γ − [erf (β ) 1] β 1
22 { lim Reαz +αβ− 21 z Im( β+ zz) 2ln =+∞ , lim Re( α=±∞z) }. zz→∞ ( ) →∞ ∞ cos γ 5. sin(βz + γ ) [erf (α z + β ) erf (α z + β )] dz = [erf (β ) erf (β )]+ ∫ 1 1 2 2 β 1 2 0
2 1 β + 4iα1ββ1 2α1β1 − iβ + exp iγ − A − erf 2 1 2β α 2α1 4 1
2 β + 4iα 2ββ 2 2α 2β 2 − iβ exp iγ − A − erf + 2 2 α 2α 2 4 2
2 β − 4iα1ββ1 2α1β1 + iβ + exp − iγ − A1 − erf α 2 2α 4 1 1
2 β − 4iα 2ββ 2 2α 2β 2 + iβ exp − iγ − A − erf 2 2 α 2α 2 4 2
α22 + αβ − β + = { lim Re1 z 211 z Im( zz) 2ln z→∞ ( )
164 = α+αβ−22 β+ =+∞ lim Re2 z 222 z Im( zz) 2ln ; z→∞ ( )
Ak = 1 if lim Re(α k z ) = +∞ , Ak = −1 if lim Re(α k z ) = −∞ ; z→∞ z→∞
lim Re(α12zz) ⋅ lim Re( α) = ±∞ }. zz→∞ →∞ 2.20.2.
+∞ 0 22 22 1. ∫∫sin (a z +γ) 1 − erf (a11 z) dz =− sin ( a z +γ) −1 − erf (a z) dz = 0 −∞
1 21a++( ia) 21a++( ia) = (sinγ− cos γ) Re ln 11+(sin γ+ cos γ) Imln 22πa21a11−+( ia) 21a−+( ia)
{a1 > 0}.
+∞ 22 1 21a1 ++( ia) 2. ∫ sin (a z+γ) erf( a12 z) − erf ( a z) dz = (cosγ− sin γ) Re ln − 0 22πa 2a1 −+( 1 ia)
21a++( ia) 21a++( ia) 21 a++( ia) −Re ln 2+(sin γ+ cos γ) Im ln 21−Im ln 21a2−+( ia) 21a21−+( ia) 21 a−+( ia)
{a1 > 0, a2 > 0} . 3. +∞ 2π b 2 sin(a 2 z 2 + bz + γ ) [erf (a z + b ) − erf (a z + b )] dz = cos − γ − ∫ 1 1 2 2 2 −∞ 2a 4a
22 b2 (21ab11−+ ab) ( i) ( 2ab2− ab 2) ( 1 + i) −sin −γ Re erf −Re erf + 2 4a 88a a22++ ia a a22 ia 12
2 2π bb22 (21ab22−+ ab)( i) + sin−γ +cos −γ Imerf − 22 2a 44aa 8a a22+ ia 2
2 (21a b11−+ ab) ( i) − { > > } Imerf a1 0, a2 0 . 8a a22+ ia 1
∞ 2 1+ i 21α−1 ( +i) α 4. sin αz + γ ±1 − erf ( α1z) dz = exp(iγ) ln + ∫ ( ) π α α+ + α 0 42 21 ( 1i)
1− i 21α−( −i) α + exp(−γi ) ln 1 42π α 2α+1 ( 1 −i) α
165 22 2 { lim Reα1 z − Im α zz + 2ln = +∞ , z→∞ ( ) ( )
α 2 ≠ α α 2 ≠ − α ± α> → ∞ 1 i , 1 i ; Re( 1z) 0 if z }. ∞ 2 1+ i 2α1 + (1+ i ) α 5. ∫ sin(αz + γ ) [erf (α1 z ) erf (α 2 z )] dz = exp(iγ ) ln 0 4 2π α 2α1 − (1+ i ) α
2α 2 + (1+ i ) α 1− i ln + exp(− iγ )× 2α 2 − (1+ i ) α 4 2π α
2α + (1− i ) α 2α + (1− i ) α × ln 1 ln 2 2α1 − (1− i ) α 2α 2 − (1− i ) α
α22 −α 2 + = { lim Re1 z Im zz 2ln z→∞ ( ) ( )
22 2 =lim Re α2z − Im α zz + 2ln = +∞ , z→∞ ( ) ( )
α 2 ≠ α α 2 ≠ α α 2 ≠ − α α 2 ≠ − α ± α ⋅ α > → ∞ 1 i , 2 i , 1 i , 2 i ; Re( 1 z ) Re( 2 z ) 0 if z }.
∞(T2 ) 2 21π−( i) 6. ∫ sin (α+β+γz z ) erf ( α+β11z ) erf ( α+β= 2z 2) dz × 4 α ∞(T1)
β 2 (2αβ − α β)(1− i ) (2αβ − α β)(1− i ) × expi γ − A erf 1 1 A erf 2 2 + 1 2 4α α 2 − α α α 2 − α α 8 1 i 8 2 i
21π+( i) β2 + exp i−γ × α 4 α 4
(21αβ − α β)( + ii) ( 2αβ − α β)( 1 + ) × 11 2 2 BB12erf erf α22 +α α α +α α 8812ii
{ α22 + α β − α2 +β + = lim Re( 1z 2 11 z) Im( zz) 2ln z zT→∞( 1)
= α22 + α β − α2 +β + = lim Re( 2z 2 22 z) Im( zz) 2ln z zT→∞( 1)
= α22 + α β − α2 +β + = lim Re( 1z 2 11 z) Im( zz) 2ln z zT→∞( 2 )
= α22 + α β − α2 +β + lim Re( 2z 2 22 z) Im( zz) 2ln z= +∞ ; zT→∞( 2 )
166 22 Ak = 1 if lim Reαkk −αiz =− lim Re α −αiz =+∞, zT→∞( 21) zT→∞( )
22 Ak = −1 if lim Reαkk −αiz =− lim Re α −αiz =−∞, zT→∞( 21) zT→∞( )
22 Ak = 0 if lim Reαkk −αiz ⋅ lim Re α −αiz =+∞, zT→∞( 12) zT→∞( )
22 Bk = 1 if lim Reαkk +αiz =− lim Re α +αiz =+∞, zT→∞( 21) zT→∞( )
= − if α 2 + α = − α 2 + α = −∞ , Bk 1 lim Re k i z lim Re k i z z→∞(T2 ) z→∞(T1 )
22 Bk = 0 if lim Reαkk +αiz ⋅ lim Re α +αiz =+∞; zT→∞( 12) zT→∞( )
±Re( α11zz +β) ⋅ Re( α 2 +β 2) >0 if z → ∞(T1 ) and z → ∞(T2 )} .
ν 2 7. ∫ sin(αz ) [erf (α1z ) erf (α 2 z )] dz = 0 . −ν
2.20.3.
+∞ 0 2m 2m 1. ∫ z sin(bz + γ )[1− erf (az + b1 )]dz = ∫ z sin(bz − γ )[−1− erf (az − b1 )]dz = 0 −∞ (2m)! = − m γ − − γ ⋅ (20 ) + ( 1) cos [1 erf (b1 )] (2m)! [sin ReW1 (2m, a, b, b1 , 1) b 2m+1 ( ) + γ ⋅ 20 > cos ImW1 (2m, a, b, b1 , 1)] {a 0}.
+∞ 0 2m+1 2m+1 2. ∫ z sin(bz + γ )[1− erf (az + b1 )]dz = − ∫ z sin(bz − γ )[−1− erf (az − b1 )] dz = 0 −∞ (2m +1)! = − m γ − + + γ ⋅ (20 ) + + ( 1) sin [erf (b1 ) 1] (2m 1)![sin ReW1 (2m 1, a, b, b1 , 1) b 2m+2
( ) + γ ⋅ 20 ( + ) > cos ImW1 2m 1, a, b, b1 , 1 ] {a 0}. +∞ 2m 3. ∫ z sin(bz + γ ) [erf (a1 z + b1 ) − erf (a 2 z + b2 )] dz = 0
167 m (2m)! = (−1) cos γ [erf (b1 ) − erf (b2 )]+ b 2m+1
+ γ (20 ) − (20 ) + (2m)!{ sin [ReW1 (2m, a1 , b, b1 , 1) ReW1 (2m, a 2 , b, b2 , 1)]
(20) (20) +γcosImW( 2,,,,1 ma bb) − Im W( 2, ma ,, bb ,1) {a > 0, a > 0} . 1111 2 2 } 1 2
+∞ 2m+1 4. ∫ z sin(bz + γ )[erf (a1 z + b1 ) − erf (a 2 z + b2 ) ]dz = 0
m+1 (2m + 1!) =( −1) sinγ erf(b12) − erf( bm) +( 2 + 1) ! × b22m+
× γ (20 ) ( + ) − (20 ) ( + ) + {sin [ReW1 2m 1, a 2 , b, b2 , 1 ReW1 2m 1, a1 , b, b1 , 1 ]
+ γ (20 ) + − (20 ) + cos [ImW1 (2m 1, a 2 , b, b2 , 1) ImW1 (2m 1, a1 , b, b1 , 1)]}
{a1 > 0, a2 > 0} .
+∞ n 5. n +γ +− + =−× ∫ zsin( bz) erf( a11 z b) erf ( a 2 z b 2) dz( 1) n ! −∞
× γ ⋅ (20 ) ( ) + γ ⋅ (20 ) ( ) − [sin ReW2 a1 , b, b1 cos ImW2 a1 , b, b1 ( ) ( ) − γ ⋅ 20 − γ ⋅ 20 > > sin ReW2 (a2 , b, b2 ) cos ImW2 (a2 , b, b2 ) ] {a1 0, a2 0} . ∞ n exp(iγ ) 1 6. z n sin(βz + γ ) [±1− erf (αz + β )]dz = (−1) n! [erf (β ) 1]− ∫ 1 2i n+1 1 0 (iβ) (20) n+1 exp(−γi ) −Wn( , αββ ,, , ± 1) +( − 1) n ! × 1 1 2i
n+1 i (20) × erf( β) 1 −Wn( , α , −β , β , ± 1) β 111
22 { lim Reαz +αβ− 21 z Im( β−− zn) ( 2) ln z =+∞ , lim Re(αz) = ±∞ }. z→∞ ( ) z→∞
∞ + exp(iγ ) 7. z n sin(βz + γ ) [erf (α z + β ) erf (α z + β )] dz = (−1)n 1 n! × ∫ 1 1 2 2 2i 0
1 × [erf (β ) erf (β )]− + 1 2 (iβ)n 1
168 (20) (20) −±Wn( ,α ,, ββ , AWn) ( ,α ,, ββ , A) + 1 1 11 1 2 2 2 n+1 n exp(−γi ) i +( −1) n ! erf( β12) erf ( β) − 2i β
(20) (20) −W11( n,,,, α1 −β β 11 AW) ±( n ,,,, α2 −β β 2 A 2)
22 { lim Reα1z + 2 αβ 11 z − Im( β zn) −( − 2) ln z = z→∞ ( )
22 =lim Re α+αβ−β−−2z 2 22 z Im( zn) ( 2) ln z =+∞ ; z→∞ ( )
Ak =1 if lim Re(α k z ) = +∞ , Ak = −1 if lim Re(αk z) = −∞ ; z→∞ z→∞
lim Re(α12zz) ⋅ lim Re( α) = ±∞ }. zz→∞ →∞
2.20.4.
+∞ 0 2mm 22 2 22 1. ∫∫zsin ( a z +γ) 1 − erf (a11 z) dz =− zsin ( a z +γ) −1 − erf (a z) dz = 0 −∞ (2m)! ( ) ( ) = γ ⋅ 20 2 − γ ⋅ 20 2 >> [sin ReW3 (a , a1 ) cos ImW3 (a , a1 )] {aa0,1 0}. m! π +∞ 2m 2 2 2. ∫ z sin(a z + γ ) [erf (a1 z ) − erf (a 2 z )] dz = 0
(2!m) (20) (20) =γ−+22 sin ReW33( aa ,21) Re W( aa , ) m! π {
(20) (20) +γcos ImW aa22 ,− Im W aa , {a > 0, a > 0, a > 0}. 33( 12) ( ) } 1 2
+∞ 2m+1 2 2 3. ∫ z sin (a z + γ ) [1− erf (a1 z + b)]dz = 0 0 2m+ 1 22 = ∫ zsin ( a z + γ) −1 − erf (a1 z − b) dz = −∞ mm!! = − γ⋅mm − γ⋅ + × + 1 erf(bi) cos Re( ) sin Im( i) 2a22m 2
169 (20) (20) ×cos γ⋅ ImW a22 , ab , ,1− sin γ⋅ Re W a , ab , ,1 {a > 0}. 44( 11) ( ) 1
+∞ 2m+1 2 2 4. ∫ z sin (a z + γ ) [erf (a1 z + b1 ) − erf (a 2 z + b2 )]dz = 0 m! = − γ⋅mm − γ⋅ + + erf(bb12) erf( ) cos Re( i) sin Im( i) 2a22m
m! (20) 22(20) +γsinReW aab ,,,1Re− W aab ,,,1 + 2 44( 11) ( 2 2)
m! (20) 22(20) +γcosImW aab ,,,1Im− W aab ,,,1 {a1 > 0, a2 > 0} . 2 44( 2 2) ( 11)
+∞ n 2 2 5. ∫ z sin(a z + bz + γ ) [erf (a1 z + b1 ) − erf (a 2 z + b2 )] dz = −∞
= γ (20 ) 2 − (20 ) 2 + sin [ReW5 (n, a , a1 , b, b1 ) ReW5 (n, a , a 2 , b, b2 )]
+ γ (20 ) 2 − (20 ) 2 cos [ImW5 (n, a , a2 , b, b2 ) ImW5 (n, a , a1 , b, b1 )]
{a > 0, a1 > 0, a2 > 0}. ∞ 2m 2 (2m)! 6. ∫ z sin(αz + γ ) [±1− erf (α1 z )]dz = × 0 m! π
exp(iγ ) (20 ) exp(− iγ ) (20 ) × W (− α, α ) − W (α, α ) 2i 3 1 2i 3 1
22 2 2 2 { lim Reα1 z − Im α zmz −( 2 − 2) ln = +∞ , α≠0 , α1 +i α≠0 , α1 −i α≠0; z→∞ ( ) ( )
±Re( α>1z) 0 if z → ∞}.
∞ 2m 2 (2m)! 7. ∫ z sin(αz + γ )[erf (α1 z ) erf (α 2 z )] dz = × 0 m! π
exp(− iγ ) (20 ) (20 ) × [W (α, α ) W (α, α )]− 2i 3 1 3 2
exp(iγ) (20) − −α α(20) −α α WW33( ,,12) ( ) 2i
22 2 { lim Reα1 z −α−− Im zmz( 2 2) ln = z→∞ ( ) ( )
170 22 2 =lim Re α2z − Im α zmz −( 2 − 2) ln = +∞ , z→∞ ( ) ( )
α≠ 2 2 α2 + α≠ α2 − α≠ 0 , α1 +i α≠0 , α1 −i α≠0 , 2 i 0 , 2 i 0 ;
± Re(α1 z )⋅ Re(α 2 z ) > 0 if z → ∞}.
∞ m! ( ) 8. z 2m+1 sin(αz 2 + γ ) [±1− erf (α z + β)] dz = exp(− iγ ) [W 20 (α, α , β, A ) + ∫ 1 4i 4 1 1 0
erf (β) 1 m! ( ) erf (β) 1 + − exp(iγ)W 20 (− α, α , β, A ) + + 4 1 2 + (iα)m 1 4i (− iα)m 1
22 2 { lim Reα11z +αβ− 2 z Im α z −( 2 mz − 1) ln =+∞ ; z→∞ ( ) ( )
2 2 A1 = 1 if lim Reα1 +iz α = +∞ , A1 = −1 if lim Reα1 +iz α = −∞ , z→∞ z→∞
2 2 A2 = 1 if lim Reα1 −iz α = +∞ , A2 = −1 if lim Reα1 −iz α = −∞ ; z→∞ z→∞
±Re( α1z +β) > 0 if z → ∞}.
∞ m! exp(− Aiγ ) 9. z 2m+1 sin(Aiα 2 z 2 + γ ) [±1− erf (αz + β)]dz = A ⋅ × ∫ m+1 4i 0 (− α 2 )
(20) exp( Aiγ) erf( β) 1 (20) ×erf β 1 +W β − Am! +W −α i 2 , α , β , ± 1 ( ) 64( ) + ( ) 4i α22m
22 { lim 2Re αz + αβ z −(2 m − 1) ln z = lim Re( αβzmz) − ln = +∞ ; zz→∞ ( ) →∞ A = 1 or A = −1; ±Re( αz +β) > 0 if z → ∞}.
∞ 21m+ 2 m! 10. zsinα+γ z erf ( α+β11z ) erf ( α+β= 2z 2) dz erf( β1) erf ( β× 2) ∫ ( ) 4i 0
exp(iγ ) exp(− iγ ) m! × − + exp(iγ )× + + { (− iα)m 1 (iα)m 1 4i
× (20 ) − α α β (20 ) − α α β − [W4 ( , 1 , 1 , A1 ) W4 ( , 2 , 2 , A2 )]
171 − (− γ ) (20 ) (α α β ) (20 ) (α α β ) exp i [W4 , 1 , 1 , B1 W4 , 2 , 2 , B2 ] }
22 2 { lim Reα1z + 2 αβ 11 z − Im α z −( 2 mz − 1) ln = z→∞ ( ) ( )
22 2 =lim Re α+αβ−α−2z 2 22 z Im z( 2 mz − 1) ln =+∞ ; z→∞ ( ) ( )
2 2 Ak = 1 if lim Reαk −iz α = +∞ , Ak = −1 if lim Reαk −iz α = −∞ , z→∞ z→∞
2 2 Bk = 1 if lim Reαk +iz α = +∞ , Bk = −1 if lim Reαk +iz α = −∞ ; z→∞ z→∞
±Re( α11zz +β) ⋅ Re( α 2 +β 2) >0 if z → ∞}.
∞ 2m+1 2 2 11. ∫ z sin(Aiα z + γ ) [erf (αz + β) erf (α1 z + β1 )]dz = 0
m! βerf( ) erf ( β) (20) (20) =Aexp Aiγ 1 +Wi −α22,,, α β BWi−α,,, α β B + ( )22m+ 44( ) ( 111) 4i α exp(− Aiγ ) + (− )m (β) (β ) + (20 ) (β) ± 1 [erf erf 1 W6 ] α 2m+2
(20) ± − γ α2 αβ exp( Ai) W4 ( i,,,111 A )
{ lim 2Re α22z + αβ zm −( 2 − 1) ln z = z→∞ ( )
22 22 =lim Re α1z + 2 αβ 11 z − Re α zmz −( 2 − 1) ln = z→∞ ( ) ( )
=lim Re( αβzmz) − ln = +∞ ; z→∞
22 A = 1 or A = −1; A1 = 1 if lim Reα1 − αz = +∞ , z→∞
22 A1 = −1 if lim Reα1 − αz = −∞ , B = 1 if lim Re(αz ) = +∞ , z→∞ z→∞
22 B = −1 if lim Re(αz) = −∞ , B1 = 1 if lim Reα1 + αz = +∞ , z→∞ z→∞
22 B1 = −1 if lim Reα1 + αz = −∞ ; ±Re( αzz +β) Re( α11 +β) >0 if z → ∞}. z→∞
172 ∞ 2m+ 1 22 12. ∫ zsin ( Aiα z +γ) erf( α+β−z ) erf ( α+βz1) dz = 0
Am! m = expAiγ erf β− erf β+− 1 exp −γ×Ai + { ( )( ) ( 1) ( ) ( ) 4iα22m ( ) ( ) Am! × [erf (β) − erf (β ) +W 20 (β) −W 20 (β )] }+ exp( Aiγ )× 1 6 6 1 4i
× (20 ) − α 2 α β ± − (20 ) − α 2 α β ± [W4 ( i , , , 1) W4 ( i , , 1 , 1)]
{ lim 2Re α22z + αβ zm −(2 − 1) ln z = z→∞ ( )
2 2 = lim [2 Re(α z + αβ1 z )− (2m −1)ln z ]= lim [Re(αβz ) − mln z ] = z→∞ z→∞
=lim Re( αβ1zmz) − ln = +∞ ; A = 1 or A = −1; z→∞ lim Re(αz) = ±∞ }. z→∞
∞(T2 ) n 2 exp(iγ) 13. zsin α+β+γ z z erf ( α+β11z ) erf ( α+β= 2z 2) dz × ∫ ( ) 2i ∞(T1)
(20 ) (20 ) exp(− iγ ) × [A W (n, − α, α , − β, β ) A W (n, − α, α , − β, β )]− × 1 5 1 1 2 5 2 2 2i
× (20 ) α α β β (20 ) α α β β [B1W5 (n, , 1 , , 1 ) B2W5 (n, , 2 , , 2 )]
α22 +αβ− α+β−−2 = { lim Re( 1z 2 11 z) Im( zzn) ( 2) ln z zT→∞( 1)
= α22 +αβ− α+β−−2 = lim Re( 1z 2 11 z) Im( zzn) ( 2) ln z zT→∞( 2 )
= α22 +αβ− α+β−−2 = lim Re( 2z 2 22 z) Im( zzn) ( 2) ln z zT→∞( 1)
= α22 +αβ − α+β−−2 =+∞ lim Re( 2z 2 22 z) Im( zzn) ( 2) ln z , α ≠ 0 ; zT→∞( 2 )
22 Ak = 1 if lim Reαkk −αiz =− lim Re α −αiz =+∞, zT→∞( 21) zT→∞( )
22 Ak = −1 if lim Reαkk −αiz =− lim Re α −αiz =−∞, zT→∞( 21) zT→∞( )
173 22 Ak = 0 if lim Reαkk −αiz ⋅ lim Re α −αiz =+∞, zT→∞( 12) zT→∞( )
22 Bk = 1 if lim Reαkk +αiz =− lim Re α +αiz =+∞, zT→∞( 21) zT→∞( )
22 Bk = −1 if lim Reαkk +αiz =− lim Re α +αiz =−∞, zT→∞( 21) zT→∞( )
22 Bk = 0 if lim Reαkk +αiz ⋅ lim Re α +αiz =+∞; zT→∞( 12) zT→∞( )
±Re( α11zz +β) ⋅ Re( α 2 +β 2) >0 if z → ∞(T1 ) and z → ∞(T2 )} .
ν 2m 2 14. ∫ z sin (αz ) [erf (α1 z ) erf (α 2 z )]dz = 0 . −ν
Introduced notations:
2 ( ) β + 4iαββ 2αβ − iβ 1) W 20 (n, α, β, β , A) = exp − 1 erf 1 − A × 1 1 2 4α 2α
k −2l n 1 E (k / 2) (2αβ − iβ) 1 n 1 × 1 + exp(− β 2 ) × ∑ k n+1−k ∑ 2k −2l 1 ∑ k n+1−k k =0 2 (iβ) l=0 l!(k − 2l )!α π k =12 (iβ)
kEk− ( /2) lr+−1 k−−12 lr + 2 Ek( /2) l!1l 42( αβ −i β) × ∑∑1 −× ∑ (2!l) ( k+− 12! l) 2k−− 12 lr + 2 lk!( − 2! l) lr=11= (r −α1!) l=1
l k −1−2l+2r 2 (r −1)!(2αβ − iβ) (20) iββ × ∑ 1 , Wn( ,αβ , , , A ) = exp × 2k −1−2l+2r 1 α 2 r=1 (2r −1)!α 2 4α
− ( ) ( ) 2 n E n / 2 k! E n / 2 1 × − A ; ∑ n+2−2k 2k −1 ∑ n+1−2k 2k π k =1 (2k )!(iβ) α k =0 k!(iβ) (2α)
(20) (20) (20) = −− 2) ReW21( abbkk ,,) Re W( nabb ,,,,1Rekk) W 1( nabb ,,,k k ,1) ,
(20) (20) (20) = +− ImW21( abbkk ,,) Im W( nabb ,,,,1Imkk) W 1( nabb ,,,k k ,1) ;
174 ( ) 1 2α − (1− i ) α 3) W 20 (α, α ) = ln 1 − α × 3 1 2m+1 m 1 2 i(iα) iα 2α1 + (1− i ) α
m−1 (k!) 2 × ; ∑ k +1 k =0 + α m−k α 2 + α (2k 1)!(4i ) ( 1 i )
2 ( ) α iαβ α β 4) W 20 (α, α , β, A)= 1 exp − A − erf 1 × 4 1 2 α 2 + α α + iα α 2 + α 1 i 1 1 i
22kl− mk(2!k ) (αβ) α × 1 +1 exp −β2 × ∑∑m+−12 k kl− ( ) kl=00ki!( α) = l 2 π 4lk !2( − 2 l) !( α1 +α i)
m kl2k−− 12 lr + 2 (2!kr) 1 !(αβ1 ) ×−∑ ∑∑ m+−12 k lk!2( − 2 l) ! − klr−+ k=1ki!( α) lr= 11= 4lr( 2!ri) α2 +α ( 1 )
kl2k−− 12 lr + 2 (l −1!) (αβ1 ) −∑∑ , (2l− 1!2) ( kl +− 1 2) ! 2klr−+ lr=11= (ri−1!) α2 +α ( 1 )
m (20) α (2!k ) Wαα, , 0, AA = 1 ; 4 ( 1 ) ∑ k α2 +αi k=0 k 21mk+− 2 1 4!(ki) ( α) ( α1 +α i)
2 ( ) n! π iβ 2iαβ − iα β 5) W 20 (n, α, α , β, β ) = exp erf 1 1 × 5 1 1 n 2 iα 4α α α 2 + α 2 i 1 i
E (n / 2 ) n−2k 2 2 (− β) n! β + 4iαβ − 4iα1ββ1 × + exp − 1 × ∑ k n−k 2 k =0 k!(n − 2k )!i α α n α 2 + α 4α + 4iα (2 ) 1 i 1
n+−12 k kl+−1 n− En( /2) k!(−β) k (−4il) ( 2 − 2!) ××∑∑ +− − kl=11(2!kn) ( 12! k) = ( l 1!)
l−1 k −l+2r 2l−1−2r 2l−2−2r α α (2αβ1 − α1β) × 1 + ∑ 2l−2−r r=0 − − α 2 + α r!(2l 2 2r )!( 1 i )
175 En( /2) nk−2 kl−1 kl−+2 r 22 l − r 2l−− 12 r (−β) kl−−1 α α1(2 αβ 11 − α β) +∑∑(li−−1) !( ) ∑ , kn!( − 2! k) 21lr−− kl=11= r= 0rl!2( − 1 − 2 r) ! α2 +α i ( 1 )
(20) (2!m) π iαβ Wm(2 ,αα , , 0, β) = erf 1 + 5 11 m mi!4( αα) i α α2 +α ii1
(2!m) iαβ2 m + exp−1 ∑ (ki −−1) ! ( )mk−−1 × 2m 2 α2 +αi mi!2( α) α1 +α 1 k=1
2kl−− 12 k−1 αmk−+222 l α k − l(2 αβ ) × 11, ∑ 21kl−− l=0 2 lk!2( − 1 − 2 l) !( α1 +α i)
2 (20) m! iαβ1 Wm5 (2+ 1, αα ,11 , 0, β) = exp − × mm++12 1 2 α2 +αi 4 α α1 +αi 1
m+−1 k 22kl− mk(2!4ki) (− ) αmk− +2 l α 2 k +− 12 l(2 αβ ) × 11, ∑∑ 2kl− kl=00k! = 2 lk!2( − 2 l) !( α1 +α i)
2 (20) ααn! iβ Wn(,,,,)αα β11 β= exp× 5 1 24ααnnα+12 α +α 2 1 i
n+−12 k kl− n− En( /2) k!(−β) k−1 (2!li) (− 4) αkl+ × ; ∑∑l kl=10(2!kn) ( +− 12! k) = 2 2 (li!) (α1 +α)
2 ( ) exp(− β ) m (2k )! 6) W 20 (β) = . 6 ∑ k 2k +1 π k =0 (− 4) k!β
176 APPENDIX Some useful formulas for obtaining other integrals
νν1 1. ∫∫f( z) erfi (αβ z+=) dz f( z) erf ( iα z + i β) dz . µµi
νν n 2. ∫∫fz( ) erfinn(α+β z) dz =−( i) fz( ) erf ( izα+β i) dz. µµ
νν∂ 3. f( z) cos α z22 +β z +γ dz = f( z) sin α z +β z +γ dz . ∫∫( ) ∂γ ( ) µµ
νν 4. ∫∫f( z) cos(α z22 +β+γ z) dz = f( z) sin( α z +β+γ+π z2) dz . µµ
νν m 5. ∫∫fz( ) sinhmm(α z22 +β+γ z) dz =−( i) fz( ) sin ( izα +β+γ izi) dz. µµ
νν 6. ∫∫fz( ) cosh(α z22 +β+γ z) dz = fz( ) sin( iz α +β+γ+π izi2) dz. µµ
νν 7. ∫∫fz( ) coshmm(α z22 +β+γ z) dz = fz( ) cos ( izα +β+γ izi) dz. µµ
νν(2!m) 8. f( z) sin22m α z +β z +γ dz = f( z) dz + ∫∫( ) m 2 µµ4!(m )
k (2!m) m (− 1) ν + ∑ f( z) cos 2 kα z2 + 2 k β+ z 2 k γ dz . 21m− −+∫ ( ) 2 k=1(mk)!!( mk) µ
k ν (2m +− 1!) m ( 1) 9. f( z) sin21m+ αz 2 +β z +γ dz = ∑ × ∫ ( ) m − ++ µ 4 k=0 (mk)!( m 1! k)
ν × +α++β++γ2 . ∫ f( z)sin (2 k 1) z (2 k 1) z (2 k 1) dz µ
νν(2!mm) ( 2!) 10. f( z) cos22m α z +β z +γ dz = f( z) dz + × ∫∫( ) m 2 21m− µµ4!(m ) 2
177 ν m 1 ×∑ f( z) cos 2 kα z2 + 2 k β+ z 2 k γ dz . −+∫ ( ) k=1(mk)!!( mk) µ
ν (2m + 1!) m 1 11. f( z) cos21m+ αz 2 +β z +γ dz = ∑ × ∫ ( ) m − ++ µ 4 k=0 (mk)!( m 1! k)
ν × +α++β++γ2 . ∫ f( z)cos (2 k 1) z (2 k 1) z (2 k 1) dz µ
mk− νν(2!mm) (2!) m (− 1) 12. f( z) sinh22m α z +β z +γ dz = f( z) dz + ∑ × ∫∫( ) mm2 21− −+ µµ(−4!) (m ) 2 k=1(mk)!!( mk)
ν ×∫ f( z)cosh( 2 kα z2 + 2 k β+ z 2 k γ) dz . µ
mk− ν (2m +− 1!) m ( 1) 13. f( z) sinh21m+ αz 2 +β z +γ dz = ∑ × ∫ ( ) m − ++ µ 4 k=0 (mk)!( m 1! k)
ν × +α++β++γ2 . ∫ f( z) sinh(2 k 1) z (2 k 1) z (2 k 1) dz µ
νν(2!mm) (2!) 14. f( z) cosh22m α z +β z +γ dz = f( z) dz + × ∫∫( ) m 2 21m− µµ4!(m ) 2
ν m 1 ×∑ f( z) cosh 2 kα z2 + 2 k β+ z 2 k γ dz . −+∫ ( ) k=1(mk)!!( mk) µ
ν (2m + 1!) m 1 15. 21m+ α2 ++ βγ = × ∫ f( z) cosh ( z z) dz m ∑ µ 4k=0 (mk−) !( m ++ 1! k)
ν × +α++β++γ2 . ∫ f( z) cosh(2 k 1) z (2 k 1) z (2 k 1) dz µ
ν 22 16. ∫ f( z) sin(α1 z +β+γ 11 z ) sin( α 2z +β+γ 2 z 2) dz = µ
νν 1122 = ∫∫f( z) cos(α3 z +β 33 z +γ) dz − f( z) cos(α4 z +β 44 z +γ ) dz 22µµ
{α3=−=−=− α 1 αβ 2, 3 β 1 βγ 23 ,, γ 1 γ 2
α4=+=+=+ α 1 αβ 2,, 4 β 1 βγ 24 γ 1 γ 2}.
178 ν 22 17. ∫ f( z) sin(α1 z +β+γ 11 z ) cos( α 2z +β+γ 2 z 2) dz = µ
νν 1122 = ∫∫f( z)sin(α3 z +β 33 z +γ) dz + f( z)sin(α4 z +β 44 z +γ ) dz 22µµ
{α3=−=−=− α 1 αβ 2, 3 β 1 βγ 23 ,, γ 1 γ 2
α4=+=+=+ α 1 α 2,, β 4 β 1 βγ 24 γ 1 γ 2}.
ν 22 18. ∫ f( z) cos(α1 z +β+γ 11 z ) cos( α 2z +β+γ 2 z 2) dz = µ
νν 1122 = ∫∫f( z) cos(α3 z +β 33 z +γ) dz + f( z) cos(α4 z +β 44 z +γ ) dz 22µµ
{α3=−=−=− α 1 αβ 2, 3 β 1 βγ 23 ,, γ 1 γ 2
α4=+=+=+ α 1 αβ 2,, 4 β 1 βγ 24 γ 1 γ 2}.
ν 22 19. ∫ f( z) sinh (α1 z +β+γ 11 z ) sinh ( α 2z +β+γ 2 z 2) dz = µ
νν 1122 = ∫∫f( z) cosh(α3 z +β 33 z +γ) dz − f( z) cosh(α4 z +β 44 z +γ ) dz 22µµ
{α3=+=+=+ α 1 αβ 2, 3 β 1 βγ 23 ,, γ 1 γ 2
α4=−=−=− α 1 α 2,, β 4 β 1 βγ 24 γ 1 γ 2}.
ν 22 20. ∫ f( z) sinh(α1 z +β+γ 11 z ) cosh( α 2z +β+γ 2 z 2) dz = µ
νν 1122 = ∫∫f( z) sinh (α3 z +β 33 z +γ) dz + f( z) sinh (α4 z +β 44 z +γ ) dz 22µµ
{α3 =α−α 1 23,,, β =β−β 1 23 γ =γ−γ 1 2
α4=+=+=+ α 1 α 2,, β 4 β 1 βγ 24 γ 1 γ 2}.
ν 22 21. ∫ f( z) cosh(α1 z +β+γ 11 z ) cosh ( α 2z +β+γ 2 z 2) dz = µ
179 νν 1122 = ∫∫f( z) cosh(α3 z +β 33 z +γ) dz + f( z) cosh (α4 z +β 44 z +γ ) dz 22µµ
{α3=−=−=− α 1 αβ 2, 3 β 1 βγ 23 ,, γ 1 γ 2
α4=+=+=+ α 1 αβ 2,, 4 β 1 βγ 24 γ 1 γ 2}.
ν 1 22. fz( ) sinm α+β+γ zz22cos α+β+γ zzdz = × ∫ ( ) ( ) + µ m 1
∂ ν × f( z) sinm+12α z +β z +γ dz . ∂γ ∫ ( ) µ
ν 1 23. fz( ) cosm α+β+γ zz22sin α+β+γ zzdz =− × ∫ ( ) ( ) + µ m 1
∂ ν × f( z) cosm+12α z +β z +γ dz . ∂γ ∫ ( ) µ
ν 1 24. fz( ) sinhm α+β+γ zz22cosh α+β+γ zzdz = × ∫ ( ) ( ) + µ m 1
∂ ν × f( z) sinhm+12αz +β z +γ dz . ∂γ ∫ ( ) µ
ν 1 25. fz( ) coshm α+β+γ zz22sinh α+β+γ zzdz = × ∫ ( ) ( ) + µ m 1
∂ ν × f( z) coshm+12αz +β z +γ dz . ∂γ ∫ ( ) µ
ν 1 26. fz( ) sinm α+β+γ zz2cos 22 α+β+γ zzdz = × ∫ ( ) ( ) + µ m 1
ν 1 × f( z) sinm+22α z +β z +γ dz + × ∫ ( ) ++ µ (mm12)( )
∂2 ν × f( z) sinm+22α z +β z +γ dz . 2 ∫ ( ) ∂γ µ
180 ν 1 27. fz( ) cosm α+β+γ zz2sin 22 α+β+γ zzdz = × ∫ ( ) ( ) + µ m 1
ν 1 × f( z) cosm+22α z +β z +γ dz + × ∫ ( ) ++ µ (mm12)( )
∂2 ν × f( z) cosm+22α z +β z +γ dz . 2 ∫ ( ) ∂γ µ
ν 1 28. fz( ) sinhm α+β+γ zz2 cosh22 α+β+γ zzdz = × ∫ ( ) ( ) ++ µ (mm12)( )
νν ∂2 1 × f( z) sinhmm++22αz +β z +γ dz − f( z) sinh 22αz +β z +γ dz . 2 ∫∫( ) m +1 ( ) ∂γ µµ
ν 1 29. fz( ) coshm α+β+γ zz2 sinh22 α+β+γ zzdz = × ∫ ( ) ( ) ++ µ (mm12)( )
νν ∂2 1 × f( z) coshmm++22αz +β z +γ dz − f( z) cosh 22αz +β z +γ dz . 2 ∫∫( ) m +1 ( ) ∂γ µµ
ν k m! m (−1) 30. f( z) sinm α z2 +β z +γ dz = ∑ × ∫ ( ) m − µ (2i) k=0 kmk!!( )
ν × − α+−2 β+− γ . ∫ f( z) exp( m 2) ki z ( m 2) kiz ( m 2) ki dz µ
ν m!1m 31. f( z) cosm α z2 +β z +γ dz = ∑ × ∫ ( ) m − µ 2 k=0 kmk!!( )
ν × − α+−2 β+− γ . ∫ f( z) exp( m 2) ki z ( m 2) kiz ( m 2) ki dz µ
ν k m! m (−1) 32. f( z) sinhm α z2 +β z +γ dz = ∑ × ∫ ( ) m − µ 2 k=0 kmk!!( )
ν × − α+−2 β+− γ . ∫ fz( ) exp( mkz 2) ( mkzmk 2) ( 2) dz µ
181 ν m!1m 33. f( z) coshm α z2 +β z +γ dz = ∑ × ∫ ( ) m − µ 2 k=0 kmk!!( )
ν × − α+−2 β+− γ . ∫ fz( ) exp( mkz 2) ( mkzmk 2) ( 2) dz µ
Note that the formulas in this Appendix can obviously be written also for improper and indefinite integrals. Also note that in formulas 3 and 22–29, fz( ) does not depend on γ .
182