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 2.20. Integrals of z n sin (β z + γ )[±1− erf (α z + β )], 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.