DOCSLIB.ORG

A Table of Integrals of the Error Functions*

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Table of Integrals of the Error Functions* JOURNAL OF RESEAR CH of the Natiana l Bureau of Standards - B. Mathematical Sciences Vol. 73B, No. 1, January-March 1969 A Table of Integrals of the Error Functions* Edward W. Ng** and Murray Geller** (October 23, 1968) This is a compendium of indefinite and definite integrals of products of the Error function with elementary or transcendental functions. A s'ubstantial portion of the results are ne w. Key Words: Astrophysics; atomic physics; Error functions; indefi nite integrals; special functions; statistical analysis. 1. Introduction Integrals of the error function occur in a great variety of applicati ons, us ualJy in problems involvin g multiple integration where the integrand contains expone ntials of the squares of the argu­ me nts. Examples of applications can be cited from atomic physics [16),1 astrophysics [13] , and statistical analysis [15]. This paper is an attempt to give a n up-to-date exhaustive tabulation of s uch integrals. All formulas for indefi nite integrals in sections 4.1, 4.2, 4.5, a nd 4.6 below were derived from integration by parts and c hecked by differe ntiation of the resulting expressions_ Section 4.3 and the second hali' of 4.5 cover all formulas give n in [7] , with omission of trivial duplications and with a number of additions; section 4.4 covers essentially formulas give n in [4], Vol. I, pp. 233-235. All these formulas have been re-derived and checked, eithe r from the integral representation or from the hype rgeometric series of the error function. Sections 4.7,4_8 and 4_9 ori ginated in a more varied way. Some formulas were derived from multiple integrals involving ele mentary functions, others from existing formulas for integrals of conflue nt hypergeometric functions, and still others, a s mall portion, were compiled directly from existing literature. In connecti on with the last three sections, the reader should refer to [3] and [4], Vol. II, pp. 402,409-411. Throughout this paper, we have adhered to the notations used in the NBS Handbook [9] and we have also assumed the reader's familiarity with the properties of the error functions, for whi c h he is referred to [5]. In addition , the reader should also attend to the following conventions: (i)z=x+iy=r exp (iO) is a complex variable, ~ (z)=x, J1(z) = y, Iz l=r, argz = O; (i i) the parameters a, b, and c are real and positive except where otherwise stated; (iii) unless otherwise specified, the parameters nand k represent the integers 0, 1, 2 ... , whereas the parameters p, q, and v may be nonintegral; (iv) the integration constants have been omitted for the indefinite integrals; (v) when x is used (in stead of z) as the integration variable, it means that the formula has been established only for real x, though it may still be valid for certain complex values; (vi) the integration symbol f denotes a Cauchy principal value. *An invited pape r. This pape r presents th e results of one phase of research carried out at th e Jet Propu lsion Laboratory , California Ins titut e of Techn ology, under Co ntract #NAS7- IOO. sponsored by th e National Aeronautics and Space Administration. "Present address: Jet Propulsion Laboratory, California Institute of Technology, Pasadena, Calif. 91103. 1 Figures in bra c k e t ~ indicat e the lit erature references at th e end of thi s paper. 1 2. Glossary of Functions and Notation A (x) Gaussian Probability Function C(Z) Fresnel Integral Ci(z) Cosine Integral -f '" cos t dt z t DvCz) Parabolic Cylinder Function n k e,,(z) Truncated Exponential 2~ k=O k! "'e- t Exponential Integral f -dt z t '" e-t Ei(x) Exponential Integral - f -dt - x t IFI (a; b; z) == M(a, b, z) Confluent Hypergeometric Function (ak)n zn kFi(al ... ak; bl ... bi ; z) Generalized Hypergeometric Function (b i )" n! H,,(x) Hermite Polynomial Hv(x) Struve Function Iv(z) Modified Bessel Function Jv(z) Bessel Function Kv(z) Modified Bessel Function L~ Generalized Laguerre Polynomial Mp , q(z) Whittaker Function Yv(z) Neumann Function (Bessel Function of Second Kind) P(x) Probability Function (p )11 Pochhammer's Symbol f(p+n)/f(p) PI/(x) Legendre Polynomial P~(z) Associated Legendre Function of the First Kind S(z) Fresnel Integral oo. dt si(z) Sine Integral - f smt- z t U(a, b, z) == 'I'(a, b, z) Confluent Hypergeometric Function Wp,q(z) Whittaker Function 2 y Euler's Constant 0. 5772156649 ... f(p) Gamma Function y(p, z) Incomplete Gamma Function f(p, z) Incomplete Gamma Function ~(z) Riemann's Zeta Function d !/J(z) Psi Function dz [In f(z)] 3. Definition and Integral Representations 3.1. Definitions and Other Notations 1. erf (z) == _2_ J i!: e- /2dt, y;. 0 2. erfc (z ) == ...;; L"e- 12dt= 1- erf (z ), 3. erfi (z) == - i erf (iz) = ...;; r el2dt. Some authors use the above notations without the factor ...;;, and some use <J:l(z) for erf (z). 4. w(z) == e- z2 erfc (- iz). For real x, Dawson's integral is defin ed as 5. F (x) == V; e- x2 erfi (x)= V; Y [w(x)]. The error fun cti on is al so closely related to the Gaussian probability functi ons: 6. erf (x) =2P(xV2)-1=A(x Y2). 3.2. Integral Representations 1. erf (az)= -2az- II e- a'Z'I' dt y;. 0 2. erfc (az)=-y;.2- e - a 2z2 Jo( ; - U'+ 2azt Jdt 3. erf ( az+~)= ~ exp (c- ::) f e-((h' +2wz+c) dz. 3 - -------- 4. erfc (-Z) =-2a exp (Z2)c-- i00 e-(a212+2zl+c)dt a ~ a2 0 5. e2ab erf ( ax + ~) + e- 2ab erf (ax - ~) =~ J e-a2x2- b2/x2dx 2a _ 22j "' e- a2(2tdt, ~(a»O ,~(z»O. 6. erf c (az ) = - e a z ~ 0 (Z2+t2)1 /2 2z (00 e-a2(2dt 7. e rfc (az) = 7r e- a2z2 Jo (t2+ Z2), ~(a) > 0, I arg zl < 7r, Z# O. 4 (I e-x212dt x2 8. 1- [erf (x)]2 =; e- Jo (t 2 + 1)' x> 0 9. erf (x + in + erf (x - in = ~ ey2/4 J e- x2 cos xydx 10. erf (x+ in - erf (x- iJ) = i~ ey2/4 Je - x2 sin xydx 11. erf (x)=~ (" ex2 cos (J cos (x2 sin 0+ 0/2)dO , x # 0 V; Jo 12. erf (x)=.!.. ( '" e- I sin (2xVt) dt. 7r Jo t 4. Integrals of Product of Error Functions With Other Functions 4.1. Combination of Error Function With Powers 1. Jerf (az)dz=z erf (az) + a~ exp (-a2z2) 2. J erfc (az)dz=z erfc (az) - a~ exp (-a2z2) 3. ( 00 erfc (ax)dx= ,II' Jo aV7r 4. Jerf (az)zdz=~ Z2 erf (az) - 4~2 erf (az) + 2a~ exp (- a2z2) 5. J erfc (az)zdz=~ Z2 erfc (az) + 4~2 erf (az) - 2a~ exp (-a2z2) ( 00 1 6. Jo erfc (ax)xdx= 4a2' 4 ZIHI e- a2z2 I - I r (~+2 1) zn-2k 7. f erf (az)zndz=--=t=l erf (az) + ,I L ( ) ~ n aV7T(n+l) k =O r ~-k+l a 2 I-j r (l+4) --+1 ,I erf (az), j = 0 or 1, 2l - j = n + 1 n a,,+1 V7T (n+2)(n+l)f ( n+2) Zll+1 8. f erf (az)z1l+ 2 dz= 2(n+3)a2 erf (az)zndz+ Z2- 2a2 (n+3) erf (az) r (~+ 1) 9. f erfc (az)zndz= (z:'l) erfc (az) _ ,~a2z2 ~ _(-,-2_-,-_) Z:~:k n a v 7T (n + 1) k = O r ~ - k + 1 2 1-j r (l+~) +--1 . ,I erf (az), j=O or 1, 2l - j=n+l n+ an+lv7T (n+2)(n+l)f ( n+2) zn+1 10. f erfc (az)zn+2 dz= 2(n+3)a2 erfc (az)z"dz+ Z2 - 2a2 (n+3) erfc (az) " r (~+ 1 ) 11. { erfc (ax)xndx= v:;;. ' Jo (n+ 1) 7Ta n+1 12.2 f erf (az)z- 'dz= In z erf (az) - ~ f In Ze- 0 2Z2dz 13. f erfc (az)z-'dz= In z erfc (az) + ~ f In ze- a2z2dz erf (az) 2a f 1 14. f erf (az)z-ndz=- . + - e- a2z2dz (n-1)zn- 1 (n - l)v:;;. zn- I ' erfc (az) 2a f 1 15. f erfc (az)z-"dz=- - - e-a2z2dz n~2 (n-l)zn- , (n-l)v:;;. zn-I ' ZP+I 1 (p 16. f erf (az)zPdz=--l erf (az) - ,/"Y -2+ 1, a2z2) , p>-2,p~-1 p + (p + 1) aP+ I V 7T 2 See appendix for integrals on th e right-hand sides of eqs (12 to 15). 5 ------ 17. Jerfe (az)zpdz= zP+11· erfe (az) + 1 v:;;. 'Y (E2 + 1, a2z2 ) , p >-1 p + (p + 1) aP+ 1 7r 7r 18. [ 00 erfe (ax)xpdx= 1 v:;;. r (E2 + 1), largal < 4",p > -l Jo (p + l)ap+ 1 7r oo al - p (p) 7r 19. erf (ax)xp- 2dx= , I r -2 ' largal < 4"'o < p<1. fo V7r(l-p) 4.2. Combination of Error Functions With Exponentials and Powers 3 1. Jerf (az)ebzdz=i ebz erf (az)-i exp (:;2) erf (az- ;a) 2. Jerfe (az)ebZdz=i ebz erfe (az)+i exp (:;2) erf (az- ;a) 00 3. 1 erf (ax)e- bXdx=i exp (:;2) erfe (2bJ, PJ(b) > 0, Iarg al < 7r/4 5. Jerf (az)e bZzdz=1: erf (az)ebz (z-1:) -1: exp (~) { (~_1:) erf (t)- _1_ e- t2 }, t= az-!!"'- b b b 4a2 2a2 b a v:;;.
Load more

Recommended publications
  • Mathematics 412 Partial Differential Equations
    Mathematics 412 Partial Differential Equations c S. A. Fulling Fall, 2005 The Wave Equation This introductory example will have three parts.* 1. I will show how a particular, simple partial differential equation (PDE) arises in a physical problem. 2. We’ll look at its solutions, which happen to be unusually easy to find in this case. 3. We’ll solve the equation again by separation of variables, the central theme of this course, and see how Fourier series arise. The wave equation in two variables (one space, one time) is ∂2u ∂2u = c2 , ∂t2 ∂x2 where c is a constant, which turns out to be the speed of the waves described by the equation. Most textbooks derive the wave equation for a vibrating string (e.g., Haber- man, Chap. 4). It arises in many other contexts — for example, light waves (the electromagnetic field). For variety, I shall look at the case of sound waves (motion in a gas). Sound waves Reference: Feynman Lectures in Physics, Vol. 1, Chap. 47. We assume that the gas moves back and forth in one dimension only (the x direction). If there is no sound, then each bit of gas is at rest at some place (x,y,z). There is a uniform equilibrium density ρ0 (mass per unit volume) and pressure P0 (force per unit area). Now suppose the gas moves; all gas in the layer at x moves the same distance, X(x), but gas in other layers move by different distances. More precisely, at each time t the layer originally at x is displaced to x + X(x,t).
    [Show full text]
  • Integration in Terms of Exponential Integrals and Incomplete Gamma Functions
    Integration in terms of exponential integrals and incomplete gamma functions Waldemar Hebisch Wrocªaw University [email protected] 13 July 2016 Introduction What is FriCAS? I FriCAS is an advanced computer algebra system I forked from Axiom in 2007 I about 30% of mathematical code is new compared to Axiom I about 200000 lines of mathematical code I math code written in Spad (high level strongly typed language very similar to FriCAS interactive language) I runtime system currently based on Lisp Some functionality added to FriCAS after fork: I several improvements to integrator I limits via Gruntz algorithm I knows about most classical special functions I guessing package I package for computations in quantum probability I noncommutative Groebner bases I asymptotically fast arbitrary precision computation of elliptic functions and elliptic integrals I new user interfaces (Emacs mode and Texmacs interface) FriCAS inherited from Axiom good (probably the best at that time) implementation of Risch algorithm (Bronstein, . ) I strong for elementary integration I when integral needed special functions used pattern matching Part of motivation for current work came from Rubi testsuite: Rubi showed that there is a lot of functions which are integrable in terms of relatively few special functions. I Adapting Rubi looked dicult I Previous work on adding special functions to Risch integrator was widely considered impractical My conclusion: Need new algorithmic approach. Ex post I claim that for considered here class of functions extension to Risch algorithm is much more powerful than pattern matching. Informally Ei(u)0 = u0 exp(u)=u Γ(a; u)0 = u0um=k exp(−u) for a = (m + k)=k and −k < m < 0.
    [Show full text]
  • Introduction to Analytic Number Theory the Riemann Zeta Function and Its Functional Equation (And a Review of the Gamma Function and Poisson Summation)
    Math 229: Introduction to Analytic Number Theory The Riemann zeta function and its functional equation (and a review of the Gamma function and Poisson summation) Recall Euler’s identity: ∞ ∞ X Y X Y 1 [ζ(s) :=] n−s = p−cps = . (1) 1 − p−s n=1 p prime cp=0 p prime We showed that this holds as an identity between absolutely convergent sums and products for real s > 1. Riemann’s insight was to consider (1) as an identity between functions of a complex variable s. We follow the curious but nearly universal convention of writing the real and imaginary parts of s as σ and t, so s = σ + it. We already observed that for all real n > 0 we have |n−s| = n−σ, because n−s = exp(−s log n) = n−σe−it log n and e−it log n has absolute value 1; and that both sides of (1) converge absolutely in the half-plane σ > 1, and are equal there either by analytic continuation from the real ray t = 0 or by the same proof we used for the real case. Riemann showed that the function ζ(s) extends from that half-plane to a meromorphic function on all of C (the “Riemann zeta function”), analytic except for a simple pole at s = 1. The continuation to σ > 0 is readily obtained from our formula ∞ ∞ 1 X Z n+1 X Z n+1 ζ(s) − = n−s − x−s dx = (n−s − x−s) dx, s − 1 n=1 n n=1 n since for x ∈ [n, n + 1] (n ≥ 1) and σ > 0 we have Z x −s −s −1−s −1−σ |n − x | = s y dy ≤ |s|n n so the formula for ζ(s) − (1/(s − 1)) is a sum of analytic functions converging absolutely in compact subsets of {σ + it : σ > 0} and thus gives an analytic function there.
    [Show full text]
  • The Error Function Mathematical Physics
    R. I. Badran The Error Function Mathematical Physics The Error Function and Stirling’s Formula The Error Function: x 2 The curve of the Gaussian function y e is called the bell-shaped graph. The error function is defined as the area under part of this curve: x 2 2 erf (x) et dt 1. . 0 There are other definitions of error functions. These are closely related integrals to the above one. 2. a) The normal or Gaussian distribution function. x t2 1 1 1 x P(, x) e 2 dt erf ( ) 2 2 2 2 Proof: Put t 2u and proceed, you might reach a step of x 1 2 P(0, x) eu du P(,x) P(,0) P(0,x) , where 0 1 x P(0, x) erf ( ) Here you can prove that 2 2 . This can be done by using the definition of error function in (1). 0 u2 I I e du Now you need to find P(,0) where . To find this integral you have to put u=x first, then u= y and multiply the two resulting integrals. Make the change of variables to polar coordinate you get R. I. Badran The Error Function Mathematical Physics 0 2 2 I 2 er rdr d 0 From this latter integral you get 1 I P(,0) 2 and 2 . 1 1 x P(, x) erf ( ) 2 2 2 Q. E. D. x 2 t 1 2 1 x 2.b P(0, x) e dt erf ( ) 2 0 2 2 (as proved earlier in 2.a).
    [Show full text]
  • Error Functions
    Error functions Nikolai G. Lehtinen April 23, 2010 1 Error function erf x and complementary er- ror function erfc x (Gauss) error function is 2 x 2 erf x = e−t dt (1) √π Z0 and has properties erf ( )= 1, erf (+ ) = 1 −∞ − ∞ erf ( x)= erf (x), erf (x∗) = [erf(x)]∗ − − where the asterisk denotes complex conjugation. Complementary error function is defined as ∞ 2 2 erfc x = e−t dt = 1 erf x (2) √π Zx − Note also that 2 x 2 e−t dt = 1 + erf x −∞ √π Z Another useful formula: 2 x − t π x e 2σ2 dt = σ erf Z0 r 2 "√2σ # Some Russian authors (e.g., Mikhailovskiy, 1975; Bogdanov et al., 1976) call erf x a Cramp function. 1 2 Faddeeva function w(x) Faddeeva (or Fadeeva) function w(x)(Fadeeva and Terent’ev, 1954; Poppe and Wijers, 1990) does not have a name in Abramowitz and Stegun (1965, ch. 7). It is also called complex error function (or probability integral)(Weide- man, 1994; Baumjohann and Treumann, 1997, p. 310) or plasma dispersion function (Weideman, 1994). To avoid confusion, we will reserve the last name for Z(x), see below. Some Russian authors (e.g., Mikhailovskiy, 1975; Bogdanov et al., 1976) call it a (complex) Cramp function and denote as W (x). Faddeeva function is defined as 2 2i x 2 2 2 w(x)= e−x 1+ et dt = e−x [1+erf(ix)] = e−x erfc ( ix) (3) √π Z0 ! − Integral representations: 2 2 i ∞ e−t dt 2ix ∞ e−t dt w(x)= = (4) π −∞ x t π 0 x2 t2 Z − Z − where x> 0.
    [Show full text]
  • Fresnel Integral Computation Techniques
    FRESNEL INTEGRAL COMPUTATION TECHNIQUES ALEXANDRU IONUT, , JAMES C. HATELEY Abstract. This work is an extension of previous work by Alazah et al. [M. Alazah, S. N. Chandler-Wilde, and S. La Porte, Numerische Mathematik, 128(4):635{661, 2014]. We split the computation of the Fresnel Integrals into 3 cases: a truncated Taylor series, modified trapezoid rule and an asymptotic expansion for small, medium and large arguments respectively. These special functions can be computed accurately and efficiently up to an arbitrary preci- sion. Error estimates are provided and we give a systematic method in choosing the various parameters for a desired precision. We illustrate this method and verify numerically using double precision. 1. Introduction The Fresnel integrals and their simultaneous parametric plot, the clothoid, have numerous applications including; but not limited to, optics and electromagnetic theory [8, 15, 16, 17, 21], robotics [2,7, 12, 13, 14], civil engineering [4,9, 20] and biology [18]. There have been numerous works over the past 70 years computing and numerically approximating Fresnel integrals. Boersma established approximations using the Lanczos tau-method [3] and Cody computed rational Chebyshev approx- imations using the Remes algorithm [5]. Another approach includes a spreadsheet computation by Mielenz [10]; which is based on successive improvements of known relational approximations. Mielenz also gives an improvement of his work [11], where the accuracy is less then 1:e-9. More recently, Alazah, Chandler-Wilde and LaPorte propose a method to compute these integrals via a modified trapezoid rule [1]. Alazah et al. remark after some experimentation that a truncation of the Taylor series is more efficient and accurate than their new method for a small argument [1].
    [Show full text]
  • INTEGRALS of POWERS of LOGGAMMA 1. Introduction The
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 139, Number 2, February 2011, Pages 535–545 S 0002-9939(2010)10589-0 Article electronically published on August 18, 2010 INTEGRALS OF POWERS OF LOGGAMMA TEWODROS AMDEBERHAN, MARK W. COFFEY, OLIVIER ESPINOSA, CHRISTOPH KOUTSCHAN, DANTE V. MANNA, AND VICTOR H. MOLL (Communicated by Ken Ono) Abstract. Properties of the integral of powers of log Γ(x) from 0 to 1 are con- sidered. Analytic evaluations for the first two powers are presented. Empirical evidence for the cubic case is discussed. 1. Introduction The evaluation of definite integrals is a subject full of interconnections of many parts of mathematics. Since the beginning of Integral Calculus, scientists have developed a large variety of techniques to produce magnificent formulae. A partic- ularly beautiful formula due to J. L. Raabe [12] is 1 Γ(x + t) (1.1) log √ dx = t log t − t, for t ≥ 0, 0 2π which includes the special case 1 √ (1.2) L1 := log Γ(x) dx =log 2π. 0 Here Γ(x)isthegamma function defined by the integral representation ∞ (1.3) Γ(x)= ux−1e−udu, 0 for Re x>0. Raabe’s formula can be obtained from the Hurwitz zeta function ∞ 1 (1.4) ζ(s, q)= (n + q)s n=0 via the integral formula 1 t1−s (1.5) ζ(s, q + t) dq = − 0 s 1 coupled with Lerch’s formula ∂ Γ(q) (1.6) ζ(s, q) =log √ . ∂s s=0 2π An interesting extension of these formulas to the p-adic gamma function has recently appeared in [3].
    [Show full text]
  • Global Minimax Approximations and Bounds for the Gaussian Q-Functionbysumsofexponentials 3
    IEEE TRANSACTIONS ON COMMUNICATIONS 1 Global Minimax Approximations and Bounds for the Gaussian Q-Function by Sums of Exponentials Islam M. Tanash and Taneli Riihonen , Member, IEEE Abstract—This paper presents a novel systematic methodology The Gaussian Q-function has many applications in statis- to obtain new simple and tight approximations, lower bounds, tical performance analysis such as evaluating bit, symbol, and upper bounds for the Gaussian Q-function, and functions and block error probabilities for various digital modulation thereof, in the form of a weighted sum of exponential functions. They are based on minimizing the maximum absolute or relative schemes and different fading models [5]–[11], and evaluat- error, resulting in globally uniform error functions with equalized ing the performance of energy detectors for cognitive radio extrema. In particular, we construct sets of equations that applications [12], [13], whenever noise and interference or describe the behaviour of the targeted error functions and a channel can be modelled as a Gaussian random variable. solve them numerically in order to find the optimized sets of However, in many cases formulating such probabilities will coefficients for the sum of exponentials. This also allows for establishing a trade-off between absolute and relative error by result in complicated integrals of the Q-function that cannot controlling weights assigned to the error functions’ extrema. We be expressed in a closed form in terms of elementary functions. further extend the proposed procedure to derive approximations Therefore, finding tractable approximations and bounds for and bounds for any polynomial of the Q-function, which in the Q-function becomes a necessity in order to facilitate turn allows approximating and bounding many functions of the expression manipulations and enable its application over a Q-function that meet the Taylor series conditions, and consider the integer powers of the Q-function as a special case.
    [Show full text]
  • Hydrology 510 Quantitative Methods in Hydrology
    New Mexico Tech Hyd 510 Hydrology Program Quantitative Methods in Hydrology Hydrology 510 Quantitative Methods in Hydrology Motive Preview: Example of a function and its limits Consider the solute concentration, C [ML-3], in a confined aquifer downstream of a continuous source of solute with fixed concentration, C0, starting at time t=0. Assume that the solute advects in the aquifer while diffusing into the bounding aquitards, above and below (but which themselves have no significant flow). (A homology to this problem would be solute movement in a fracture aquitard (diffusion controlled) Flow aquifer Solute (advection controlled) aquitard (diffusion controlled) x with diffusion into the porous-matrix walls bounding the fracture: the so-called “matrix diffusion” problem. The difference with the original problem is one of spatial scale.) An approximate solution for this conceptual model, describing the space-time variation of concentration in the aquifer, is the function: 1/ 2 x x D C x D C(x,t) = C0 erfc or = erfc B vt − x vB C B v(vt − x) 0 where t is time [T] since the solute was first emitted x is the longitudinal distance [L] downstream, v is the longitudinal groundwater (seepage) velocity [L/T] in the aquifer, D is the effective molecular diffusion coefficient in the aquitard [L2/T], B is the aquifer thickness [L] and erfc is the complementary error function. Describe the behavior of this function. Pick some typical numbers for D/B2 (e.g., D ~ 10-9 m2 s-1, typical for many solutes, and B = 2m) and v (e.g., 0.1 m d-1), and graph the function vs.
    [Show full text]
  • Chapter 9 Some Special Functions
    Chapter 9 Some Special Functions Up to this point we have focused on the general properties that are associated with uniform convergence of sequences and series of functions. In this chapter, most of our attention will focus on series that are formed from sequences of functions that are polynomials having one and only one zero of increasing order. In a sense, these are series of functions that are “about as good as it gets.” It would be even better if we were doing this discussion in the “Complex World” however, we will restrict ourselves mostly to power series in the reals. 9.1 Power Series Over the Reals jIn this section,k we turn to series that are generated by sequences of functions :k * ck x k0. De¿nition 9.1.1 A power series in U about the point : + U isaseriesintheform ;* n c0 cn x : n1 where : and cn, for n + M C 0 , are real constants. Remark 9.1.2 When we discuss power series, we are still interested in the differ- ent types of convergence that were discussed in the last chapter namely, point- 3wise, uniform and absolute. In this context, for example, the power series c0 * :n t U n1 cn x is said to be3pointwise convergent on a set S 3if and only if, + * :n * :n for each x0 S, the series c0 n1 cn x0 converges. If c0 n1 cn x0 369 370 CHAPTER 9. SOME SPECIAL FUNCTIONS 3 * :n is divergent, then the power series c0 n1 cn x is said to diverge at the point x0.
    [Show full text]
  • Analytical Evaluation and Asymptotic Evaluation of Dawson's Integral And
    Analytical evaluation and asymptotic evaluation of Dawson’s integral and related functions in mathematical physics V. Nijimbere School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada Abstract Dawson’s integral and related functions in mathematical physics that in- clude the complex error function (Faddeeva’s integral), Fried-Conte (plasma dispersion) function, (Jackson) function, Fresnel function and Gordeyev’s in- tegral are analytically evaluated in terms of the confluent hypergeometric function. And hence, the asymptotic expansions of these functions on the complex plane C are derived using the asymptotic expansion of the confluent hypergeometric function. Keywords: Dawson’s integral, Complex error function, Plasma dispersion function, Fresnel functions, Gordeyev’s integral, Confluent hypergeometric function, asymptotic expansion 1. Introduction Let us consider the first-order initial value problem, D′ +2zD =1,D(0) = 0. (1) arXiv:1703.06757v1 [math.CA] 14 Mar 2017 Its solution given by the definite integral z z2 η2 daw z = D(z)= e− e dη (2) Z0 Email address: [email protected] (V. Nijimbere) Preprint submitted to Elsevier March 21, 2017 is known as Dawson’s integral [1, 17, 24, 27]. Dawson’s integral is related to several important functions (in integral form) in mathematical physics that include Faddeeva’s integral (also know as the complex error function or Kramp function) [9, 10, 22, 18, 27] z z2 2i z2 z2 2i η2 w(z)= e− 1+ e daw z = e− 1+ e dη , (3) √π √π Z0 Fried-Conte function (or plasma dispersion function) [4, 11] z z2 2i z2 z2 2i η2 Z(z)= i√πw(z)= i√πe− 1+ e daw z = i√πe− 1+ e dη , √π √π Z0 (4) (Jackson) function [14] G(z)=1+ zZ(z)=1+ i√πzw(z) z2 2i z2 =1+ i√πze− 1+ e daw z √π z z2 2i η2 =1+ i√πze− 1+ e dη , (5) √π Z0 and Fresnel functions C(x) and S(x) [1] defined by the relation z iπz2 e 2 daw √iπz = eiπη dη = C(x)+ iS(x), (6) √iπ Z0 where z z C(x)= cos(πη2)dη and S(x)= sin (πη2)dη.
    [Show full text]
  • Multiple-Precision Exponential Integral and Related Functions
    Multiple-Precision Exponential Integral and Related Functions David M. Smith Loyola Marymount University This article describes a collection of Fortran-95 routines for evaluating the exponential integral function, error function, sine and cosine integrals, Fresnel integrals, Bessel functions and related mathematical special functions using the FM multiple-precision arithmetic package. Categories and Subject Descriptors: G.1.0 [Numerical Analysis]: General { computer arith- metic, multiple precision arithmetic; G.1.2 [Numerical Analysis]: Approximation { special function approximation; G.4 [Mathematical Software]: { Algorithm design and analysis, efficiency, portability General Terms: Algorithms, exponential integral function, multiple precision Additional Key Words and Phrases: Accuracy, function evaluation, floating point, Fortran, math- ematical library, portable software 1. INTRODUCTION The functions described here are high-precision versions of those in the chapters on the ex- ponential integral function, error function, sine and cosine integrals, Fresnel integrals, and Bessel functions in a reference such as Abramowitz and Stegun [1965]. They are Fortran subroutines that use the FM package [Smith 1991; 1998; 2001] for multiple-precision arithmetic, constants, and elementary functions. The FM package supports high precision real, complex, and integer arithmetic and func- tions. All the standard elementary functions are included, as well as about 25 of the mathematical special functions. An interface module provides easy use of the package through three multiple- precision derived types, and also extends Fortran's array operations involving vectors and matrices to multiple-precision. The routines in this module can detect an attempt to use a multiple precision variable that has not been defined, which helps in debugging. There is great flexibility in choosing the numerical environment of the package.
    [Show full text]