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:;;.