A Table Of Integrals Of The Error Function. II. Additions .

JOURNAL OF RESEAR CH of the Noti ona l Bu rea u of Stand a rd s- B. Mathemotical SciencesVo l. 75 B, Nos. 3 a nd 4, July- Decembe r 1971A Table of Integrals of the Error Function.II. Additions and Corrections*Murray Geller** and Edward W. Ng**(September 28,1971)Thi s is a n exte nsion of a co mpe ndium of ind e finit e a nd d e finit e integra ls of produ c ts of th e erro rfun ction with e le me ntary or tra nsce nd e ntal fun c tions rece ntl y publi s he d by the a uthors.Ke y word s : Error fu nc ti ons; indefinit e integra ls; s pec ia l fun ctions.1. IntroductionSince th e present a uthors publi shed an exte nsive compe ndium of integrals involvin g th e errorfun ction [1J ,] numerou s co mme nts and s ugges tions have bee n received. In parti c ular , we havebe en advi sed by P . I. Hadji (II. II. XA,I1)RII) of hi s publi cation in Ru ssia n [2] , whi c h cont ainsformulas not included in [1]. Careful examination of [2] has r evealed num erou s mathe mati caland typogra phi cal errors. Beann g in mind also th e m access ibilit y of hi s re port , th e a uth or s beli e veit to be appropriate to publi sh an exte nsion of [1] based largely on corrected versions of Hadji 'sformulas but also co ntaining new res ults. It should be noted that the authors have verified and/ orcorrected aU of Hadji 's formulas in additi on to rederiving and c heckin g all of the formulas in [1].Throu ghout thi s arti cle, we conform to the format and notation of [1] , including secti on a ndequation numb ers. A s upple me ntary glossary is also included. Within eac h s ubsec tion, typograph ·ical e rrors, equivale nt form s a nd errors de tected in [1J will be li s ted under correction s.2. Supplementary GlossaryB(p, q)Beta Fun ctionf3(X)f (p )f(q ) /f(p q)H",(t h) - ",(h)]Gegenbauer PolynomialGCatalan's Constant0.915965594 .Jacobi PolynomialTn(X)Chebyshe v PolynomialAMS S ubject Classifiw tio1l ; Prim ary 6505.* An in vit e d pa per. Thi s pa pe r prese nt s th e res ult s of one phase of resea rc h carri ed out at the Je t Propul sion Labora tory, C alifornia In stitut e of T ec hn o logy,und er C ontra ct #NA S7- 100 , s p o n so fl d by th e Na ti ona l Aerona uti cs a nd S pace Admini stration .** Present a ddress: J et P ro pul sio n Laborat ory, Ca li fo rni a Ins titut e of T ec hno logy, Pasade na, CaHfornia 91103.IFi gures in b rackets indi ca te the lit e rature references a t the e nd of thi s pa pe r.149

3.* Integrals of Products of Error Functions with Other Functions3.1 . Combination of Error Function with PowersAdditionsf(20)1e a erf(x)x-pdx p-1[erf(a)a p- I1 v-;e- a2 /2]aP/2 W P/4 ,(P 2)/4(a 2) ,Jdxo [erf(ax c) -erf(bx c)] ---; In(21)OOp l(a)b erfc(c)Jo( oo [berf(ax) -aerf(bx)] dxx2 2abv-; In (b)-;;(22)JOO(23)odx1[Yxerf(Yx)-erf(x)] 2 . (2-y)x1 2] dX l (2-y)r oo [erf(x) 2xV;l x(24)y1TJo:; V;x[1- r( 2 ( r4e-a/(2U) W P-{5/4) '( )(25)Jou(u - x)Perf[ V (a/x) ]dx (26)Jo[(b iX) - P- ' - (b - ix) - P- I]erf(ax)dx - 2i (a Y2)Pea2b2/2D p(ab V2),p(27)Jooox[(b ix) -P-' (b-ix) - p-']erf(ax)dx002p (p -1) (a/b )p- Iea2b2/2[(p -l)aD p(a) - pDt pea) J,Joo(28)-xJoo(29)(30)(31)(32)oJerf[1/4J.p -lp 0p 1, a ab Y2erf(ax), f--'-----'-- dx 2a Y 1Te - a2b2e rfi (ab )(x b)2erfc[aO x)I/4] d x --e4-a2- 4e rfc ( a )(1 x)3/4aV;. x-b2Yx. . Y (ax)] Yx(X b)2 dx - x b erf[ v (ax)] -2 Y (a/1T)e abE,(ab ax)lou X2V-I(U2-X2)JL-I erf (ax)dx( 00 x erf (ax) d Jo (x2 b2)2 x- 2b·Section 3 co rres pond s to section 4 of referencey:;;fll·1501T ea 2b 2erf c(ab)

lf'"' xerf(ax) d x a V7TejU' 2(33)o (x 2 1)2(34)4x e rf-:( ax-,-) dx f'!:, (x2 1)24[f' ()]l - er2a --, erf()a4V7i e [1 U2erf(a)] -'- erf(a)41'"erf (ax)- 12 2o (X2 b2)3/2 dx - b2 [l exp (a b ) erf c (ab)](36)(37)(38)(39)3 .2 . Combination of Error Function with Exponentials and PowersCo rrections(10)Replace (bz" nz'l-l) in last exp ression by (bz" - ' (n - 1)zn - 2) .(12)Replace (bz" nz'l-l) in last expressio n by ' 1 (bz'l-l(13)Replace (2a)(k - II)/2 by (2a 2 )(k -v:;;a (n -1 )ZIl- 2) .") /2 .Additions(b)( '" [ er f(e - ax ) - e rf(e - ox)] dx erf (l) i n Jo(15)(1 6)(17)f[ ae- iPX ]oefoe-00(1 8)7T[ ae PC]( a) .(a)7T erf - - .- e- 1PX - .dx- - ? - e pc erf - - ,p 0 , b "'" c for the lower sianb LXx 2 ccb c"foe- 00(19)dxerf (b ix)a x2 c2 e rf (b c)n ,p O,a O ,b""'cforth e lower s i gn-00,dx7T(ix) - p erf (ae- to :r ) - ?- - erf (ae - oc ), Iplx- c2cp 1 J oo erf (ae:t iDX) T-oox-c151 i7T erf (ae :t ioc) 1

r"(20)Jodx[erf (1 ix) erf (1- ix) ] 2·sin hhrx(1 iX)2(1- ix)2 rf (1)e3.3. Combination of Error Function with Exponentials of More Complicated ArgumentsCorrectionsi"erf(ax)e- b 2x 2 dx -1- tan- 1(a)(2)Equivalent form:(3)Remove condition "b may be complex" and replace by.9t(a2 ) .9t (b 2 )(8)For the form given , add the condition .9t (b 2 ) .9t (a 2 )oby;.-bEquivalent form:.9t (a 2 ) 0 ,.9t (b 2 ) 0,p -2.(16)(35)Add the condition .9t (c 2 ) .9t (a 2 )Additionsf"1x exp (x 2 ) erf(x )dx 2 ,exp (x 2 ) erf (x) -(39)1-:y; x(40)f(41)f(42)(43)(44)2b 2fVrr[erf(ax)]n exp (-a 2 x 2)dx -:--:--- [erf (ax)] 11 12a(n 1)[a erf (bx) exp (- a 2x 2 ) b erf(ax) exp (- b 2x2 ) Jdx V;;erf(ax c)exp(-b 2x 2 )xdxfx 2n erf (ax) exp( - a 2 x 2 ) dx 152 ----erfl(ax) erf (bx)(2n)!y;.---;;r:- (2a)2n 1 A,

where12a,11 - 1k !A - erf2 (ax) - - x erf (ax) exp (- a x )(4a 2x 2 ) h'2y:;.'. 0 (2k 1) !2:exp(-2a 2x2) 1I 12"L.7Tf(45)X 211 1 erf(ax)"' 0hk!k!eh' (2a 2x 2 )(2k 1)-,------,-,n!exp(- a 2x 2 )dx - 211 - 2 Ba '.where8 2 - 3 / 2 erf (ax \12)2: (2n 8 - "'- :2 erf(ax)e - a' x' e (a 2x 2 )II]ll1.' 0- -axV;(46)26 f x I l erf(ax)exp(-6 2x 2 )dx 2n fX 211 - 1e - 2a 'X 2,, 1f(48)f exp{ - 62 [ erf (ax) ] "}exp (-a 2x 2) [ erf (ax) ] Qdxdx1erf(ax)exp(-a2x 2 ) o --erf(ax)exp(-a2 x 2 )x-x ::;f(49)(1n1!k - I8-k2:. , (8a 2x2)I1 0(2/ 1),erf(ax)exp(- 62 x 2 ) dx-x 2 I1 erf(ax)exp(-6 2x 2 )(47)2:II faV;---6-'4 1 Q) /P'}' (2X 2 11exp[- (a b2) x2]dxae rf2 (ax)- - E 1 (2a 2x 2 )y:;.q; 1, 62 (erf (ax)]p), q -e- a2x2V;erf(ax) dx 2a In[ erf(ax)](50)(51))0{oo erf [ ]. e- b / x dxx2(52) b1 [1- e -2V(abf](53)(54){oo [a erf (ex exp [- ce ax ])Jol-e - ax6 erf (ex ex p [- cebX ] ) ] dx Inl-ebx(55)153(l!.-)erf (exeac)1,p 0

(56)foo X-erf i (ax) e- a2x2 - - dx - erf c (ab ) ea2b2ox 2 b22(57)3.4. Definite Integrals from Laplace Transforms Involving Erf ( )Corrections(12) Replace erf ( ) in integrand by erfc ( ) and remove the condition b a.(13) First term inside square brackets in integrand should read (2ax l)xeax erf c ( ) insteadof (2a l / 2x l /2 1 )xeax erf ( ) and remove the condition b a.(14)Last term inside square brackets in integrand should read- 2 (ax3/ ) 1/2 (2ax 5) instead of -2(a3x5/ )1/2(2ax 5).(15)Replace erfc (v;;-) in integrand by erf (v;;-) and add the condition b c.(16)First term inside square brackets in integrand should read(18)First term inside brackets in integrand should readAdditionsroo e- X erf (y ) dx l Jo(l e- x j2(19)i(20)00ofOO(21)e- X erf ( )[(v'2 1) (1)1 - e - nx] dx21- -x . r , Ievx v o1L ,{,tanknk II[V(a/k) ]Vdx-e- bx erf ( Vax) e- ncx [sin hcx] n2-Vx 2 L.n (-I)k(V 2nk 0n) (b 2ck)-1/2 tan--kI( a )1 /2b 2ck3.5. Combination of Error Function with Trigonometric FunctionsCorrections(10)For the form given a b.Fora b:2 b 2 abv'2) 21 [1 n (aa 2 b2ab v'2 tanroo f ( ) . b2 2d Jo er ax sm x x 4bV(2 )154-I (abv'2)]b2-a 2 2

(11)Change the condition on p to Yt (p) - 2.(12)Change the condition on p to Yt (p ) -1.1;(13)Iooo(16)erf( ) sin bxdx -l; [l-e -10 erf( ) cos bxdx -l;e 00(17)(24): E, (: 2) aV; erf (;a)·erf (ax) sin bx1000erf c(ax)x sin x sin hxdx AAcos A],sinA,A (2ab)1/2A (2ab)1/2 [:2 sin (2 2) cos (2 2) -1]Additions2i(b 2 c 2)(31)I ic)A - -erf (ax) e- bx sin cxdx (b(b - ic)A ,whereA ", (exp [( b iC)2]2aerfaxI [tan erf (ax)]e - a(32) b iC)2a-2 2xV;dx - 2a In {cos [erf (ax)]}V;'2b(33)Joerf (ax)e -(34)fTerf (a sin x) cos (2n l)x dx 00ifx2sin cxdx (27TJo(35)Le- C?)/ 4b-2 (erf i[2b(a2DT erf (a sin x) acb2)1/2]s in 2nxdx Oerf (a sin x b cos x)dx O27T(36)exp [ - (b ic)x ] err (ax)erf (ae ix ) sin nx dx i02 v;.-1n 2k 2k 1 ika2k 1n 2k 1n 2k(37)n 2k 1[7T/2(38)Joerf (a tan x) sin 2x dx V"ii ae a2 erf c (a)(39)155

(40)folerf (ax) cos [(2n 1) COS-I x](l- X2) - 1/2dx10"(42)[erf (tan - I ax) - erf (tan-I bx)] :x In( )erf( )3 .6. Combination of Error Function With Logarithms and PowersCorrectionsReplace tz2 erf (az) In z on the R.H.S. by t z2 erf c(az) In z.(5)Additions(10)xdx fo"erf (ax) In (2 2 cos x) -2--2x cbf"oerf (a In x)(a)Ierf 2c dx ! [f3 (2.) - f3 (1-2.)]1 XCcccerf(1f/2Jo1[r (ta) Perf (b sin 2x) In (a tan x)dx "2 (In a) erf (2b) r(a)Jo1f/2(15)(16)(17)(18)(19)( ),2c( ) (c d)Jo" In rc derf(ax)]dxc d erf (bx) x - In b Inc(13)(14)In (1 e- c ) erf i (ac)J(I0 erf (a lnx) 1X - X C dx --;;1 f3 (b) (11)(12)1Terf (a In tan x) dx 0dxb -:;J'" erf (x2a xb2) In (x)opx)Jo{ " erf ( x 2p b2pInf"f'"oo(x)b 0dxx 2 b2 0dxerf (xP x - p ) In x - 0xerf (x P x- p ) In x156dx1 x 2-- 0c b

100(20)oerf (a In x) OIn xl-x 21dx00o erf (a In x) 1(21)(22)2a 2Loo x bx x 20 , Ib I 2In x erf c (x) e - a2x2 dx3.7. Combination of Two Error FunctionsCorrections(3)2bLas t term on R.H.S. should read V; E, (2ab)(4)Last term on R.H .S . should read -f(7)2bV; E 1 (2ab)erf (x) erf [(l-X 2)1 /2]xdx (2e) - '(8)Remove the entire express ion and replace by(9)Add the condition c a.Additions(10)f erf (ax) erf (bx)dx x ert (ax) erf (bx) -(11)(2n 1) f X211 erf (ax) erf (bx)dx X211 1 erf (ax) erf (bx)(12)f f X211 1V;,1 r;;;. (a 2 b 2)1 /2 ert [xY(a 2 b 2)]ab v 1Terf (ax)e - b 2X2 dx -n'2a f x2n , erf (bx)e - a2x2 dxV;X2 11 14X211 erf2 (ax)dx -- erf2 (ax ) - - .A,2n l2n l V;a2n 1157

whereL (2k) 8 - La::.- e - 2a2x2 11-V7Tk Okkk-I1 0I'.,(21 1).(8a 2x 2) (erf2 (ax) x erf (ax)e - a2X2 e - 2a2x2Jx erf2 (ax)dx (X22 - )4aa y:;;.2r.a(13)Jx 211 1 erf2 (ax )dx-- [ x(14)271 22(2n I)!] erf2 (ax)n! (2a)211 22n 2 2(2a)271 2(2n I)! B(n I)! 'where(16)(""xp-1erfc(ax)erf(bx)dJox r(l pI2).Fa P I 7T(p l)a b,(17)132(1.2' p lp 2. P 3. b2 ' 2 '2' 2 ' a 22)'p -100 erfax( ) erf (bx)e- c 22dxx 2xo J;{a In [(a2b c ) 1/ 2 ] bIn [ (b2a : ) 1/2] -c tan- I (;!)} ,d (a 2 b 2 c 2 )1 / 2(18)(19)(20)(006( 2)1/2(21)Jo [1- erf3 (x) ]dx ;:;:(22)(0012 (2 )1/2Jo [1-e r f 4 (x)]dx -:;;:tan-I (8- 1/2 )158tan- I (2- 1 / 2 )

3.S. Combination of Error Function with Bessel FunctionsCorrections(12)b2 ) on th e R.H.S. by I ( - pR eplace f ( - p -1 , 4a2(13)ReplaceW-fJ /2,fJ /2(::2)bychange the condition -1 P (19)Multiply the R.H.S. by1 4ab2 )-2'2 W-P/2, P/2(4b: 2);f to - t p f,J;;change the condition on p to- t p O.J;(20)Multiply the R.H.S. by(21)Re place th e condition A tp 0 by A tp -tAdditions(22)(23)Jo( 00 Ko(ax) erf (bx)xdx 8b1(24)2ec[Kt(c)-Ko(c)], c a 2 j (8b 2 )(25)Jo((26)00x 2 fJ- 2e - x Kv(x2)erf(x)dx 2(1)P 2f (p v) f (p - v)2pfp (a)2V2"b2v- 21.- tf(A - v l),;v7T(2v-2A-1)f(2v 1)(3F211 31)p v,P - v' 2;P 2'2 ; -2 'Ivi11 a bZ;2v 1,v- A, V-A Z;- 4 2tF 3 (v- A-2 )a)2A 2 bf(v-A-1)(1 3a 2b 2 ),;tF 3 2;2,A-V 2,A V 2;-- , ( v7Tf(v A 2),45V A - 4159

3.9. Combination of Error Function with Other Special Functions!Corrections(3)Replace si(2p x) in integrand by si ((5)Replace e a2/ 4 on R.H.S. by e a2 / 2 ; ).change second index of W from (1 21L)/4 to (1 2v)/4;change argument of W from a 2 to a 2 /2;change the condition IL v to IL v. -t.(6)Add the condition v(7)Add the condition v -to(9)Change the R.H.S. to.); 2 (2a-2b 1)-1 2F, (a, a-b t; a-b i; -1),(10)Change the R.H.S. to(11)Change the R.H.S. to(16)Change the condition on p to p O.(18)Integrand should contain erfc (a ! b x ) instead of erfc ( x );coefficient on R.H.S. should read2V/2 1a- 3/2/ [ ( 1 )r ( 1 - )](20)Replace pV I /2 on the R.H.S. by a V 1/2(21)First term inside 3F2 expression should readinstead of 2V/2 2a-3/2/ (lJ7T)- A. IL t instead of A. IL t;argument of 3F2 should read a insteadof - a;add the condition a:S; 1; a 1, A. tp.160IJ