Im λ + D + C Re λ A B η = λ 1 ln λ. Im η E + F + B + A + C + D + A D. C Re η A C

Transcription

1 Uniform Asymptotics for the Incomplete Gmm Functions Strting From Negtive Vlues of the Prmeters N. M. Temme CWI, P.O. Box 9479, 9 GB Amsterdm, The Netherlnds e-mil: nicot@cwi.nl Abstrct We consider the symptotic behvior of the incomplete gmm functions ( ; z) nd ( ; z) s!. Uniform expnsions re needed to describe the trnsition re z, in which cse error functions re used s min pproximnts. We use integrl representtions of the incomplete gmm functions nd derive uniform expnsion by pplying techniues used for the existing uniform expnsions for (; z) nd (; z). The result is compred with Olver's uniform expnsion for the generlized exponentil integrl. A numericl veriction of the expnsion is given in nl section. AMS Mthemtics Subject Clssiction (99): 33B, 4A6. Keywords & Phrses: incomplete gmm functions, uniform symptotic expnsion, error function.. Introduction The incomplete gmm functions re dened by the integrls (; z) = Z z t e t dt; (; z) = Z z t e t dt; (:) where nd z re complex prmeters nd t tkes its principl vlue. For (; z) we need the condition < >, for (; z) we ssume tht j rg zj <. Anlytic continution cn be bsed on these integrls, or on series representtions of (; z). We hve (; z) + (; z) = (). Another importnt function is dened by (; z) = z (; z) = () () Z u e zu du: (:) This function is single-vlued entire function of both nd z, nd is rel for positive nd negtive vlues of nd z. For (; z) we hve the dditionl integrl representtion (; z) = e z ( ) Z e zt t dt; < < ; <z > ; (:3) t +

2 which cn be veried by dierentiting the right-hnd side with respect to z. For other informtion on the incomplete gmm functions we refer to Chpter IX of the Btemn Mnuscript Project (953) or Chpter of Temme (996). The purpose of the pper is to derive new uniform symptotic expnsions for the functions ( ; z); ( ; z) nd ( ; z). The uniform expnsions for (; z) nd (; z) of our erlier ppers, which will be summrized in the next section, re not vlid for negtive vlues of. Recently there is much interest in symptotic properties of incomplete gmm functions. The function (; z), or the relted exponentil integrl E p (z) = z p Z z e t t p dt = zp ( p; z); (:4) plys n importnt role in Berry's smooth interprettion of the Stokes phenomen for certin integrls nd specil functions; see Berry (989). Olver (99) investigted E p (z) in prticulr t the Stokes lines rg z = nd used the results in Olver (99b); see lso Olver (994). We summrize Olver's result in the next section. In Dunster (994) new expnsion for the function E p (z) is given for complex vlues of z nd lrge positive vlues of p nd in Dunster (994b) error bounds re given, in prticulr for the Stokes' smoothing pproximtions. In Pris (994) the uniform symptotic expnsions of the next section re used for complex vlues of the prmeters in new lgorithm for computing the Riemnn zet function on the criticl line. The computtionl problem of the incomplete gmm functions for complex vlues of nd z is not well solved in the softwre literture. In prticulr when the complex prmeters hve lrge negtive rel prts existing computer progrms my give relly flse nswers. We expect tht the new expnsions for ( ; z); ( ; z); ( ; z) nd Olver's expnsion for E p ( z) will be of vlue to solve this problem.. Uniform expnsions for incomplete gmm functions. We summrize the known uniform expnsions for the incomplete gmm functions; see Temme (975), (979) nd more recently Olver (99), (994)... Uniform expnsions for P nd Q. Let be the rel number dened by = ln ; > ; sign() = sign( ): (:) Extend the reltion between nd to complex vlues by nlytic continution. We hve Then we write, with = z=, ( ) 3 ( ) + : : : ;! : (; z) Q(; z) = () (; z) P (; z) = () = erfc( p= ) + R (); = erfc( p= ) R (): (:)

3 3 The error functions re dened by erf z = p Z z Z e t dt; erfc z = erf z = p e t dt (:3) The error functions re the dominnt terms in (.) s tends to innity, nd they describe the trnsition t = z. We hve R () = e p S (); S () X n= z C n () ; (:4) n s!, uniformly with respect to j rg j ; nd ; j rg j, where ; re rbitrrily smll positive constnts. More informtion on the coecients C n () is given in the next subsection. In this pper we derive similr new expnsions for (; z) nd (; z), where we concentrte on negtive vlues of nd z. Becuse simple reection formuls re missing tht relte ( ; z) with (; z), etc., we cnnot simply trnsform the bove expnsions to this cse... Further detils on the previous results. Becuse the new expnsions (see x3) re closely relted with the one in (.4), we give some detils on computing the coecients nd on the mpping! () dened in (.). The expnsion in (.4) for S () cn be obtined by dierentiting one of the eutions in (.) with respect to, which gives where d f() = d S () S () = [ f()= ()] ; d d = ; () = p =() e (): (:5) Substituting the symptotic expnsion for S () one nds for the coecients the reltions C () = ; C n() = d d C n () + n f(); n ; (:6) where n re the coecients in the reciprocl gmm function expnsion X () n= n n! ; j rg j < ; (:7) of which the rst few re given by = ; = ; = 88 ; 3 = ; 4 = : Next we summrize the properties of the mpping! () dened in (.). More detils re given in Temme (979). Write = + i. The mpping is one-to-one for < rg < ; 6=. The corresponding domin is given in Figure. The mpping is singulr t

4 4 Im λ F + π E + D + E D A B C Re λ F π E + Im η F + η = λ ln λ B + A + C + D + A D A C B E B F C Re η Figure. Corresponding points in the nd plnes for the mpping! () dened in (.). The points B re singulr points in the plne. In the plne they correspond to exp(i). The points D ; E correspond to points D ; E hving phses i. The thick brnches of the hyperbols re brnch cuts. The prbol shped curve in the plne corresponds to the imginry xis in the plne.

5 = exp(i) with corresponding points B = p ( i). The positive rel xis (with rg = ) is mpped to brnches of the hyperbols = ; < p. When we cut the plne long the two thick prts of the hyperbols in Figure we hve one-to-one interprettion of the mpping in (.), nd the function () is singulr t = p ( i). The prbol shped curve in the plne is the set of points =( ln ) =, nd given by the eution () = = sin ; < < ; () =, where ; re the polr coordintes of : = exp(i). The corresponding set in the plne is the imginry xis Olver's expnsion. Olver (99) investigted the generlized exponentil integrl E p (z) dened in (.4). Let F p (z) = (p) E p (z) z p nd z = e i ; = n + p with lrge prmeter, p xed. Then F n+p (z) ( ) n ie i " erfc c() i e fc()g p X s= s g s(; ) s # ; (:8) uniformly with respect to [ +; 3 ] nd bounded vlues of jj; denotes n rbitrrily smll positive constnt. Furthermore, c() = fe i + i( ) + g ; with the choice of brnch of the sure root tht implies c() ( ) s! ; the coecients g s (; ) re continuous functions of nd. A similr expnsion for F n+p (z) is given when [ 3 + ; ]. As Olver remrks, this expnsion unties the Stokes phenomenon, tht is, the rpid but smooth chnge in form of other expnsions s psses through the common intervl of vlidity of the other expnsions. In the present pper we give n expnsion tht is lso vlid in domin round the negtive z xis nd tht lso contins the error function. Our expnsion is uite dierent, however, nd is strongly relted to the uniform expnsion given in (.){(.4). 3. Strting from negtive vlues. We strt with (.3) nd replce with : ( ; z) = e z ( + ) Z e zt t dt; < > ; <z > : (3:) t + By turning the pth of integrtion nd invoking the principle of nlytic continution we cn enlrge the domin of z. For exmple, when we consider the pth in (3.) long the positive imginry xis, the integrl is dened for < rg z <, nd in the overlpping domin = < rg z < its vlue is the sme s in (3.). We turn the pth from the positive imginry xis rg t = = to the negtive xis rg t =, voiding the pole t t = by

6 6 using smll semi-circle. The integrl is then dened for 3 < rg z <. We chnge the vrible of integrtion nd the result is: ( ; ze i ) = ez e i ( + ) Z e zt t t dt; < > ; < rg z < (3:) nd we void the pole by integrting under the pole t t =. By turning the pth in (3.) clockwise we obtin: ( ; ze +i ) = ez e i ( + ) Z e zt t t dt; < > ; < rg z < (3:3) nd we void the pole by integrting bove the pole t t =. An esy conseuence is e i ( ; ze +i ) e i ( ; ze i ) = i ( + ) ; (3:4) which follows from computing the residue of the integrl over the full circle round the pole t t =. It lso follows from the fct tht (; z) = ()[ z (; z)] nd (; z) is entire. We proceed with (3.3) nd ssume, for the time being, tht nd z re positive. We chnge the vrible of integrtion t! t=z, nd obtin ; ze i = ez e i ( + ) = e i ( + ) Z e t t dt t Z e g() d where = () is dened in (.), = z=, = (t), tht is, nd = t ln t ; sign = sign(t ); g() = dt d t = t t ; t : (3:5) In the integrl the pth psses bove the pole t =. When z, tht is,, the pole is ner the sddle point =, nd for lrge vlues of we need n error function to describe the symptotic behvior. We cn split o the pole by writing g() = [g() g()] + g(), where, s is esily veried, g() =. Using the representtion Z ez e t erfc iz = dt z C; i t z where the pth psses bove the pole t t = z, we obtin (introducing suitble normliztion) ( + ) ei i ; ze i = erfc i i e p T (); (3:6)

7 nd, for instnce by using the reltion in (3.4), where ( + ) e i i r T () = ; ze i = erfc i Z e h() d; 7 + i e p T (); (3:7) g() g() h() = : (3:8) Stndrd methods for integrls cn be used now (see Wong (989)) to obtin n symptotic expnsion of T () in negtive powers of. It is esier, however, to use dierentil eution stised by T (). In this wy we cn identify the coecients of the expnsion with those in (.4). Dierentiting (3.6) with respect to, nd using (.), (.) nd (.3) we obtin the dierentil eution d d T () + T () = [f() () ] ; (3:9) where f() nd () re given in (.5). As in (.7) we hve (with the sme coecients n indeed) () X n= Substituting this nd the expnsion ( ) n n T () n ;! ; j rg j < : X n= ( ) C n() n ; (3:) n in (3.9) we nd for C n () the sme recursion s in (.6), nd we conclude tht in (.4) nd (3.) the sme coecients occur. 3.. The domin of vlidity of expnsion (3.). We return to the cse tht, z nd = z= re complex prmeters nd clim tht expnsion (3.) holds s!, uniformly with respect to j rg j nd j rg j, with ; rbitrrily smll positive constnts. To verify this we observe tht () is nlytic nd univlent in the sector j rg j <. This follows from x.. The function (t) is in fct (t) nd the untity t = t() occurring in g in (3.5) is singulr t the points B shown in the lower prt of Figure. The functions g nd h (given in (3.8)) re singulr in the sme points. It follows tht h() is nlytic in the disk jj < p nd in the sectors j=j < j<j. When is complex we cn turn the pth of integrtion in (3.8) inside the sectors j=j < j<j nd < cn be kept positive on the pth of integrtion s long s j rg j <. This gives the domins for nd. We remrk tht the function h() is bounded t innity inside the sectors j=j < j<j. This property is not needed to verify the bove result, however.

8 8 3.. A rel expnsion for ( ; z). It is of interest to hve result for the function (; z) (see (.)) for negtive vlues of the prmeters. In tht cse (; z) is rel nd we hve rel symptotic representtion: ( ; z) = z cos e sin F + T () ; (3:) where Z z F (z) = e z e t dt which is Dwson's integrl. The reltion with the error function is: F (z) = ip e z erf iz: Representtion (3.) cn be veried by using (3.3) nd the reltions ( ; ze +i ) ( ) z e i ( ; z) ; ( + ) ( ) = sin : In (3.) we see tht the oscilltory behvior of ( ; z) is described by two terms, one with cos nd one with exp( ) sin (the fctor contining Dwson's integrl is slowly vrying when the prmeters nd z re positive). This complicted oscilltory behvior is one of the problems in writing relible softwre for the functions (; z) nd (; z) when the prmeters hve lrge negtive rel prts. Dwson's integrl becomes dominnt when we consider complex vlues of the prmeters. The function F (z) is n entire odd function nd hs for <z the symptotic behvior F (z) 8 < : z if rg z < 4 ; sign(=z) ip e z elsewhere. jzj! ; 3.3. Compring Olver's expnsion with our expnsion. In Olver's expnsion (.9) the modulus of z is the lrge prmeter, wheres in our expnsion (3.) is the lrge prmeter. In ddition, in Olver's expnsion the trnsition occurs t =, wheres in our expnsion the trnsition tkes plce t =, tht is, t =, with = z=. We hve, s!. Hence, when z = + i", then i"=. When is positive nd " increses from negtive to positive vlues, ze i crosses the negtive xis with incresing rgument, nd p= erfc(i ) of (3.6) chnges rpidly from to. Olver's expnsion hs more prmeters thn ours becuse his nlysis strted by considering F n+p (z) s reminder in the expnsion for F p (z). However, one cn tke n = ; p = +. Then Olver's nottion grees better with our nottion. In his result the prmeter = p is ssumed to be bounded, nd restriction like this is not present in our method.

9 Remrk. We verify wht hppens when the prmeters ; z in (.) re tken with negtive signs. First let =; =z be positive, nd replce in Q(; z) the prmeters ; z with e i ; ze i, respectively. The untity dened in (.) does not chnge by this opertion, wheres the expnsion for S () becomes the expnsion for T () given in (3.). When we formlly write S () = T (), which certinly is not true, it follows tht the right-hnd side of the rst line in (.) formlly becomes which is the right-hnd side of (3.7). But Q( ; z i ) = ; ze i p i erfc = + i e p T (); ( ) which mens tht the left-hnd side of (3.7) euls ( + ) e i i ; ze i = = ( + ) sin ( ; ze i ); e i Q ; ze i : Hence, by proceeding formlly from the reltion for Q in formul (.), we miss the fctor =[ exp(i)], this fctor being negligible when = is positive nd lrge, s we ssumed here. A similr conclusion holds when =; =z be negtive nd we replce ; z with e +i ; ze +i Numericl veriction of the results. To verify numericlly the uniform expnsion we hve used the rel representtion (3.) for ( ; z) with lrge positive vlues of nd z. We hve used the recursion formul ( ; z) + z ( ; z) = sin ez ( + ) (4:) for testing the results. This reltion esily follows from the well-known recursion (; z) ( + ; z) = z e z nd the reltion between (; c) nd (; z) given in (.). To void overow nd strong oscilltions we hve normlized the function (; z) by using untity e (z) dened by ( ; z) = z cos + sin () e z e (z): (4:) The recursion reds for this new function reds e + (z) + e (z) + = ; = z : (4:3) Lrge untities in the right-hnd side of (3.) re z (3.) by e z () we cn use the reltion nd z exp( ), nd when dividing z e e z () = r () ;

10 where () is dened in (.5). This gives e (z) = () F + T () ; (4:4) Dwson's function F nd () re computed with extended precision (bout 9 relevnt digits). We hve used expnsion (3.) in (4.4) nd truncted the series fter the term C 6 ()= 6, with the expecttion tht the order of mgnitude of the reminder is bout 4 if. We hve used the representtions of the coecients C ; : : : C 6 s given in Section 3 of Temme (979). For jj we hve used Mclurin expnsions of the coecients. An extensive set of Mclurin coecients is given in Didonto & Morris (986). We hve used enough Mclurin coecients in order to obtin 4 digits ccurcy for C on the intervl jj, digits for C, etc., with the intention to use the symptotic expnsion if. In the numericl veriction we xed = nd took z in intervls round the point, the trnsition point. In fct we used rndom numbers for z in intervls [( k ); (+ k )], for k = ; ; : : : 6. All computed left-hnd sides of the recursion reltion (4.3) were (in modulus) less thn 5:5 8. Vlues of e (z) nd ( ; z) round the turning point = z = re given in Tble 4., together with the error in the computtion of e (z). z e (z) ( ; z) error e e{ e+96.7e{ e e{ e+97 7.e{ e e{ e e{ e+98.4e{ e e{ e+99.e{ e+99.e{ e+ 7.e{9. { e+ 8.e{9. { e+ 9.5e{9 3. { e+ 4.6e{9 4. { e+ 8.7e{9 5. { e+ 5.4e{ 6. { e+.e{8 7. { e+.e{8 8. { e+3 4.9e{9 9. { e+3.7e{9. { e+3.e{8 Tble 4.. Vlues of e (z) of (4.) nd ( ; z) for = :5 with z vlues round the trnsition point = z; the "error" is the computed left-hnd side of (4.3). Nottion: e+96 mens : , 5.3e{8 mens 5:3 8.

11 Bibliogrphy [ ] Btemn Mnuscript Project (953). Higher trnscendentl functions, Vols. I, II, III. A. Erdelyi et. l. (eds.) McGrw-Hill, New York. [ ] Berry, M.V. (989). Uniform symptotic smoothing of Stokes' discontinuities, Proc. R. Soc. Lond. A, 4, 7{. [ 3] DiDonto, A.R. & Morris Jr, A.H. (986). Computtion of the incomplete gmm function rtios nd their inverse. ACM Trns. Mth. Softw., [ 4] Dunster, T.M. (994). Asymptotic solutions of second-order liner dierentil eutions hving lmost colescent turning points, with n ppliction to the incomplete gmm function. To pper in: Royl Society of London Proceedings: Philosophicl Trnsctions. [ 5] Dunster, T.M. (994b). Asymptotics of the generlized exponentil integrl, nd error bounds in the uniform symptotic smoothing of its Stokes' discontinuities. To pper in: Royl Society of London Proceedings: Philosophicl Trnsctions. [ 6] Olver, F.W.J. (99). Uniform, exponentilly improved, symptotic expnsions for the generlized exponentil integrl. SIAM J. Mth. Anl., 46{474. [ 7] Olver, F.W.J. (99b). Uniform, exponentilly improved, symptotic expnsions for the conuent hypergeometric function nd other integrl trnsforms. SIAM J. Mth. Anl., [ 8] Olver, F.W.J. (994). The generlized exponentil integrl. In: Approximtion nd Computtion: A Festschrift in Honor of Wlter Gutschi, 497{5, R.V.M. Zhr (ed.), ISNM 9, Birkhuser. [ 9] Pris, R.B. (994). An symptotic representtion for the Riemnn zet function on the criticl line. Proc. R. Soc. Lond. A 446, 565{587. [ ] Temme, N.M. (975). Uniform symptotic expnsions of the incomplete gmm functions nd the incomplete bet function. Mth. Comput., 9, 9{4. [ ] Temme, N.M. (979). The symptotic expnsions of the incomplete gmm functions. SIAM J. Mth. Anl., [ ] Temme, N.M. (996). Specil functions: An introduction to the clssicl functions of mthemticl physics. Wiley, New York. [ 3] Wong, R. (989). Asymptotic pproximtions of integrls. Acdemic Press, New York.