UNIFORM ASYMPTOTIC EXPANSIONS R. Wong Department of Mathematic City Univerity of Hong Kong Tat Chee Ave Kowloon, Hong Kong for NATOèASI Special Function 2000 1
The method of teepet decent i probably the bet known procedure for ænding aymptotic behavior of integral of the form Z è1è Ièè = gèzè e f èzè dz; C where fèzè and gèzè are analytic function, i a large poitive parameter, and C i a contour in the z-plane. It wa introduced by Debye è1909è in a paper concerning Beel function of large order. Debye' baic idea i to deform the contour C into a new path of integration C 0 o that the following condition hold: èaè C 0 pae through one or more zero of f 0 èzè. èbè the imaginary part of fèzè i contant onc 0. If we write z = x + iy and fèzè =uèx; yè+ivèx; yè and uppoe that z 0 = x 0 + iy 0 i a zero of f 0 èzè, then it i known that èx 0 ;y 0 è i a addle point of uèx; yè and the new curve vèx; yè = vèx 0 ;y 0 è give the teepet path on the urface u = uèx; yè in the Carteian pace èx; y; uè. For implicity, we hall aume that z 0 i a imple zero of f 0 èzè o that f 00 èz 0 è 6= 0. On the teepet path C 0,wehave è2è fèzè =fèz 0 è, t 2 ; where t i real and uually increae monotonically to +1. Changing variable from z to t give Z 1 Ièè =e f èz 0è gèzè dz dt e,t2 dt: By expanding fèzè into a Taylor erie at z 0 and ubtituting it into è2è, we have by reverion 2 z, z 0 =,f 00 èz 0 è t + c 2 t 2 + æææ Thu, a a ært approximation, we obtain Ièè gèz 0 èe f èz 2 0è,f 00 èz 0 è Z 1 e,t2 dt; which in turn yield è3è Ièè gèz 0 èe f èz,2 0è f 00 èz 0 è : 2
If fèzè ha more than one addle point, then the full contribution to the aymptotic behavior of the integral Ièè can be obtained by adding the contribution from all relevant addle point. For intance, if fèzè ha two imple addle point, ay z + and z,, then the aymptotic behavior of Ièè i given by è4è Ièè gèz e f èz,2 +è f 00 èz + gèz,è e f èz,2,è f 00 èz, è : We hall aume that Re fèz =Refèz, è, for otherwie one of the term on the right-hand ide of formula è4è will dominate the other. For a detailed dicuion of the teepet decent method, we refer to Copon ë2, Chapter 7ë or Wong ë6, Chapter II, Sec. 4ë. The above ituation i completely changed when the function fèzè i allowed to depend on an auxiliary parameter æ; the very form of the aymptotic approximation in è4è change when the two addle point z + and z, coalece. To be more peciæc, we conider the integral Z è5è Iè; æè = gèzè e f èz;æè dz; C and uppoe that there exit a critical value of æ, ay æ = æ 0, uch that for æ 6= æ 0, the two ditinct addle point z + and z, in è4è are of multiplicity 1,but at æ = æ 0, thee two point coincide and give a ingle addle point z 0 of multiplicity 2. Thu f z èz 0 ;æ 0 è=f zz èz 0 ;æ 0 è=0; f zzz èz 0 ;æ 0 è 6= 0; and f z èz + ; æè =f z èz, ;æè=0; f zz èz æ ;æ 0 è 6= 0 for æ 6= æ 0. Since z æ! z 0 and hence f zz èz æ ;æè! 0 a æ! æ 0, the approximation in è4è i not valid in a neighborhood of æ 0. To obtain an aymptotic expanion for Iè; æè a!1, which hold uniformly for æ in a neighborhood of æ 0, Cheter, Friedman and Urell è1957è introduced in what i now regarded a a claic paper, the cubic tranformation è6è fèz;æè = 1 3 u3, u + ; where and are function of æ. Thee function are determined by the condition that the tranformation z! u i one-to-one and analytic in a neighborhood of z 0 for all æ in a neighborhood of æ 0, i.e., in a neighborhood of the two addle point. Making the tranformation è6è, 3
the integral in è5è i reduced to the canonical form è7è where C æ i the image of C and Iè; æè =e Z C æ ' 0 èuè e èu3=3,uè du; ' 0 èuè =gèzè dz du : To obtain an aymptotic expanion for the lat integral, we ue a method of Bleitein è1967è and write è8è ' 0 èuè =a 0 + b 0 u +èu 2, èè 0 èuè; where the coeæcient a 0 and b 0 can be determined by etting u = æ p on two ide of the equation. Inerting è8è in è7è give è9è Iè; æè =e V è 2=3 è a 0 + V 0 è 2=3 è b 0 1=3 + I 1è; æè ; 2=3 where è10è and è11è Z V èè = e v3=3,v dv C æ Z I 1 è; æè = èu 2, è e èu3=3,uè è 0 èuèdu: C æ For implicity, let u aume that the coeæcient in è6è i real, and that the contour C æ can be deformed into one which begin at 1 e,i=3, pae through p, and end at 1 e i=3. Thu, we have è12è V èè = Ai èè; where Aièè i the Airy function. To the integral I 1 è; æè we apply an integration by part, and the reult i I 1 è; æè = 1 Z è13è ' 1 èuè e èu3=3,uè du; C æ 4
where ' 1 èuè =è 0 0 èuè. In view of the factor 1 in è13è, it i anticipated that the integral I 1è; æè i of a lower aymptotic order than the ært two term on the right-hand ide of equation è9è. Hence, a a ært approximation, we obtain from è9è and è12è è14è Iè; æè e Ai è 2=3 è a 0 +Ai0 è 2=3 è b 0 : 1=3 2=3 The integral I 1 è; æè i exactly of the ame form a the one in è7è. Hence, the above procedure can be repeated, and will lead to an inænite aymptotic expanion. A detailed dicuion of thi method can be found in Bleitein and Handelman ë1, Chapter 9ë or Wong ë6, Chapter VIIë. In recent invetigation of aymptotic behavior of ome orthogonal polynomial, we have encountered ituation in which there are two critical value of æ, ay æ + and æ,, uch that for æ 6= æ æ there are two ditinct addle point z + and z, of multiplicity 1,but at æ = æ æ, thee two point coincide and give addle point æ of multiplicity 2. Thu, è15è f z è æ ;æ æ è=f zz è æ ;æ æ è=0; f zzz è æ ;æ æ è 6= 0; and è16è f z èz + ;æè=f z èz, ;æè=0; f zz èz æ ;æè 6= 0 for æ 6= æ æ. The following example provide concrete illutration of uch ituation. EXAMPLE 1. one can get the integral repreentation where x = næ; æ 2 è0; 1è, è17è Meixner polynomial m n èx; æ;cè; ee ë3ë. From their generating function, 1 n! m nèx; æ;cè = 1 Z, nf èz;æè dz e zè1, zè æ ; fèz;æè =æ log 1, z,æ logè1, zè, log z c and, i a circle centered at the origin with radiu le than min è1; jcjè. EXAMPLE 2. Meixner-Pollaczek polynomial M n èx; æ;è; ee ë4ë. By the ame argument, one alo ha the integral repreentation 1 n! M nèx; æ;è= 1 Z ëè1 + æzè 2 + z 2 ë,=2 nf èz;æè dz e z ; C 5
where x = næ; æ 2 è0; 1è, è18è z fèz;æè =æ tan, log z 1+æz and C i a circle centered at the origin with radiu 1= p 1+æ 2. If we put z 0 =,æ + i 1+æ 2 and z 0 = r 0 e i 0 with r 0 = 1 p 1+æ 2 ; then è18è can alo be written a è19è fèz;æè = æ 2i log èz, z 0è, æ 2i logèz, z 0è, log z + æè, 0 è: EXAMPLE 3. repreentation where p=q; æ x=n; n=n, Krawtchouk polynomial K èn è n èx; p; qè; ee ë5ë. They have the integral Kn N èx; p; qè = pn,æn Z C Nfèz;æè dz e z ; è20è fèz;æè =è1, æè logè1, zè+æ logè + zè, log z and C i a mall cloed contour urrounding z =0. In all three example above, the large variable i n èor Nè. A imple function which exhibit two addle point coalecing at two ditinct place i given by è21è èèu; è =, log u + u, u2 2 : The addle point occur at è22è u æ = æ p 2, 4 ; 2 and they coincide when = æ2 p. If we put d =,, 1 2 èé0è and z = p in the integral repreentation of the parabolic cylinder function è23è Uèd; zè =,è 1 2, dè e,z2 =4 e zu, 1 2 u2 u d, 1 2 du; 6
we obtain è24è Uè,, 1 2 ; p è=,è +1è e,2 =4,=2 e èèu;è du u : We now return to the integral Iè; æè in equation è5è, and uppoe that fèz;æè atiæe the condition in è15è and è16è. To derive an aymptotic expanion for Iè; æè, a!1, which hold uniformly in a region containing both æ + and æ,,we compare it with the integral in è24è. Thi ugget that we make the tranformation z $ uèzè deæned by è25è fèz;æè =èèu; è+æ; where æ i a contant to be determined, and require uè0è = 0. Changing variable from z to u, the integral in è5è become è26è Iè; æè =e æ èuè e èèu;è du u ; where è27è èuè =gèzè è uèu; è f z èz;æè u: The contour C in the z-plane hould ært be deformed into a teepet decent path; it will than be mapped into the loop path hown in è26è in the u-plane. Put 0 èuè =èuè, and write è28è 0 èuè =a 0 + b 0 u +èu, u èu, u, èh 0 èuè; where u + and u, are given in è22è. By etting u = u + and u = u, on two ide of the equation, one ænd that the coeæcient a 0 and b 0 can be expreed in term of 0 èu and 0 èu, è. For implicity, let u deæne the new function è29è W èx; è e x2 =4 Uè,, 1 2 ;xè: Clearly and from è23è W è p ;è= W x è p ;è=,è +1è,è +1è,=2 1=2,=2 e èèu;è du u ; e èèu;è du: 7
Subtituting è28è in è26è give è30è where Iè; æè = " 1 =,è +1è =2 e æ a 0 W è p ;è+ b 0 p W x è p ;è+" 1 ;,è +1è An integration by part give,=2 e èèu;è èu, u èu, u, èh 0 èuè du u : " 1 = 1,è +1è,=2 e èèu;è 1 èuè du u ; where 1 èuè =uh 0 0 èuè. Neglecting the error term " 1,wehave from è30è, a a ært approximation, Iè; æè,è +1è =2 e æ a 0 W è p ;è+ b 0 p W x è p ;è : Thi proce can again be repeated to yield an inænite aymptotic expanion. REFERENCES 1. N. Bleitein and R. A. Handelman, Aymptotic Expanion of Integral, Holt, Rinehart and Winton, New York, 1975. èreprint in 1986 by Dover Publication, New York.è 2. E. T. Copon, Aymptotic Expanion, Cambridge Tract in Math. and Math. Phy. No. 55, Cambridge Univerity Pre, London, 1965. 3. X. -S. Jin and R. Wong, Uniform Aymptotic Expanion for Meixner Polynomial, Contr. Approx., 14 è1998è, 113 í 150. 4. X. -C. Li and R. Wong, On the Aymptotic of the Meixner-Pollaczek Polynomial and Their Zero, Contr. Approx., to appear. 5. X. -C. Li and R. Wong, A Uniform Aymptotic Expanion for Krawtchouk Polynomial, J. Approx. Theory, to appear. 6. R. Wong, Aymptotic Approximation of Integral, Academic Pre, Boton, 1989. 8