Temme, N.M. (Nico)

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Temme, Nico

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Traces to Tricomi in recent work on special functions and asymptotics of integrals

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stichting mathematisch centrum AFDELING TOEGEPASTE WISKUNDE (DEPARTMENT OF APPLIED MATHEMATICS) N.M. TEMME TW 239/83 APRIL TRACES TO TRICOMI IN RECENT WORK ON SPECIAL FUNCTIONS AND ASYMPTOTICS OF INTEGRALS Preprint ~ MC kruislaan 413 1098 SJ amsterdam BlBLlOTHEEK MATH!i'.~1V1TISCH CENTRUM -AMST EP.D11,M ___,.. Printed at the Mathematical Centre, Kruislaan 413, Amsterdam, The Netherlands. The Mathematical Centre, founded 11 February 1946, is a non-profit institution for the promotion of pure and applied mathematics and computer science. It is sponsored by the Netherlands Government through the Netherlands Organization for the Advancement of Pure Research (Z.W.O.). 1980 Mathematical subject classification: 41A60, 33AIS, 33A30, 33A40, 33A65, 65D20, 30E15 Copyright© 1983, Mathematisch Centrum, Amsterdam Traces to Tricomi in recent work on Special Functions and Asymptotics of *) Integrals by N.M. Tennne ABSTRACT This tribute to Tricomi contains a selection of examples which refer to his work on special functions and asymptotics of integrals. The examples include: a class of polynomials introduced by Tricomi and which is used in uniform expansions of Laplace integrals; Watson 1 s lemma for loop integrals; uniform asymptotic expansions of incomplete gamma functions; computational aspects of Tricomi's f function (confluent hypergeometric function). KEY WORDS & PHRASES: special functions, Laplace integrals, Tricomi poly-nomials, uniform asymptotic expansions of integrals, Watson's lemma, Tricomi's ~ function, inC!omplete gamma function This paper will be submitted for publication in a volume honoring F.G. Tricomi. I 1. A CLASS OF POLYNOMIALS In TRICOMI (1951) a set of polynomials related to Laguerre po-lynomials is introduced, The definition is (1. 1) l (x) = (-l)n L(x-n)(x), n n n = 0, 1 , 2, ••• , where L (a) (z) is the classical orthogonal Laguerre polynomial. Ob-n serve that in (1.1) the parameter a depends on x, giving a polyno-mial essentially different from the classical case. For instance, the degree of l is not n but the greatest integer [n/2] in n/2. n Tricomi gives a first attempt at a systematic study of the po-lynomials. The motivation for the investigations is the occurrence in several situations. For instance in an expansion of a confluent hypergeometric function in terms of Bessel functions, and in an I 2 asymptotic expansion of the incomplete gamma functions. In this section we mention two later applications in asymptotic expansions of Laplace type integrals. Before doing so we remark that in an interesting paper by GARLITZ (1958) an orthogonality relation is given for polynomials related to l (x). Carlitz defined n which is a polynomial of degree n. He proved 00 I where ~(x) is a step function with the jump i·j-1 -j/'' :2J e J• at the point I.I. Uniform asymptotic expansions +•-! -J , j = 1,2, •.•• The following integral is important in this connection 00 ( I • 2) I J A-I -t n Pn(A) = f(A) t e (t-).) dt. 0 By expanding (t-A)n it easily follows that n P (A) = I (n) r (A +m) (-).) n-m. n m=O m f(A) By comparing this with the known representation L (a) (x) = n n m \ (n+a) (-x) l n-m m! m=O it follows from (I.I) that l (x) = P (-x)/n!, n n n=0,1, •.•• We have PI (A) = 0, and there is a recursion P +l (1t) = n[P (1t)+AP l (1t) ], n n n-n ~ 0. This follows from (1.2) or from a well-known relation for L (a) (x). n The polynomials P (A) are used by the present author to obtain n an asymptotic expansion for the Laplace integral 00 ( 1 • 3) 1 ( 1t-l -zt F"(z) = f(1t) j t e f(t)dt, 0 Re z > O, Re A> O, z large, and where" may be large as well. It is shown in TEMME (1982a) that an expansion is possible,uniform with respect to the parameterµ := A/z, µ ! [0, 00), with extension to com-plex values ofµ. The expansion is obtained by expanding f(t) at t =µ,at which • A -zt point the dominant part of the integrand of (1.3), that ·is, t e , attains its maximal value (considering only real parameters, for the moment). We write 00 (1 .4) f(t) = I s=O s a (µ) ( t-µ) s and obtain by substituting this in (1 .3) the formal result 00 ( I • 5) FA (z) ~ 1 s=O a (µ) p ()..) s s -s-1t z z + 00. To investigate the nature of this expansion we suppose that f is holomorphic in a connected domain n of the complex plane with the following conditions satisfied: (i) the boundary an is bounded away from [O,00); 3 4 (ii) n contains a sector S 0 , with vertex at t = 0, defined by a,µ S = {t Et I - a< phase (t) < S}, a,S where a and Sare positive numbers; (iii) f(t) = O(tp) as t • 00 in S , where pis a real number. a,S The uniformity of the expansion (1.5) holds with respect to µ in a closed sector, with vertex at t = O, properly inside S O and con-a,µ taining t = 0 as a boundary point. This is proved in TEMME (1982a). Moreover, error bounds for the remainders are given. A simple example is f(t) = 1/(l+t), in which event (1.3) is an s s+l exponential integral and a(µ) = (-1) /(!+µ) • The sector S O is s a,µ defined with a= S = n-€ (€ small). We have ( l • 6) I s=0 (z+>-.) s+l where EA(z) is the well-known exponential integral. Expansion (1.5) is very useful when, apart from large values of z, also large values of A are used. In fact it is useful too, when A+ oo, uniformly with respect to z, z ~ z0 > 0. The example (1.6) shows quite well why the uniformity in A (or inµ) holds. The degree of the Tricomi polynomial P (>-.) is [s/2], which is amply s absorbed by the denominator. An important feature of the uniform expansion (1.5) is that it modifies Watson's lemma for Laplace integrals in which f is ex-panded at t = 0, giving FA (z) 00 'i' (0) r(),+s) l as r (>-.) s=0 -s-:>-z This expansion is useless when A is large, say A= O(z). The con-ditions on fin Watson's lemma are less restrictive, however, than for the uniform case. The relation between the polynomials P (>-.) in (1.2) and (1.5) n and the l (x) introduced by Tricomi was kindly brought to my n attention by Prof. E. Riekstin~ (Riga, Russia). 1.2. Expansions and estimations for remainders 1n asymptotic series The polynomials l (x) of (I .1) or P (A) are used for so called n n converging factors in asymptotic expansions. In this technique the remainder of an expansion is re-expanded. For information on this point, with some historical details, we refer to OLVER (1974, Ch.14). The use of the polynomials {P} is pointed out by BERG n (1977) and quite recently by RIEKSTIN~ (1982). To describe the idea, we follow the latter paper and consider We suppose 00 FA (z) = f 0 that there n-1 f(t) = I s=O tA-le -ztf(t)dt. is an expansion A A at s + R (t)t n, s n which approximates fat t = O. That is, R (t) = 0(1) as t + 0, n Re As+l > Re As. Furthermore it 1s asstnned that Re(A+A 0) > 0. Then n-1 I s=O 00 a s f(A+A ) ___ s_ + E (z), ;\+;\s n z n -zt J A+;\ -1 En(z) = t e Rn(t)dt. 0 We suppose that the integrals FA (z) and En(z) are defined when Re z 1s large enough. Re-expanding R (t) as in (1.4) withµ= (;\+;\ )/z n n gives again an expansion involving the polynomials P (A). n Riekstin~' expansion is somewhat different; he takesµ= I and sup-poses A+ A - z ~ 0 (when A,A and z are real), assuming that there n n exists certain relation between z and the index n of the remainder. As an example the confluent hypergeometric function of the second kind (also called Tricomi's ~ function) is treated. An important aspect 1n Riekstin~' approach is that an exact error bound is given for the remainder in the expansion for 5 6 the original remainder E (z). n 2. LOOP INTEGRALS FOR SPECIAL FUNCTIONS In TRICOMI & ERDELYI (1951) an expansion is given for a ratio of gamma functions. The asymptotic expansion is ( 2. I) r(z+a) r(z+S) 0:, I j=O (-I)Jr(S-a+j) (a-S+l) a-S-j r(S-a)j~ Bj (a)z ' as z + "", Jarg(z+a) J < n, and the B~µ)(z) are the generalised J Bernoulli polynomials defined by ezt(t/(et-l))µ = ; tJ (µ) ( ) l -----r- B. z , j=O J. J It I < 2n. The proof is based on a loop integral for the Beta function, viz. ( 2. 2) r(z+a) r(z+S) (O+) = r(a+l-S) f 2ni --00 vz e f(v)dv, f( ) av( v l)s-a-1 v = e e - • + The notation f(O) is a loop integral where the path of integration --00 starts at t = --oo (arg t = -n), encircles the origin once in the counterclockwise direction, and returns to --oo (arg t = n). FIELDS (1966) has shown that a shift of z in the inte3ral (2.2) produces an expansion in negative powers of (z+½(a+S-1)) 2, this ex-pansion essentially being an even one. This is interesting from a computational viewpoint. These results are summarized in LUKE (1969). OLVER (1974) has generalized the analysis leading to (2.1) into a useful result known as Watson's Zerrma for loop integrals, by considering (2.2) with more general function f. In this section we point out that for several special functions a loop integral representation exist, which is in some sense reciprocal to a Laplace type integral for a related function. A beautiful example l.S the pair 00 r(z) I -t z-1 = e t dt, Re z > 0 0 (2.3) (O+) f t -z I/ r ( z) = e t d t, 27Ti Z E (C. The second integral 1.s known as Hankel's loop integral for the re-ciprocal of the gamma function. TRICOMI & ERDELYI (1951) used it for (2.1) by expanding f of (2.2) in powers of v. OLVER (1974, p.120) used it for the general case. Observe that the second of (2.3) has no restriction with respect to z; in this way it is more powerful than the first one. Another example is the pair representing Riemann's zeta func-tion 00 r(z) = I I [et-I J-1 z-ld ? r(z) t t, Re z > I 0 (2.4) -z -t -I t [e -!] dt, z "f o. -00 The contour does not enclose the poles at t = ±2mri, n E JN • The restriction z f O in the second integral 1.s due to the pole of r(z) at z = O; all remaining singularities at z = -n, n E JN, are re-moved by the integral. The relation between the integrals is now given by the functional equation z-1 z;(z) = 2(21r) sin(½,rz)r(l-z)z;(l-z). For the ratio of Gamma functions we have the pair 7 8 00 r (z) f(z+A) = r(~) f e-zt(l-e-t)A-ldt, 0 (2.5) r) r (z+A) = f(A) e zt (1-e-t) ->,_dt r (z+ l) 2'!Ti -0, where the second one is the same as (2.2). For modified Bessel functions we have 00 -1 -v z.. I I -zt >i.-1 'IT 2 (½z) e1\(z) = r(>i.) e [t(t+2)] dt, (2.6) 0 (0 +) 1 v-1 -z r(>i.) J zt ->,. '1T 2 (!z) e Iv_ 1(z) = 2ni e [t(t+2)] dt, where v =>,_+½.The reciprocity between the integrals is now re-flected in the Wronskian relation ( 2. 7) I 1 (z)K (z) + I (z)K 1(z) = 1/z. v- \) \) v-A pair of parabolic cylinder functions satisfies 2 -v !z z e D ( z) -v In the second integral the contour cuts the real t-axis at a posi-tive t-value. A relation between the functions is again given by a Wronskian determinant D (z)D' (-z) - D (z)D' (-z) -v -v -v -v Finally we mention the pair '¥(a;b;z) -zt a-I b-a-1 e t (l+t) dt (2.8) b-1 -x.. x e M(a;b;z) -oo for Tricomi's \\' function and the other confluent hypergeometric function (Kununer's function). Again there is a Wronskian relation '¥(a;b;z)M' (a;b;z) - ll''(a;b;z)M(a;b;z) As in TRICOMI & ERDELYI (1951), asymptotic expansions for the loop integrals in (2.6), (2.7) and (2.8) follow by expanding part of the integrand at t = 0 (for (2.7) an elementary transformation is needed before doing so). This leads to well-known asymptotic expansions of the special functions concerned. 3. INCOMPLETE GAMMA FUNCTIONS TRICOMI (1950) opens his paper on incomplete gamma funcions with the remark that, for some time past, he used to call the in-complete gamma function y(a,x) as the Cinderella of special func-tions. In that paper he gives interesting results for the asympto-tic behaviour of y(a,x), for instance when a and x are both large and of the same order. He found, among others, (3. I) l y(a+l,a+y(2a) 2 ) __ ½ r(a+l) 2 l 2 -y -I erfc(-y)-l/3(2/aTI) 2 (1+y )e +O(a ), y,a real, a++ 00 ; the expansion is uniformly valid in yon compact intervals of lR. The function erfc is the complementary error func-tion defined by 00 (3. 2) erfc(x) = 2rr-½ I e-t2dt. X Some results of Tricomi are corrected and used by KOLBIG (1972) for 9 10 the construction of the zeros of the incomplete gamma functions. DINGLE (1973) has generalized the results of Tricomi, but the same restrictions on y must hold as in (3.1). Tricomi used ad-hoc methods to derive results as (3.1). Later on, important general methods were introduced to derive uniform asymptotic expansions of integrals, from which his results follow as special cases. In fact by the new methods more powerful results were obtained than by the method of Tricomi. In TEMME (1975) we used a loop integral for y(a,x) where a saddle point coincides with a pole in the 11 transition11 case a = x. The method of VAN DER WAERDEN (1951) was used to give an expansion of y(a,Aa) for a • w, which holds uniformly in A~ 0 (here we con-sider real parameters). When comparing this with (3.1) we observe l that A= I+ y(2/a) 2 • Since y ranges in a compact set we infer that in Tricomi's expansion A ranges in an interval around unity of length O(a-½). In TEMME (1982b) we used a different method, which yielded error bounds for the remainders and in which complex vari-ables can be used. We shall give a short description of the last method, which can be used for many more probability functions. It is convenient to consider the normalized function X P (a,x) y(a,x) 1 f a-1 -t := -r(-a_)_ = r(a) j t e dt. 0 A simple transformation gives (we consider a> O, x > O) x/a -a a j e a P(a,x) = r(a) -a(t-1-lnt) -I e t dt. 0 Next we define the mappings 1 r 2 -- 0 2.,, t-1-,c..nt, sign r,; = sign(t-1) (3 .3) 1,,, 2 -- 0 2,, >..-l""'l...n>.., sign ri = sign(A-1), A = x/a. With the r,; and n transformations the integral is (3 .4) P(a,x) -00 a standard form for the integrals considered in TEMME (1982b). The function f (?;) is given by a (3 .5) -a a ½ f () = e a (2n/a) a s r (a) cp (?;), cp ( ?;) = t: 1 • The asymptotic problem is to give for the integral in (3.4) an asymp-totic expansion for a+ w, which is uniformly valid with respect to n, For fixed values of n, the integral has three different asympto-tic expansions, according whether n < 0, n = O, n > 0. It is pos-sible to combine these three expansions into one, in which the er-ror function (3.2) describes the transition. An integration by parts procedure is used by taking into account the contributions in the integral (3.4) at?;= 0 (the saddle point) and at?;= n (end point of integration.) Since cp(O) = I we write n (~)½ f 2TI -w -lar2 <j>(?;)e 2 "' d?; = ! erfc(-n/a/2) - (2na)-! j -w with <1> 1(,) = d/d?;[{<j>(?;)-1}/?;]. Repeating this process we obtain (3.6) P(a,x) 2 -e_-½_a_n_ [n~l = ½ erfc(-n(a/2)½) + l hna1 s=O Bs(n) B (a,n)) ---+ n s n a a 1 B (n) = - - -o n ).-1' B() 1 + 1 + 1 1 1 n = O,-l)3 0,,-1)2 12(:>.-1) - n3' -For the remainder B bounds are available. In (3.6) the variable n n is defined in (3,3). In terms of the original variables x and a we can say that (3.6) is a uniform expansion for a+ oo, holding uni-formly with respect to x, x ~ 0, especially for x ~a.An interesting I I 12 point is that for the other incomplete gannna function Q(a,x) := r(a,x)/r(a) a similar expansion exists, viz. (3. 7) Q(a,x) 2 -1an rn-1 ~ e 2 \' = ! erf(nv'a/2) - ; 2;ra, 1 l Ls=O B8 :n) + B0 (a:n) ] a a where the B (n) are the same as in (3.6). Hence the addition rule s P(a,x) + Q(a,x) = I is reflected in the expansion for both func-tions, since erfc(z) + erfc(-z) = 2. 4. COMPUTATIONAL ASPECTS OF TRICOMI' s 11' FUNCTION The '!'-function can be defined by <X> (4. I) 1 j a-1 -zt b-a-1 'l'(a;b;z) = r(a) t e (l+t) dt, 0 this representation being valid for Re a> 0, Re z > O, b E ~- In general, i.e., for general a and b, 'l'(a;b;z) is singular at z = 0. With respect to a and b, 'l'(a;b;z) is an entire function. For a= 0,-1,-2, ... , it can be written in terms of Laguerre polynomials 4' (-n;a.+ 1; z) = (-l)nn~ L (a.) (z) n ' hence Tricomi's polynomial (I.I) is a special case of Tricomi's 11' function. If b-a-1 = n, n = 0, 1,2, ... , it also reduces to an elemen-tary function. It easily follows from (4.1) that 11'(a;a+n+1 ;z) = where (a)k = r(a+k)/r(a), k = 0,1,2, ... Information about the 4' function can be found in TRICOMI (1954), BUCHHOLZ (1953) and SLATER (1960). Computational aspects are con-sidered by FIELDS & WIMP (1970), WIMP (1974), LUKE (1969) and TEMME (1983). In the latter a computer program is given for compu-K ting a set of functions {4'(a+k;b;z)}k=O' for b E JR, z > O, a 2c'. O. In the publications of Fields, Wimp and Luke the emphasis is on ex-pansions of the~ function, which enable computation for a wide range of the parameters. An interesting result is 00 (wz)a~(a;b;wz) = l n=O * C (z)T (1/w) n n where r*(x) is a shifted Chebyshev polynomial and in which the co-n efficients C (z) easily follow from a four-term recurrence relation. n The expansion is given in LUKE (1969), together with a device for computing C (z) (a Miller algorithm). n REFERENCES BERG, L., (1977), Zur Abschi:itzung des Restgliedes in der asympto-tischen Entwicklung des Exponentialintegrals, Computing, ~. p.361-363. BUCHHOLZ, H., (1953), Die konfluente hypergeometrische Funktion, Springer Verlag, Berlin/ Gottingen / Heidelberg. GARLITZ, L., (1958), On some polynomials of Tricomi, Boll. Un. Math. Italiana, Ser. III, _!l, p.58-64. DINGLE, R.B., (1973), Asymptotic expansions: Their Derivation and Interpretation~ Academic Press, London/ New York. FIELDS, J.L., (1966), A note on the asymptotic expansion of a ratio of gamma functions, Proc. Edinb. Math. Soc. II, Ser . ..!2_, p.43-45. FIELDS, J.L. & J. WIMP, (1970), Rational approximations to Tricomi's ~ function, in: Analytical Methods in Mathematical Physics (R.P. Gilbert & R.G. Newton (eds.)), Gordon and Breach, New York. KOLBIG, K.S., (1972), On the zeros of the incomplete gamma function, Math. Comp. 26, p.751-755. 13 14 LUKE, Y.L., (1969), The special functions and their approximations, Vols. land 2, Academic Press, London/ New York. OLVER, F.W.J., (1974), Asyrrrptotics and special functions~ Academic Press, London/ New York. RIEKSTIN~, E., (1982), The method of Stieltjes for error bound of the remainder in asymptotic exapnsions (Russian, pre-print), Akademiya Nauk Latviiskoi SSR, Institut Fiziki, Lafi-052. SLATER, L.J., (1960), Confluent hypergeometric functions, Cambridge University Press, Cambridge. TEMME, N .M., ( 1975), Uniform asymptotic expans1,ons of the incom-plete gamma functions and the incomplete beta function, Math. Comp. ~. p.1109-1114. TEMME, N.M., (1982a), Uniform asyrrrptotic expansions of Laplace in-tegrals (preprint), Report TW 224/82, Mathematisch Centrum, Amsterdam. TEMME, N.M., (1982b), The unifom asyrrrptotic expansion of a class of integrals related to curmtlative distribution func-tions, SIAM J. Math. Anal. _!l_, p.239-253. TEMME, N.M., (1983), The numerical computation of the confluent hypergeometric function U(a,b,z), Nu.mer. Math. (to appear). TRICOMI, F.G., (1950), Asyrrrptotische Eigenscha.ften der unvollstan-digen GaJJTmafunktion, Math. Z., ~, p.136-148. TRICOMI, F.G., (1951), A class of non-orthogonal polynomials re-lated to those of Laguerre, J. Analyse Ma th . , _!_, p • 209-23 I. TRICOMI, F.G. & A. ERDELYI, (1951), The asymptotic expansion of a ratio of gamma functions, Pacific. J. Math._!_, p.133-14 2. TRICOMI, F.G., (1954), Funzioni ipergeometriche confluenti, Edizioni Cremonese, Rome. WAERDEN, B.L. VAN DER, (1951), On the method of saddle points, Appl. Sci. Res. Ser. BI, p. 33-45. WIMP, J., (1974), On the computation of Tricomi's ~ funotion, Com-puting ..!l., p.195-203. 15 

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