Temme, N.M. (Nico)

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stichting mathematisch centrum AFDELING TOEGEPASTE WISKUNDE (DEPARTMENT OF APPLIED MATHEMATICS) N.M. TEMME A NOTE ON TWO INTEGRALS RELATED WITH BESSEL FUNCTIONS Preprint ~ MC TW 199/80 MAART 2e boerhaavestraat 49 amsterdam PJunte.d a;t the. Ma.:the.ma,t,Lcal Ce.ntJz.e., 49, 2e. Boe.tt..ha.a.ve/4.tnaAt, Arn.tde.Jtdam. The. Ma.:the.ma,t,Lc.a..l Ce.n:tJte., ooun.de.d .t.he. 11-.t.h 06 Fe.btt..u.aJLy 1946, .l6 a. n.on-ptt..06,i.:t 1..no:tU:u,tfon. a,,i.m,Ln.g a;t .t.he. ptt..omoUon. o,6 puJte. ma.:the.ma.UC6 a.n.d .l.t6 a.ppUc..a.:t,i,ono. I.t ,L6 .t.iponoott..e.d by .t.he. Ne.thvif.a..n.d.o Gove.tt..n.me.n:t .t.htt..ou.gh .t.he. Ne..thvif.a..nd.o Ott..ga.Mza.Uon oott.. the. Adva.nc.eme.n-t on PUite. Re/4e.a.tt..c.h (Z .W.0) • 1980 Mathematics subject classification: 33A30 44A20 A note on two integrals related with Bessel functions*) by N.M. Temme ABSTRACT Two integrals are considered that can be expressed in terms of Bessel functions. Mellin transform techniques are used for proving the identities. KEY WORDS & PHRASES: Bessel function integrals, confluent hypergeometric function, Mellin transform This report will be submitted for publication elsewhere. 1 1. INTRODUCTION In this note we consider the integrals 00 J 5 3 -),_2t"2 G(r,>-.) = s sinrs e "d~ 0 (1. 1) 00 f 5 3 -A.2~2 H(p,>,.) = s J 0 (p~ )e ds. 0 2 We assume that rand pare real and A E ~ such that Re>,_ > O; J 0 is the. ordinary Bessel function. We consider first the integrals as given above, after that we treat more general forms of these integrals. The functions G(r,>-.) and H(p,>-.) were presented to the author by dr. R.A. Pasmanter of Rijkswaterstaat (The Hague) who met them in his re-search on diffusion in highly turbulent fluids (see [3]). They occur as terms in expansions of the Green functions for the solutions of two and three dimensional turbulent diffusion problems. We write these functions first as confluent hypergeometric functions and we give expansions for the numerical evaluation. Then the relation with modified Bessel functions is described. The functions G and H can be viewed as Laplace transforms or as sine and Hankel transforms. Standard tables do not give information about them and it was rather surprising to the author that the combination of ~5 , ~3 and ~2 terms yielded a confluent hypergeometric function. Apparently other combinations of s-powers will do the same. 2. MELLIN TRANSFORM TECHNIQUES The results of this note can be obtained in several ways. The Mellin-approach was found to be most convenient. The Mellin-transform of a func-tion f is written as 00 ( 2. 1) -f(z) = I z-1 t f(t)dt. 0 2 If f is continuous on (0, 00 ) and if we know that f (t) t -+ (X) where a and Sare real numbers (a< S) then f exists for a< Re z < S, and perhaps for a wider domain of Re z. . + The behaviour of G(r,A) and H(p,A) for r,p-+ O is as follows r, p -+ oo we G(r,A) = O(r), H(p,)..) = 0(1), from the behaviour of sin(rs 3 ) and 3 J 0 (Ps > for r, p -+ 0. For obtain by partial integration co G(r,A) = ;~ J 3 -A2s 2 3 s e a cos rs = 0 00 00 1 I 2 3 -A 2s2 n2 I 4 3 -A2s 2 = s cos rs e ds s cos rs e d.; r 3r 0 0 = 0 (r -l) , 00 1 J A. 2 2 3 3 H(p,)..) = e- s d[s J 1 (Ps )] = 3p 0 00 = 2A2 I 3p 0 -1 0 (p ) , where we used dd {z J (z)} = z VJ 1 (z) . From this we conclude that the z \) \)-Mellin transforms of G and H with respect tor and pare 00 G(z,)..) = J z-1 r G (r, A) dr, -1 < Re z < 1 (2. 2) 0 00 H(z,A) = I z-1 p H {p, A) dp, 0 < Re z < 1, 0 where the ranges of Re z may be larger than indicated (for instance, by a refinement of the integration by parts process it follows that H(p,A) = O(p-8/3)' P +co). The next step consists in substituting (1.1) in (2.1). This leads to the question under which conditions is the equation 00 00 00 00 (2. 3) f {f f(x,y)dx}dy = f {f f(x,y)dy}dx 0 0 0 0 correct. A well-known result of de la Vallee Poussin is (see for instance BROMWICH [2, §177]): (2.3) is correct, provided that both the integrals 00 J lf(x,y) lax, 0 00 J If (x, y) I dy 0 are convergent and that either of the repeated integrals (2.3) converges absolutely. When using estimates for sin and J 0 in the form (2. 4) I sin x I X ~ 0, 3 where A and Bare assignable numbers, we can easily conclude that (1.1) can be substituted in (2.2) and that the order of integration may be inter-changed. Using 00 f z-1 3 -3z r sin r~ dr = ~ r (z) sin ½'ll'z, -1 < Re z < 1, 0 00 I z-1 3 P J 0 (p~ )dp 0 < Re z < 3/2, 0 we obtain 00 G(z,A) = r(z)sin~'ll'z ~5- 3ze-A ~ d~ = f 2 2 (2 .Sa) 0 00 I 2 2 H (z, :>,.) z-1 !;5-3ze-A !; d!; = 2 T(~z)/T(l-~z) (2. Sb) 0 ~2z-1:>,.-6+3z r (~z) 3 = r (1-~z) f(3- 2 z). The inversion formula for (2.1) reads as follows (2.6) f(t) = 1 27Ti y-ico -z-t f(z)dz, a.< y < 13-For our Mellin transform pairs (2.2) and (2.5) we obtain ( 2. 7) 1 -6 G(r,:>,.) = 4 7Ti :>,. 1 -6 H(p,A) = 47Ti A I 3z -z . 3 A r sin½'lfzf (z) r (3 - 2 z) dz f 3z -z z-1 3 A p 2 T(½z)r(3- 2 z)/f(l-½z)dz L2 where the contours of integration L1, L2 are such that (compare (2.2)) on L1 : -1 < Re z < 1, on L2 : 0 < Re z < 1. 4 The integrals in (2.7) can be written in a standard form, "standard" for our purposes. We want to write them so that they are recognized as spe-cial cases of ( 2. 8) a z T(a)f(l+a-b)U(a,b,z) = 1 27Ti c+i00 I c-i00 -s r(-s)r(a+s)f(l+a-b+s)z ds a~ 0,-1,-2, ••• , b-a ~ ·1,2, .•• and the contour must separate the poles of f(-s) from those of T(a+s) and r(l+a-b+s). The function U(a,b,z) is a con-fluent hypergeometric function. To write (2.7) in the form (2.8) we need some properties of the gamma function, viz. -5 r(z+1) = zr (z) sin 'll"Z 1 = 1TZ r (1-z)r (1+z) (2. 9) r(2z) = 'll"-~2 22- 1r(z)r(~+z) r (3z) When using this in the integrands of (2.7), we obtain sin z-1 ~ (5-3z) 4 5 ~7fZ2 r(~z)r(~+~z)3 f(1-~z)r(3 -~z)r(3 -½z) = TI½2'll"r(l-~z)r(1+½z) z-1 3 2 f(~z)r(3- 2 z) r ( 1-~z) 22 - 1r (½z) 3 ~ (S-Jz) r (1-½z) r c.!- ~z) r (~ - ~z) 3 3 =-------------------2rrr ( 1-½z) = 2\ 2z-13~(5-3z)r(~z)r(f-½z>r(;-½z). Furthermore we use in(2.7) the transformations ½ + ½z + -s, ½z + -s, respectively, which result in (2 .10) (2.11) and G (r, A) H (r, A) 1 21Ti -2 5/2 -1 -6 1 f =23 -rrA -2 . 1Tl. L2 n = on L 1 : -1 <Res< 0, 1 - 2 <Res<0. Finally, using (2.8) we arrive at 105 r -9 11/6 11 2 G(r,A) = 32 vrrr;,. I; U(6 , 3 ,i;;) (2.12) with i;; and n given in (2.11). 6 From known results of confluent hypergeometric functions further pro-perties of G and H can be derived, although some of them can be easily ob-tained from the definition (1.1). For instance, we mention the asymptotic expansion a z U(a,b,z) = R-1 (a) (l+a-b) t n n (-z)-n + O(lzl-R), l n! n=0 as z ➔ 00 , larg zl < 3TI/2. Expansions for small i;; and n follow from with 1T = U(a,b,z) sin Trb { M(a,b,z) 1-b M(l+a-b,2-b,z)} r(l+a-b)f(b) - z f(a)f(2-b) 00 M(a,b,z) = l n=0 n (a) z n (b) n ! . n This series converges for all finite z € ~- In these formulas (a) = n r (a+n) /r (a) . 7 Next we express G and Hin terms of Bessel functions. we use the re-lation (2.13) -1.:1 z -\) U(v+~ 1 2v+1,2z) = ~ e (2z) K.v(z), where K is the modified Bessel function. In order to bring the functions 'V in (2.12) into the form of the U-function in (2.13) we need some recurrence relations for the U-functions: (2.14) a(l+a-b)U(a+l,b,z) = aU(a,b,z) + zU' (a,b,z) (l+a-b)U(a,b-1,z) = (1-b)U(a,b,z) - zU' (a,b,z) (b-a)U(a,b,z) = zU(a,b+l,z) - U(a-1,b,z) (b-a-l)U(a,b-1,z) = (b+z-l)U(a,b,z) - zU(a,b+l,z) which can be found in the literature (see, f.i., ABRAMOWITZ & STEGUN [1, chapter 13]). They can also be derived from the integral 00 r (a)U(a,b,z) = I -zt a-1 b-a-1 e t (l+t) dt, 0 Re z > O, Re a> O. By using (2.15) we obtain uc~,f,n> = -[C6n+S}u(~,i,n) +½n<3n+2)U'C-f,i,n)J. Hence G and H can be written in terms of respectively, and K' (z) can be replaced by~ K (z) - K 1 (z). The result is \) z \) \!+ 8 9 -9 3/2 ½!; 2 G(r,A) = l6 r). 1; e [3!;(61;+5)K4/ 3 (½1;) - (181; +451;+20)K1/ 3 (~1;)] 1 -½ -6 ½ ~n 2 H(p,A) = 4 rr ). n e [3n(3n+2)K1116 c~n) - (9n +30n+20)K516 c~n)J. 1 Bessel functions of order 3 are in fact Airy functions. 3. GENERALIZATIONS Let us consider the integrals co I a 8 -).2~2 G(r,).;a,8) = E; sinr~ e d; 0 where the parameters are such that the integrals converge (the analysis will be rather formal here). The Mellin transforms are r (a+l-Sz) r ( v+z) - = L 2z-1,-a-1+8z 2 2 H (z,A;a,8) ~ A 2 \) re\/+ -z) 2 and the inversion gives G(r,A;a.,8) = 1 2rri f r-zG (z, Ai a, 8) dz Ll H\/(p,A;a,8) = 2~i f L2 -z~ p H ( z, A; a, B) dz. \) For specific combinations of a, B (and\/) the integrals may be reduced to combinations of (2.8). For 8 = 2,3,4, .•. it is possible to write them as a Meyer's G-function. For 8 = 3 we can use again -f i ' Hence if a= 3v + k, then fork= 1,3,5, H is a confluent hypergeometric V function, fork= 7,9, ... it is a combination of such functions. If a= 1,3,5 then G is a U-function if a= 7,9, •.. it is a linear combination of such functions. REFERENCES 9 [1] ABRAMOWITZ, M. & I.A. STEGUN (eds.), Handbook of Mathematical Functions, Applied Math. Ser. 55, U.S. Government Printing Office, Washington D.C., 1964. [2] BROMWICH, T.J.I'A, An Introduction to the Theory of Infinite Series, London, 1947. [3] PASMANTER, R.A., Diffusion in Highly Turbulent Fluids, (submitted for publication), Rijkswaterstaat, The Hague, 1980. To be published by Physics Letters (Febr. 1980). 

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