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2m m Therefore the coefficients a, a 0 , au ^ a m, ... T(v 40 SPECIAL FUNCTIONS [Ch. 1 1 ν We take α0 = [2 Γ(Ν+ l ) ] " and define the Bessel function J v (z) as M . 38) \ 2 / nt o η\Γ(Ν + η+ 1) the series being convergent for any complex z. 36). (')- £ ( _ , r . 39) J . Q ( Z ) = ( - I Y J Q { Z ) . The Bessel function X{Z) = ^ ν π Λ ( ζ ) - . 36), for any v. 17) we can write i m 1 (z/2YV( iv( z / 2 2 e i d? ) "f where C is the contour shown in fig. 2. The order of summation and integration can be reversed as is readily seen, so that ® ( — 1 Y» and 1 Γ /zY ε ζ _ ζ 2 / 4ζ 2 / 7 N \ Ch.

91) 58 SPECIAL FUNCTIONS [Ch. 1. Some integral representations for HankeFs functions We start from the representation 2 ν ζ6 - ( ζ / 4 ζ ) If we make the change of variable C = ^zu, the new path of integration is obtained when C is rotated by - a r g ζ round the centre 0 . However if | a r g z | < ^ we can turn back to the path C as seen by Cauchy's theorem and With u = e we have w j (z) = -^-7 f exp{z sh w— vw}dw 2πι J v L 2π J . 9 Ch. 11 60 SPECIAL FUNCTIONS [Ch. 5 — ^π, +^π;^π, — ioo+^π or any path in the strip |Re v\ <\n which joins ioo— jK to — ioo+^π.

20b) Ch. 4] RELATION BETWEEN Γ ( ζ ) AND LINEAR MAPPINGS 35 X RX(G) ( A X ) Α), Λ being a real or complex constant parameter. ^ ( Θ Ι ) ^ Χ ( 9 2 ) Φ ( Χ ) ) 2 ^ Λ( Θ Ι ) [ ^ X A ! 2 E I X = ^EE = À = ' Χ Φ ( Α 2 Χ ) ] Ç ) ( f l 1I I 2X ) ( A I B 2 + B ( P ( A 1A 2X ) E L ) X and R X ( G I ) R M = R X ( G I G 2 ) . We now turn to the connexion between this group and the gamma function following N. Ja. Vilenkin's work [64]. We proved in Chapter 1 that Mellin's transform is a linear invertible J mapping from the space LT of functions

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