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By MAURICE ROSEAU (Eds.)

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II. 1. 36) where ν is a real or complex constant, we are looking for a solution of the form y=z* £ a z . 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?

65) tmnigigi) = Σ tmkig^hnigi), k— — oo where the series is absolutely convergent. 64) + 00 É"«Jn(r) = Σ IKV2 ^ JN-k{r,)Jk{ri) k= — oo the addition formula for Bessel functions. 66) 50 [Ch. 4. T h e B e s s e l f u n c t i o n s o f p u r e l y i m a g i n a r y a r g u m e n t They are defined by = exp{ - ν \ i} J ( z e - ) = (ζ/2)' f M 2 v = exp{v^i}j (ze- '

9. 28 THE FOURIER-LAPLACE INTEGRAL [Ch. 50) P ~ Y - D S \ S H ( P ) D P = 0 . 50) that r = log s, G ( S ) = F(S)/S, H ( P ) C I Γ+ DP P C- \ ^ S H ( P ) D P = 0, OO I J G V T H ( C + IV)DV = 0 9 W. e. 1. 1. A summation formula Let / (ζ) be a meromorphic function of ζ such that ζ / (ζ) is bounded as |ζ|->οο. It is readily shown that the integral J G / ( ζ ) cotg7tCd£ taken round the square whose vertices in the complex plane are (w + i ) ( ± l ± i ) with η an integer, tends to 0 as w->oo. If / ( ζ ) has only poles of order one at £ l 5 ζ2, £fc, none of them being an integer, we can write, by Cauchy's theorem, 2 Σ - Σ AN P;cotgnf; = 0 .

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