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2/ <(t /n)e 2 valid for 0 < t 0 . l | J0 tends to zero as n->oo, with Re ζ > 0, whence the following representation Γ(ζ)= r + oo ^ο e - ^ - M t , for R e z > 0 . 13) EXAMPLE Thus Γ(ζ) appears to be the Mellin transform of the exponential function _ i e ; as oo e S + c s i i s r ( c + ii) = J e~ e ds, c > 0 , Γ+ F(c + ii) is the Fourier transform of the rapidly decreasing function - * + c s a n c j consequently is also a rapidly decreasing function of t.

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.

W . 58) ^ Ä s i n f c C . ^ ι F r o m Kepler's equation w e have sin w = - Υ i n ι^Ζs er A s i m i l a r m e t h o d c a n b e u s e d t o e v a l u a t e a n y p e r i o d i c f u n c t i o n o f w, for i n s t a n c e c o s w: w e start w i t h t h e F o u r i e r expansion 0 c o s w = b— + Xfcfcc o s KC 2 ι where 2 = 1 Γ — Η J 2 π - π η Γ 1 Γ πjn π cos wd£ = — π c o s w c o s fcCdC = J n f π/c / 0 cos w)cos wdw = ( 1 — E 2 -τ πκ cos(/cC-w)dw - Γ π s i n /e£ s i n w d w J 0 cos(/eC + w ) d w ^ ο hence QO cos w = J Σ γ ( Λ - Ι ( ^ ) - Λ + Ι ( ^ ) ) COS K C .

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