Download PDF by MAURICE ROSEAU (Eds.): Asymptotic Wave Theory

By MAURICE ROSEAU (Eds.)

ISBN-10: 0444107983

ISBN-13: 9780444107985

ISBN-10: 0444601910

ISBN-13: 9780444601919

ISBN-10: 0720423503

ISBN-13: 9780720423501

ISBN-10: 0720423708

ISBN-13: 9780720423709

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4] RELATION BETWEEN Γ ( ζ ) AND LINEAR MAPPINGS 1 Tc + ioo — ÇA ! 27) =0 &(z)H{z)az c - i oo with 1 Tci+ioo uw H(z) = r - r R(W-u)R(u-Z)b - du-R(W-Z)(b+\Y- W . Ci - i GO We can easily prove that r+oo Γ+οο \ R ( w - C I - i T ) \ - 00 \ R ( C 1 - C + I ( T - T ) ) \ D T D Z < + 00 - 00 which implies r+ 00 J |jff(c + i r ) | d t < + 0 0 . 10 we derive H(Z) = 0, or r(w-z)(^tij R(w-u)R(u-z)P-'du with Re w > d > R e z, > 0 ; changing 6, u - z , w - ζ into t, w, W respectively we get the addition formula r M( Ti)"=aC!?

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 . 1) 1 π cotg πζ = ~Ζ- ζ 00 2 Σ! 2 2 * /I — Ζ Since this series is uniformly convergent with respect to z in any bounded set we can integrate term by term : or z\ 2 K z n ( l - ^ ) e - n ( l ^ ) e - .

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 . 1) 1 π cotg πζ = ~Ζ- ζ 00 2 Σ! 2 2 * /I — Ζ Since this series is uniformly convergent with respect to z in any bounded set we can integrate term by term : or z\ 2 K z n ( l - ^ ) e - n ( l ^ ) e - .

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Asymptotic Wave Theory by MAURICE ROSEAU (Eds.)


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