# Read e-book online Asymptotic Wave Theory PDF

By MAURICE ROSEAU (Eds.)

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**Extra resources for Asymptotic Wave Theory**

**Example text**

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?

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?

2 p v + With p! 76) We now consider behavior of V(z, ν, K) as K-+oo ; under the assumption Re ν -hi > Ο, I arg ζ\<%π, we can write Ch. 4] BESSEL FUNCTIONS OF PURELY IMAGINARY ARGUMENT { ç+i Λ+0Ο + e- (t -iy~±dt zt 2 e- (l-f ) -Mi-e z r + f e- (l —f ) -*df-e 1 J 55 zr 2 v 3iÄ(v 2 v -^ i 7 r ( v "^ + 1 +Ρ - (i -l) -*di-e " -* } J zf e ^ +00 2 = -2iz sin27ive v v 2 7 t iv 4i Λ+ 00 (v e" (i -l) "^di 2 i ^ι ) 2 v Γ+1 -2iz e v c o s νπ e" '(l-i ) "*dr. 77) for Re v + - j > 0 and |arg z | <%iz. 72), it follows that =f(£i)(i)T ~'' e - ( , 2 i r i d ' ( 2 ·' 8 ) for Re V + -2 > 0 and |arg z | < \n.