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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**Additional info for Asymptotic Wave Theory**

**Example text**

48) obviously holds for complex t). 62); indeed this series is not only convergent, as asserted before, but absolutely convergent since |cj yfelU T Zl^lU = ZkllU<(l 2 2 <\\F\\-\\Txq>M\\<\\FTM\. We now apply the above arguments to addition formulas for Bessel functions following N. Ja. Vilenkin's work [64]. 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.

7. / / 2 C2 2 F(s)s -±eL (0, 1), F ( s ) s - ± e L ( l , +00) /or some c < c , then x 2 &{p) = f ^ S ^ M s 0 is holomorphic in the strip c < Re ρ < c and l F(s) = J - [ ° ζιπ J 2 s~W{p)dp , wirft c j < c < c 2 _ c i oo w/nc/z means r/zaf 1 rc+ΐω strongly tends to F(s) as ω->οο m ί/ze space 0/ measurable functions (/>(s) nor med by m c 2 i 2 = (£ °5 - i0(S)i dS) i Fwri/zermore Sup ce[ci,c 2] Γ+ 00 2 I |^(c + ii;)| di;<+00 . 8. Let ^(p) be a function holomorphic in ct < Re ρ < c2 such that r+00 Sup ce]ci,c 2[ \^(c + iv)\ dv< + 0 0 .

Ja. Vilenkin's work [64]. 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- '

### Asymptotic Wave Theory by MAURICE ROSEAU (Eds.)

by Charles

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