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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**Extra resources for Asymptotic Wave Theory**

**Example text**

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? ) "f where C is the contour shown in fig. 2. The order of summation and integration can be reversed as is readily seen, so that ® ( — 1 Y» and 1 Γ /zY ε ζ _ ζ 2 / 4ζ 2 / 7 N \ Ch. 3 If R e v ^ 0 we can replace the contour C by the straight line c — ioo, c + ioo, with c any positive number, to obtain 2 ι rc + ioo / \ v ζρ - ζ / 4 ζ * « - S i L . ( ï ) - r î - * •<«·>* « > o .

86) Ch. 5] THE HANKEL FUNCTIONS or through the definition of y. 87) Kv(ze-"" ) π 2 2 //<ν >(ζ)=^-^Κν(ζε<*' ). 89) 2 //_ ν (ζ) = ε - - ' ^ ν > ( ζ ) . 90) 2 A ( z v / /^ ) ( z ) ) = z ^ < 2 1( z ) . (2) The Hankel functions //„"(z), H {z) of the Bessel equation 2 2 are a system of fundamental solutions 2 z / ' + z / + (z -v )y = 0. 85) 2iz / d sin νπ V dz d dz T the constant value of ζ W is easily reached at point z = 0 using the series expansions for Jv and J_ v ; thus we have z(a*\z)±W(z)-#»(z)± WV>(z)) = j i .

1. 2. Some elementary properties of Jv(t). 41) let us assume z— t to be real positive and make the change of variable C = iw/2; hence L . 48) for any integer rc. 47) with respect to t Ch. 49) = I(Jv-l(t)-Jv+1(t)). 47) provides and finally νΛ(ί) = ^ ( Λ - . ( 0 + Λ . 52) ( t v _ v( t ) ) ' = - r j _ ( v _ u ( r ) . All of these relations can be extended to complex values of t by analytic continuation. 2 to t ^ J ^ t ) - * — , Ρ Rev^O. 53) 2 2 = 0, z = a(t -b )K t>b, a, b real 44 [Ch. 2 SPECIAL FUNCTIONS 1 Tc + ioo 2 C A \ T 2 - B c>0.

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