By MAURICE ROSEAU (Eds.)

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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 . 91) 58 SPECIAL FUNCTIONS [Ch. 1. 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 .

1 ; ) ) - s ( + 1 i + + - ; - + k , » i " + 1 ) > (24) known as the Euler constant. From the definition we have CLAIM 1. located The function Γ{z) is meromorphic at 0, — 1, — 2, ... and has only poles of order one . 3) written as - log Γ(ζ) = y + l o g z + f (log ( 1 + - ^ the series obtained through term by term differentiation upto the second order are uniformly convergent with respect to ζ in any bounded set; consequently we can write r z 1 '( ) - 7 i ) z dMog Γ(ζ) 2 dz ~ = 1 Λ n < \ ^ ç ( ^ - - j 1 - _ j _ + ζ 2 + ( · ) 2 ) ( 2 6 2 \ ( ζ 5 + η) ' ^ Hence [d logF(x)]/dx > 0 , if χ is real positive and we have 2 CLAIM 2.

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. 54) We now change the variable Ζ into Ξ through or Ζ = ± ( ΐ - ο ) [ Ξ + /Λ ( Ξ 2 + Α 2 ) ] where ^ ( Ξ + Α ) is the branch which takes real positive values for Ξ real positive.