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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.

If Re v-h-j > 0, it can be integrated term by term, whence Γπ J ο oo 2n 2 ch(zcos(p)sin >d

sin >d(p . 70) 52 SPECIAL FUNCTIONS cos "φ sin (pd(p = 2 2 [Ch. 6, the complex plane being cut along the part of the real axis from — 1 to + oo. 6 With the conventions a r g ( f + l ) = a r g ( r - 1 ) = 0 at A, the function l) ~* is holomorphic inside the contour AEBDA and by Cauchy's e~ (t theorem we have tz v 2 U + V = 0 Ch. 4] 53 BESSEL FUNCTIONS OF P U R E L Y IMAGINARY A R G U M E N T with U(z,v, R) = z*jAEBe-"(t -iy-idt V(z, v,R) = z»jBDAC-»(t -lY-*dt.

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 .