By MAURICE ROSEAU (Eds.)

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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. 74) 5 4 SPECIAL FUNCTIONS [Ch. 18) a p (z,v) = 2 i e ^ - ^ r ( 2 v - 2 p ) . 75) lim 17 (z, v, K) = Κ-+00 2iz-*e ""sin2nv Τ ( - l) ( ~ * > " - ( ; ' - P * ) z * T ( 2 v - 2 p ) . 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.

1 where y is a constant chosen such that Γ(1) = 1; therefore ' - ? ( ; - " » ( 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.

2. 4 KEPLER EQUATION AND BESSEL FUNCTIONS Ch. 2] 45 ellipse, let A'A, C be the major axis and the centre of symmetry; the centre of force Ο is a focus and may be taken to be the origin of polar coordinates with r = OM, φ= L_AOM\ φ is the true anomaly. Let Ρ be the point on the circle with diameter A A' chosen such that Ρ, M have the same orthogonal projections on A A' and are on the same side of AA'\ then let us introduce the eccentricity e of the ellipse, e < 1, 2a = AA' the length of the major axis, w= L· ACP the eccentric anomaly, Ζ = n(t—t0) the mean anomaly with η = 2π/Τ where Τ is the period of motion, t0 a time when the particle was at the positive end of the major axis, t the time when it is at M.