104 Mr Watson, Bessel functions of large order 



The stationary points of xi^m\iw — iv, qua function of w, are 

 given by cosh ?<; = !/«. As <\lx^l, we put x = sec/3 where 

 ^ /3 < I^TT ; and two stationary points are given by w= ± ^i. 



Now it has been shewn by Debye that a branch of the curve * 

 I{x sinh lu - w) — I {x sinh ijS — i/3) 



is a suitable contour for HJ^\ and the reflexion of this contour in 

 the real axisf is a suitable contour for HJ-K 



On making a change of variable by writing w = t + i^, we 

 have 



1 rQO+7r?-(/3 



where i tan /3 (cosh t — 1) + sinh t — t = — T. 



If we put t = 'u + iv, where u, v are real, the equation of the 

 contour is 



cosh w = (sin ^ + v cos /3) cosec (v + /3), 



and, on the contour, 



r = u — sec /S sinh u cos (v + /3). 



When V is given, cosh u is given and the sign of u is ambiguous ; 

 we take u to have the same sign as v, in order that the contour may 

 be of the requisite type. 



Next define T by the equation 



^TH tan /3 + ^T' = - r. 



We write T^U+iV, where U, V are real; a contour in the 

 T- plane on which t is positive is that branch of the cubic :|:, whose 

 equation is 



(t/^ _ T/2) tan ^ + 1 F(3[7- - V) = 0, 

 which passes from — cc — i tan ^ through the origin to x exp (^Tri). 



Taking this curve as the contour, we shall shew that an 

 approximation to 



r<x+Tri—ip rco exp (Jn-i) 



e-''^dt is e-'^^dT. 



.'-cxj-i/S J -00— i tan/3 



* This curve is derived from the curve shewu in fig. 2 (p. 540) of Debye's first 

 paper by turning it through a right angle and taking the origin at tlie node. The 

 reader will observe that tlie character of the contour has changed with the passage 

 of X through the value unity. 



t Since iJ^*^), HJ") are conjugate complex numbers when n and x are real, it 

 will be sufficient to confine our attention to HJ^^. 



X Of course t is real on the whole cubic ; as T traverses the specified portion of 

 it, T decreases from + oo to and then increases to + oo . 



