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THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 



we have spoken of the complex n-plane, but this is essentially the 

 m-plane shifted by unity. 



Since we are going to deal with a fixed value of z {i ' ^ or f " rj) but 

 with a variable value of m, we make fo and ti one-valued functions of m 

 by cutting the w-plane as shoA\ni in Fig. 10.2. 



It may be shown that ^o and /i lie in the opposite half-planes in- 

 dicated in Fig. 10.1. This restricts arg U to lie between arg z — t/2 and 

 arg z + 7r/2. Arg d is restricted to lie between arg z — t and arg 2 + x 

 by the cut shown in Fig. 10.1. It may also be shown that 



^ol ^ Ul 



arg to — arg ti\ ^ x. 



(10.7) 



t, REGION 



CUT FOR t, ^>> 



^ 



.^^ 



BOUNDARY BETWEEN 



to HALF-PLANE AND 



t, HALF-PLANE 



Fig. 10.1 — Diagram showing the half-plane regions to which the saddle points 

 to and ti are confined in the /-plane. 



One might wonder why cuts in the w-plane are required since it has 

 already been pointed out that Un(z), etc., are one-valued functions of 

 m = n + 1. The trouble is that the asymptotic expressions for Un{z) 

 are many-valued functions of m even though Un{z) itself is not. 



Now that we have considered the saddle points ta and /i, we turn to 

 a consideration of the paths of steepest descent in the /-plane which 

 pass through them.* The path of steepest descent which passes through 

 /o, for example, is that branch of the curve 



Im im - f(U)] = (10.8) 



for which to is the highest point (i.e., Re [f(t) - f(to)] ^ on it). The 



* Watson^s has studied paths corresponding to Re(n) > when z is any com- 

 plex number, and has given curves which are related to some of those shown in 

 Section 11. 



