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THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 



Table 12.2 — Leading Terms in the Asymptotic Expansions for 



Uniz), Vn{z), Wn{z) WHEN Z = I'^p AND [ 2n ] » p' 



< arg m ^ 37r/2; and in regions la and Ih arg m is approximately 

 — 7r/2 and 3x/2, respectively. Gamma functions may be introduced 

 into the expression for B with the help of (12.4). It may be verified that 

 the functions do not change, except for neghgible terms, in crossing 

 over the boundary from la to lb (ao and ai are interchanged and B is 

 changed by the factor exp { — m-wi)). 



Since the zeros of our functions, regarded as functions of ?i, occur 

 (asymptotically) when the contributions from two saddle points cancel 

 each other, we may look at Table 12.1 and pick out regions which may 

 possibly contain zeros. Thus, Ao may equal Ai along the line | Ao | = 

 I Ai I, i.e. very nearly Re\j{h) - j{t^] = 0, in the m-plane. These Hues 

 were discussed in Item 7 of Section 10 and are plotted on the auxiliary w- 

 plane in Fig. 10.3 When plotted on the ??i -plane the lines appear as 

 shown in Fig. 12.1 The condition Re\j{h) - fih exp (-2x2))] = gives 

 the line I Aq I ?i^ I i*"Ai I for some of the zeros of Wn{i'^p)- 



yn (Z) 



Fig. 12.1 — When Un(z), V„{z), and Wniz) are regarded as functions of n their 

 zeros lie on the lines indicated when z = i^'^p. The three branches coming out from 

 7n = ?pV2 are lines along which \ Ao\ ~ \ Ai\ and the branch for IF,, (2) coming 

 down from m = is a line along which | Ao 1 = 1 i*"Ai | where Ao and Ai appear 

 in Table 12.1. 



