454 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 



the path of integration L^ may be deformed into ACD without passing 

 over any singularities of the integrand of (6.2). 



In order to verify (b), we note that the entries in Table 12.3 for Wniza) 

 show that along ACD 



W^„(^:)-.4^^exp[/(^o)]. (0.7) 



Here the expressions for Wnizo) along ACD are taken to be those cor- 

 responding to the regions / b and II shown in Fig. 12.2. In making the 

 last approximation in (6.7) we have neglected the slowly varying co- 

 efficient of the exponential function in the expression (12.9) for .4n. 

 Since I ^ | « ijo we set ^ = in Un{z). Then upon using the values (9.4) 

 for Vn{^), (6.7) for T^„(2!o), and unity for w, we see that the integrand 

 of (6.2) behaves like 



rexp[-/(^o)] /Qgx 



2 sin (7rn/2)r(l -f n/2) ' 



On ACD we have, in dealing with TF„(2o), — 37r/2 < arg m ^ — 7r/2. 

 Hence, from ^o + ^i = ^~^'^^o and from J{U) + j\ti) as calculated from 

 (12.9), we have 



exp [m] = exp (![/(/(.) + fih)] + ^[.f(/o) - m]) 



(6.9) 



~ r'^^(27r)-^'-'r(-n/2) exp (^- ^ + hAm - .m] 



where we have used the second of expressions (12.10) to evaluate 

 exp [/n(l - log (w/2))/2l. Substitution of (6.9) in (6.8) shows that the 

 integrand behaves roughly like 



exp (^^ - Um - /(/i)]) . (6.10) 



The truth of statement (b) then follows from the fact that the lines of 

 steepest descent of (6.10) in the /i-plane are given by Im [f(to) — f(ii)] = 

 0. To see that C is the high point of ACD we use (12.9) to show that 

 near C we have 



f(to) - m ^ (2/32o) (zo' - 2mf' 



where m = n -\- 1. Consequently, /(/o) — f(ti) is real and positive on 

 AC [where, near C, arg (^o" — 2m) = — 7r/6] and on CD [where arg 

 {z'o^ — 2m) = — 37r/2, m being in region IT according to the convention 

 used in (6.7)]. That C is the high point now follows from (6.10). 



In accordance with statement (a), we must study the form assumed by 

 the integrand of (6.2) when n is near point C. 



