86 BELL SYSTEM TECHNICAL JOURNAL 



As Xo and 5 increase from to oo , w and / of course being fixed, we have the 

 following behavior: 



(Al-14) 



[t is seen that vi ^ and V2 ^ 0. 



Putting aside for the moment the factor (1 — z')"^ in (Al-10), the path of 

 steepest descent through the saddle point vi is one of the two curves specified 

 by equating the imaginary part of F(v) to zero. Introducing polar coordi- 

 nates gives 



ie 

 V = pe 



Real F(v) = (sp + l/p) cos 6 — log p — s — t 



Imag. F(v) = (sp — l/p) sin — 



At I'l, Q = ^, p = vx. Imag. F{v-^ = and, from (Al-12), 



Real F{v^ = (25Z'i — 1) — log zji — 5 — / 



= (1 + 4^/)i/2 _ log 1,1 - 5 - / 



(Al-15) 



(Al-16) 



The path of steepest descent through Vx may be obtained in polar form 

 by solving 



{sp - t/p) = e/s,\n e (AM 7) 



for p as a function of 6. Setting ip = d esc 6 and taking the positive value of 

 p leads to 



P = [^ + (^* + Asiyi']/2s (Al-18) 



As 6 increases from to tt, v? increases from 1 to co , and p starts from vx (as 

 it should) and ends at oo . Thus, the path of steepest descent through vx 

 comes in from v = — ^ -\- iir/s (when0 is nearly tt, p ~ ip/s, <p ~ 7r/(7r — B) 

 and p(7r — 0) ?^ tt/^), crosses the positive imaginary v axis and bends down 

 to cut the real positive v axis (at right angles) at z'l, and then goes out to 

 i) = — 00 — i-k/s along a similar path in the lower part of the plane. It thus 

 avoids the branch cut (which we take to run from — oo to 0) in the v plane 

 necessitated by the term log v in F{v). Since yn and 5 are positive the path of 

 integration K in (Al-10) may be made to coincide with the path of steepest 

 descent when vx> \. This corresponds to the case in which x^ C x as (Al-14) 



