DIFFRACTION OF RADIO WAVES BY A PARABOLIC CYLINDER 481 



paths of steepest descent may be shown to have the following prop- 

 erties : 



1. Let t = tr ■{- iti = r exp {id). Then the paths of steepest descent 

 either run out to /r = + °^ with ti -^ Im z or spiral in to i = as r = 

 (constant) exp { — irird/mi). 



2. The steepest descent path through U may be computed by a graph- 

 ical method based on* 



arg {dt) = diVgt - arg {t - h) - arg (t - h). 



(10.9) 



2ARGZ-77 



Fig. 10.2 — Diagram showing the cuts in the complex w-phme, w = n + 1. 



It' we draw the triangle to ti and bisect the interior angle at ^o by the 

 line 6o^o then 



arg (dt) at to = angle t if oho. 



(10.10) 



If one goes clockwise in traveling from the side tJi to tJh) then arg dt is 

 negati\'e. Likewise, arg (dt) at ti (on the path through ti) is the angle 

 between the side tJo and the bisector ^i6i of the interior angle at ti. 



3. When 7n has the critical value z'/2 the saddle points coincide: 

 ^0 = ^1 = z/2, and the paths of steepest descent start out from t = z/2 

 in the three directions arg (t — z/2) = (arg z)/3 + 5 where 8 is 0, 27r/3, 

 or -2x/3. 



4. Some of the paths of steepest descent change their character as m 

 goes from one region of the m-plane to another. This is illustrated in 

 Section 11 for the case z — p exp (iV/4) where it is shown that the 



* A similar method was used in 1938 by A. Erd^lyi in an unpuljlishcd study of 

 the asymptotic behavior of confluent hypergeometric functions. 



