484 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 



1 1 . DESCRIPTION OF PATHS OF STEEPEST DESCENT 



In this section we shall study the paths of steepest descent when 

 z = i^'^p in some detail because it is one of the two cases (the other is 

 iT^'^p) encountered in our diffraction problem. Curves related to some 

 of those shown here have been studied by Watson (for Re{n) > 0, as 

 mentioned in Section 10) and by Rydbeck* (for Re(n) = —}4)- 



The affixes to and ti. of the saddle points are given by (10.2) and are 

 made definite by the two cuts in the 7n-plane w^hich now run from m = 

 ip'/2 torn = i<x>^ and from m = to w = — z'co. From Figs. 10.1 and 

 10.2 (drawn for z = I'^p) we obtain, 



-7r/4 < arg {ip^ - 2mf'^ ^ 3ir/4, 



— ir/2 < arg m ^ 3x72, 



(11.1) 

 -7r/4 < arg U ^ 37r/4, 



-37r/4 < arg h ^ 57r/4. 



A convenient equation for the path of steepest descent through to 

 is obtained by setting t = r exp (id), n -\- 1 = m - nir -\- imi in (10.1) 

 and (10.8), and dividing through by p^: 



- {r/pf sin 2e + 2(r/p) sin {6 + 7r/4) - {nu/p') log {r/p) 



- {mr/p')d = Im [j{to) + m log p]/p'' 



Replacing ^o by ti gives the equation for the path through ^i. Equation 

 (11.2) and its analogue for ti were used to compute the paths of steepest 

 descent shown in Figs. 11.1 and 11.6. 



When m/p' is small, so is ti/p. It may be shown from (11.2) (for ti) 

 that 



(r/ 1 ii I ) sin {d + 7r/4) - {rm/ \m\) log (r/ | ^i | ) 



(1 1 .o ) 



— B rrir/ I w I ?:b {mi — rur arg ti)/ \ m \ 



gives the behavior near ^ = of the path through ^i. Paths computed 

 from (11.3) are shown in Figs. 11.3 and 11.5. Here ^i ;^ 'mi'^'^/2p. 



In computing the paths shown by the figures of this section, the work 

 was simiplified by taking m to be purely real or purely imaginary, or by 

 assuming m/p~ to be small. Even so, this often required the solution of 

 a rather simple transcendental equation [as (11.2) and (11.3) show]. 

 The graphical method based on (10.9) was not used, although it might 



Pages 26-36 of Reference 21 cited on page 478. 



