DIFFRACTION OF RADIO WAVES BY A PARACOLIC CYI.IXDKR 407 



The location of the zeros far out on the lines of Fig. 12.1 may be ob- 

 tained by writing the appropriate expressions of Table 12.2 as propor- 

 tional to B times a cosine or sine. Examination of the trigonometrical 

 terms shows that 



Unii^'^p) has zeros at n ;^ 2A; + 1+ i^'^k^'^/r, 



7„(i'"p) has zeros at 7i ;^ -2/c + i"%k"^/r, (12.7) 



TF„(i"V) has zeros at n ^ -2A; + {'"^k'^^Tr, 



where fc is a large positive integer. Of course, Un(z) also is zero when n 

 is a negative integer. 



So far we have been dealing with z = I'^p. Now we consider the case 

 z = i~^''p. 



Asymptotic expressions which hold when z = i'^'^p may be obtained 

 from Table 12.1 by using the relations (9.11) between functions of z 

 and of its complex conjugate z*. Thus, for example, V a+ib{i~^'" p) is equal 

 to the complex conjugate of W a~ib{i''^ p) ■ These relations, and relations 

 such as 



[^0 for z = i p, 71 = a — ib] * = U for z = i p, n == a -\- ib 



[f(to) for z = i''p, 71 = a — lb] * = f(to) for z = -T^^'p, 7i = a -\- ib 



(12.8) 



have been used in constructing Tables 12.3 and 12.4 from Tables 12.1 

 and 12.2 The interchange of V„{z) and Wn{z) should be noted. The 



rriL 



I'b 



I'd 



EI' 



Wn (Z) _,^- 



n' 



m' 



m = -L/?2/2 



^'^^ZEROS OF Un (Z) 



U' 



Fig. 12.2 — Regions in the comple.x m-plane corresponding to different asymp- 

 totic expressions when 2 = z~"V- The lines on which the zeros of the various 

 functions lie are shown bj' the dashed lines. The corresponding information for 

 z = i^'-p is shown on Figs. 11.2 and 12.1. 



