DIFFHACTIOX OF KADIO WAVES BY A TAl^ABOTJC CYLTXDKK 



1()1 



S,Oi) 



2/.I/1 (/3- D-^-ttZ 



*"'/3"7„U) 



sin Trn dWn(zo)/dn 



(7.11) 



It can be shown that (7.10) converges if /3 < 1 (see (4.18)) and that 

 (7.11) converges U ^ > 1 (see (12.13)). The term l/(^ - 1) in (7.11) 

 comes from the poles of esc irw inside the path L3 shown in Fig. 5.1 

 From (7.7) and 



(7.12) 



— X sin 6 -\- y cos d — Ti = r, 



Ti = 7,(1 + 13) cos (e/2) 



it may be sho^^'n that when ^ > 1, so that Ti is negative, (5.6) has the 

 asymptotic expression 



exp [-ix sin d + iij cos 0] + aSi -' 2iMx/{l - /S). (7.13) 



When this is added to (7.11) the 1/(1 — /3) terms cancel leaving a series 

 for E valid in the shadow where j8 > 1 : 



E 



8 = 1 



t'"/3"F„(2^) 



sin Trn dWn{zQ 



M 



o)/5nJn=n, 



(7.14) 



This series may also be obtained from the more general series (5.12) 

 for E by using (7.1) and neglecting /. 



We now take up the problem of finding the paths of steepest descent 

 for the integral in (7.5) when /3 is near unity and h is large. When |S = 1 

 and h is large, the integrand in (7.5) may be expressed in terms of 

 exp [/(^i) — /(/o)] by using Table 12.3. In Section G it has been pointed out 

 that the path of steepest descent for exp [/(/i) — jik)] is the path ACD 

 of Fig. 6.2, with C being the high point. This suggests that the path 

 ACD should be used to deal with the terms in (7.5) leading to exp [/(/i) 

 — j{k)]- These terms are U niz^) /W niz'o) (introduced by the use of (4.8)) 

 for the portion of Lo between B and C, and V n^z'^) /W „{z'^^) for the portion 

 between C and Z). As a further argument supporting the use of the path 

 AC WT note that when n is on AC, i.e., on the edge of region I'h, Table 

 12.3 gives 



Un{z,)/Wr.{z,) ~ -i{\ - I'^-W./tof" exp [/(/:) - m] . (7.15) 



Hence the variation of ^"" esc irn in the integrand of (7.5) is just cancelled 

 by that of (1 — t~*") in (7.15). Consequently f''U„(zo)/[sin irn Wn(z',)] 

 varies as exp [f(^) — f(to)] along AC (the variation of h/to is relatively 

 small). 



These considerations lead us to write (5.4) as 



