DIFFRACTIOX OF RADIO W W KS HV A I'AUAHOLIC CYLIX DKK l!)') 



Sometimes it is helpful to use 



(27r) " exp 



loy; 



m 



1/r 



, (m + 1 

 ~2 



l/r(l + n/2) for 



-7r/l^ < arg m < 7r/2, (12.4) 



r (^^')/27r = r"-^r(-n/2)/2T, 



7r/2 < arg m < 37r/2, 



where the last line is obtained by setting m exp ( — Tri) for /// in the 

 second line. 



The asj^mptotic expansions for regions 76 and III may be obtained 

 from those for la and II by using equations (9.11) and (9.13). However, 

 the work is more difficult than one might suspect at first glance. 



Incidentally, the leading terms in the asymptotic expansions (9.15) 

 and (9.17), which hold when p ^ x and ii remains fixed, may be ob- 

 tained by considering the entries for la and Ih in Table 12.1. 



It is sometimes convenient to use the limiting forms of the asymptotic 

 expressions when \ m\y> p .\n this case, for z = i ' p, 



2'o -^ 2 + i(2m) 

 2'i -> 2 + 'i{2my'^ 



1 



2m 



+ 0(>n-"'-) 



\-£\+00n-") 



(12.5) 



log (i/to — > + tT + iz{2m)' 



-1/2 



2 + 



6m 



+ 0(m-^'0, 



where the upper signs hold when —ir/2 < arg m < 7r/2 and the lower 

 ones when 7r/2 < arg m < 3x/2. Substituting (12.5) in (12.3), neglect- 

 ing the higher order terms, and setting 



D 0-3/2 -1/2 



B = 2 w exp 



I 1 -104')+. 72 



ao = exp [-p{2m/iy^^], 



«! = exp [p{2m/iy'^] = 1/ao , 



(12.6) 



converts Table 12.1 into Table 12.2. 



In this table B, ao, ai, are defined by (12.6); m = n -{- 1; — 7r/2 



