niFKH ACTION OF RADIO WAVES HY \ l'\l{A|{OLIC ( VI-I NDKH 4.")9 



only troublesome term in (5.5) most of this section will he dexoted to 

 its stiuiy. Far behind the cyUnder ^ and rj are lai'j>;e and positiNc, and the 

 corresponding tei'ins in the integrand of (5.4) are 



r(/. + i)r„(z)]\\(z') = m/r,)"i''c-''\i +/)/2W% (7.1) 



where the asymptotic expressions (9.10) and (9.17) give 



1 + f = 1 + '"^"7^^ + '^" + \^^; + ^^ + O(nVr). (7.2) 



In writing tlie "order of" term it is assumed that ^ and tj are Ijoth ()(/•''') 

 with /• » n'. 



When (7.1) is put in (5.4) we obtain 



.S,(/0 = M, / —. ';. >. dn (7.3) 



where, from the expressions (4.1) for ^ and 77, 



.1/, = [(^•/x)^V^sec {d/2)]/4v, . (7.4) 



l3 = tir/rj = ^[tan (^/2)]/r? = cot ((p/2 + 7r/4) tan (6/2). 



I Although it is not pro\'ed here, there is good reason to believe that 

 (7.3) can be written as 



r"/3"F„(.~o) 



S-2 



.m = M. f ■ ^ \:'''\ dn + 00^/r^'') (7.5) 



when r becomes large and we restrict ourselves to the region | ^ | < 7r/2 

 in order to get 0(^") = 0(77") = 0(r). The first term contains r only through 

 the factor Mi and is of order r"^'". The "order of" term assumes h to 

 be moderately large compared to unity but h' « r. When h < 1 the h 

 is to be replaced by unity. 



The general idea leading to (7.5) is that (7.2) may be used over the 

 portion of L2 where the integrand is large and important. On the portion 

 where (7.2) differs appreciably from unity the integrand is negligibly 

 small. The important portion of Lo runs from ?i = 3^ to w = —}4 — ih 

 (approximately). In particular the variation of t "Vn(eo)/sin TnWn(zo) 

 along L2 may be summarized as follows : from —]/^io-\- i ^ it decreases 

 exponentially as i'", from — K to —ih it is ecjual to — 2z plus an oscillat- 

 ing function of order unity, and from —ih to — t^o it decreases slowly 

 at first and then more rapidly until it g(x^s down like 2""" (steepest 

 descent behavior). This may be shown with the help of Fig. 12.2, the 

 entries for regions /'a and //' in Table 12.3, and the following items [see 

 (12.9) and Figs. 10.1 and 10.2 for z = r^'-ri^ with -37r/2 ^ arg {iril 



