DIFFRACTIOX OF HADIO W \Vi;s in \ I' A H A liol.lc CVIJXDliK 417 



seen that Li may be deformed into Lj and we ol)tain 



c sec 





4). 



Whether a particular integrand, such as the one shown in (5.4), con- 

 \erges at the ends n = ±.i^ of L-i can often be decided from Table 5.1. 

 This table gives a rough idea of the behavior of the various functions in 

 terms of powers of i. For example, if the integrand should turn out to 

 be proportional to z" = exp(z7rn/2) at n = f x , the integral \\ill con- 

 verge Uke exp ( — tt | w | /2). 



Table 5.1 



The approximations for r(n + 1) follow from its asymptotic expres- 

 sion, and those for the parabolic cylinder functions come from Tables 

 12.2 and 12.4. The entries for the c^^inder functions may also be sur- 

 mised from expressions (9.4) which hold for z = 0. 



Table 5.1 may be used to show that the integrand in (5.1) is of the 

 order of z" as n — » —i^c. Hence there is no hope of deforming Lj into 

 L2 in this case. On the other hand, the integrand in (5.4) is of the order 

 of t" as n — > z 3c and of i~" as n ^ — z oc , and therefore (5.4) converges 

 exponentially. In fact, it converges for all real positive values of w = 

 tan 9/2. This enables us to obtain an expression for the field which 

 holds for < 6 < T (i.e. it is not subject to the restriction | li^ | < 1 re- 

 (luired by (4.16)). This expression, which is fundamental for our woik, 

 has the form 



E = exp [-ix sin 9 + iij cos ^] + Si + S, (h). 



(5.5) 



Here Sy and »S'-)(/0 are given b}- (5.3) and (5.4), respectively. 



In working with (').')) it is sometimes convenient to u.se the expression 



