DIF^FHACTIOX OF HADIO W AVKS HY A rAUAHOTJC (AMNDI'.K 



Wlieii (1) n is near C, (2) 770 is large, and (8) | | | « r?,) we have tor tlie 

 various terms in (6.2) 



I /Sin irn '-^ 2i 



r(l + 7i/2)W,Xzo) ~ (770/4)^ '(2x)''-r'' exp [-/77o'/2].4z-(a), 



2r(l + n/2)Un{z) ^ "exp 



:^/n 



2 ^S t 



6 V2m 



(fi.ll) 



(6.12) 



wliere Ai (a) denotes the Airy integral defined by (13.12), m = w + 1, 

 arg 7)1 is near —ir/2, and 



a = (2A7?;)^'' (/n + ivl/2), (6.13) 



da = (2/ir]lY'^ dn. 



Expression (6.11) comes from (13.21) and expression (6.12) comes 

 from region la of Table 12.2 (strictly speaking, region lb should be 

 used but point C is so close to arg m = — t/2 that the simpler expression 

 for la may be used). In obtaining (6.12) it is necessary to use the terms 

 shown in the expansions (12.5) of to and log ti/to. It may be shown that 

 (6.12) also holds for negative values of ^. 



We now set w = 1 in (6.2) and change the variable of integration 

 from n to a. Substituting for m in (6.12) its expression in terms of a, 

 expanding in powers of a and neglecting higher order terms, converts 

 the argument of the exponential function into 



if/2 — i^rjo — iy'^/3 + ayi 



1/3 



(6.14) 



B '-oo 



C IS AT n = -1 -lH 



D -1.00 

 Fi^- 6.2 — Patfis of integration used in stuch'ingthe current density and dinrac- 

 tiun jjattern when /) is large. Path BCD is eciuivalont to path Ln of Fig. o.l. AC 

 and CD arc boundary lines which mark a chaiigo in the asymptotic l)cha\ ioi- of 

 H „(2o). Far out towards ^1 the line AC tends to become parallel to BC. 



