SERIES FOR WAVE FUNCTION OF RADIATING DIPOLE 



103 



which becomes, after integration by parts, 



= -r^{ 



— e' 



^\'w^ — 1 



A-2/s 



+ is 



Mils gisrw^^ -. 



(3) 



The path of integration is the straight Hne in the complex w plane 

 joining the points ^1/5 and k2/s. Arg (w — 1) and arg (w + 1) are 

 taken to be zero at the point this contour crosses the real axis. The 

 Argand diagram for a typical case is shown in Fig. 1. From the 

 definitions of ki, k^, and 5 it follows that \s\ < \ki\ < |^2|, and 

 = arg ^1 < arg 5 < arg ^2 < 7r/4. 



W^-l-l IS F 

 - ki/5 IS A 



= ka/s IS B 



Fig. 1 — Paths of integration in the w plane. 



Asymptotic Expansion 



To obtain an asymptotic expansion for ni(r, 0) we deform the linear 



path joining A and B into the path ACDB as is shown in Fig. 1. 



The lines AC and BD are both inclined to the real axis at the angle 



arg (is*) where 5* is the conjugate of s. This is the direction in which 



the exponential term e"''"" decreases most rapidly since along it the 



variable part of the exponent is real and negative.*"' The section CD 



may be displaced to infinity where its contribution to the value of 



the integral becomes zero because of this exponential decrease. 



^ To show this for the line A C we set w = ^1/5 + is*u. As w goes from A to C u 

 is real and increases from zero. The exponent then becomes isrw = ikir — j^prM 

 since j.?* = \s\^. 



