664 BELL SYSTEM TECHNICAL JOURNAL 



When ^2= °o, V= IR^'^ exp ikiR2 and £/ is zero. When 

 ^2 = ki, V = li?2~^ exp ikiR2 and E^^ just cancels the field of the 

 image dipole in £°°. 



The wave potential of a vertical dipole is '^^ ^ 



n/ = ^-^+F. (4) 



V is the function ITo analysed by Sommerfeld in Riemann-Webers 

 Differentialgleichungen der Physik. Sommerfeld transforms the inte- 

 gration from to 00 into an integration from — oo to + =» by re- 

 placing the Bessel function by its equivalent Hankel functions and 

 then wraps the real axis path of integration around the zero of kiH 

 -f ki^m and the two branch cuts from ki and ^2 to -f- * 00 . He thus 

 gets V = P -{- Qi -{- Q2 where P is the integral around the zero of 

 k2~l + ki~m, Qi is the integral around the branch cut from ki, Q2 is 

 the integral around the branch cut from k^ and the function inte- 

 grated is 



^^^]^H.^ir,)e-'Hr. (5) 



/ and m are pure imaginaries on the branch cuts, which are rectangular 

 hyperbolas, from ki and ko respectively. As one carries r counter- 

 clockwise around the branch cut from ki to -f- *'co, I travels up the 

 right hand side of the imaginary axis from — ioo -j- e to + ^=0 -f e. 

 A similar statement holds for m. 



Multiplying numerator and denominator of equation (5) by k^H 

 — k^m and then taking out a factor (^2* — ki^)~^ our integrand 

 becomes 



k2^ - ki' {t - s){t + S) 



HoKrp)e-^^'''-''^\dT, 



where 5 = -f kiki -^ ^lkr^ + k{-. Integrating around the pole at 

 T = 5, we get 



K2 — «1 ■^S 



= if k2~-^s'' - ^1- - ki--\s'' - ki' = 



~ki' 



= - 27r ■l'_'lJ U(sp)e-""''^"\ (6) 



ki^-yjs^ — k-c — k{Hs'^ — k2' = 2^2"Vi"" — k{- = likikiS. 



