108 BELL SYSTEM TECHNICAL JOURNAL 



arg (1 — /) = 0, arg (1 — {s^lkr)t) = 0. This may be verified by 

 expanding the numerator of the integrand and using 



I. 



"+' /. ■, J. ,. ON rC"! + i)r(i' + 1) 



where w is a positive integer or zero, with v = — 3/2. 

 Expression (16) now becomes 



Uh) = 2 1? E ^^" F{\, - n/2 ; 1/2 ; s^h') (18) 



and the series converges for all finite values of r since the series inte- 

 grated termwise in Equation (16) is uniformly convergent. 



We obtain the series for Tli{r, 0) given in statement of results as 

 equation (19) by putting (18) and the corresponding expression for 

 L{k2) in equation (15). 



Notation 



The following symbols are used. C.G.S. electromagnetic units are 

 used throughout the paper. 



c = velocity of light, 3 X 10'° cm. /sec. 



F{a, h;c',x) = The hypergeometric function 



«& a(a + 1)6(6+1) 



^ \\c ^ 2!c(c + 1) ^ 



-A)(^0 = Bessel function of the first kind, zero order. 



k\ = co/c. 



ki = Veco^ + i^Tvaca. The real and imaginary parts are 

 positive. 



Pn{t), Pli\~2'{t) = Legendre's polynomial, and associated Legendre's 

 function of the first kind. 



RiN, R20 = Remainder terms in asymptotic series. 



r = horizontal distance of representative point from 



dipole. 

 5 = kik2Hki' + ki" or 1/52 = l/kr^ + l/y^a^. The real 



and imaginary parts of 5 are positive. 



5* = The complex conjugate of s. 



t = time in the introduction, otherwise a complex 

 variable. 



w = complex variable. 



