SERIES FOR WAVE FUNCTION OF RADIATING DIPOLE 105 

 have \w — 1| — \(ki/s — 1) sin d\ where 



d = arg is* - arg ( Y - 1 j = ^ - arg (ki — s) > ^ • 



Similarly we have [w + 1| — \kils + 1|- These inequalities enable 

 us to deal with the integral of |e"''"'| which may be integrated to 

 show that 



(N + i)!givVcsc d 



i?i 



< 



lr{ki - s) sin 0]^+i 



(8) 



By interchanging k] and ^2 in (6) and (7) we obtain expressions for 

 7(^2) and RiN- An inequality for |i?2iv| is obtained from (8) by 

 setting 9 — -kII and interchanging kx and ki. By combining these 

 expressions in accordance with equation (4) we obtain an asymptotic 

 expansion for Ilifr, 0). 



In general, I{ki) is negligible in comparison with /(^i) because ^2 

 has a positive imaginary part which causes e'^s"" to decrease rapidly. 

 Since lijki) — R20, R20 being the remainder after zero terms, w^e may 

 obtain an inequality for 7(^2) by setting N = 0, 6 = t/2, and inter- 

 changing ki and ki in (8). Then from (4) we have the result 



ni(r, 0) = 



1 



n=l (iTsr)'^ ^ 



T^ia 



(1 - T'-)r 



where Rm satisfies the inequality (8) and |i?2o| < \ e'''^'' / (rki — rs) 

 Series for ni(r, 0) in Ascending Powers of r 



, (9) 



Put 



^2 Ji 



4 



w^ 



(10) 



and define K{ki) as being obtained from (10) by interchanging ki 

 and ki. By referring to equation (3) we see that 7 may be written 

 in the form 



1 



ni(r, 0) = 



We write 



K{kx) = e'^i^ 



1 - 



1 - 



(1 - r')r 

 ikisr 

 ikisr 



IKiki) - r'Kikin 



(11) 



Ji 4w' - 1 J 



(12) 



the infinite series being uniformly convergent. 



