446 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 



perimentation shows that if we set Un(z()) = — Vn(zo) —W„(zo) in (4.16), 

 the series splits into two series, one of Avhich may l)e summed and the 

 other may be converted into an integral along the path Lo of Fig. 5.1: 



E = exp [-ix sin 9 + iy cos 6] + Si + S->(h), 



00 



^x = r sec (6/2) E n!{-iw/2rU„(z)W,Xz'), (52) 



00 



SM = e^'^sec {d/2) J2 nK-iw/2rih{z)Wniz')V„{zo)/WM). 

 



The series for *Si may be summed by replacing Un(z) and Wn(z') by 



their expressions (9.19) in terms of definite integrals, and interchanging 



the order of summation and integration. The resulting series may be 



summed and the integrations performed. The result is Sommerfeld's 



integral for a diffracted plane wave: 



,00 



CI { • I Nl/2 — j'j- sin fl+ij/ cos 9 / — »'^ Jv 



*Si = — (t/Tr) e e df, 



Ti = rj COS - - ^ sm - = {2ry-- sm {~— + ^ ) 



(5.3) 



The inversion of the order of summation and integration may be 

 justified when | iij | = | tan (0/2) | < 1 (in w^hich case the series in (5.2) 

 converge) by using (1) the result that | Ru \ < \ a^/N! \ when a is real in 



A'— 1 / • \ n 



T- = ^ — Rn-, 



n! 



and (2) the ineciuality 

 / r'' exp [-r + 2' '6^]d/ 



< r e-'^'-^'^'e' dt f exp [-2af + ^%t\ dt 



Jo J_oo 



< A2-''h-"'N-'''N! &xp{2hN"-) 



This inequality holds when N » 6" and at the same time iV ^ 1. The 

 value of A is independent of N and ?; is a number which exceeds | ^ |. 

 In this work the parameter a has been arbitrarily introduced; and has 

 then been chosen so as to make the product of the two integrals a mini- 

 mum w^hen A^ is large. This value of a is bN~^'^. 



When the series (5.2) for S^ih) is converted into a contour integral 

 taken along the path Li, by the procedure used to obtain (5.1), it is 



