442 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 



where Hn{z) is Hermite's polynomial. When z becomes very large the 

 leading terms in (9.17) and (9.16) give 



Un{z) ^ {2zr/n!, (4.10) 



Wr.{yii-'") ~ i{^"/vy^' e-'''/2'K"\ (4.11) 



In order to obtain a series for the incident wave 

 exp [ — ix sin 6 + iy cos 6] 

 shown in Fig. 4.1 we consider the special case ^ = 0. In this case the wave 

 is simply exp{iy) or exp [i(rf — ^)/2] and may be obtained by setting 

 n = in (4.5). This suggests that the incident wave may be expressed 

 as the sum of terms like (4.5). The series turns out to be 



exp [— ix sin 6 + iy cos 6] 



= exp [ — 1^7] sin d -\- i cos d{ri' — ^')/2] 



= exp [- izz' sin d - cos e{z~ + z'')/2] (4.12) 



00 



= e'^sec (6/2) Yi n!{- iw/2yUn{z)Un{z') 



where 



w = tan {6/2), 



z = ^i''\ (4.13) 



/ —1/2 



Z = 7]l . 



This series has been studied by a number of writers. It goes back to 

 Mehler^^ who obtained it by evaluating the integral 



— 1/2 x2 



w e 



/ exp [— (,t — iy)' — {x + tat)'] d( 



J— 00 



first in closed form, and then as a series (by using the generating func- 

 tion exp [— ( — iat)' -\- 2{ — iat)x] for H„(x) and integrating termwise). 

 This leads to 



(1 - aT''" exp 



2xya — {x~ + y~)a 



1 - a^ 



(4.14) 



which is equivalent to (4.12). Since (4.14) converges when \a\ < 1, 

 (4.12) converges when \ w \ < 1 or | | < 7r/2. 



^' Reihenentwicklungen nach Laplaceschen Functionen hoher Ordnung, J. 

 Reine Angew., Math., 66, pp. 161-176, 1866. 



