4CA THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 



pressed as a series in which the paraboUc cyhnder functions in (7.14) 

 are replaced by Airy integrals. One way of obtaining this series from 

 (7.14) is to use the Airy integral approximations (13.21). The zeros 

 Hi, no, • • • of Wniz'o) go into the zeros ai, 02, • • • of Ai(a) by virtue of the 

 relation n — Wo = a{ihy ^ and we have 



E ~ i-'-"M, E "Ti'/tr"' ' (7.30) 



.J r \ I 1/ .113 ^ exp [- ttsQi .„„ s 



where g < 0. Here, as in (6.17), at = -2.338 • • and Ai'(ai) = .701 • • • 

 In obtaining these relations we have used z"'7sin irn ^ 2i, and 



Ai{asi"') = -i"Bi{a,)/2 = i"'/2TrAi'{a,), (7.32) 



where the first equation follows from (13.17) and the second from the 

 Wronskian 



Ai{a)Bi'{ci) - Ai'(a)Bi(a) = I/tt. (7.33) 



The equal sign in (7.31) holds even though the steps leading to it 

 indicate that ^ should be used. This may be seen from an alternative 

 derivation of (7.31) in which Ai{ai~'^'^) in the first integral of (7.27) is 

 replaced by the right hand side of* 



Ai(al~"') = -i'"Ai(a) - r'"Ai(ai"'). (7.34) 



In the first portion the Ai(a)'s cancel and the resulting integral con- 

 tributes — 1/g to (7.27) (g must be negative for convergence). The 

 second portion combines with the second integral in (7.27) to give a 

 contour integral which leads to the series in (7.31) w^hen the path of 

 integration in the a-plane is closed on the left. The closure may be 

 justified by the asymptotic expressions (13.19) and (13.20) for Ai(a) 

 (again g must be negative). 



Since the integrals in (7.27), and their equivalents in (2.11), converge 

 uniformly for all finite values of g, ^(g) is an integral function of g. 

 When g is negative ^(g) may be computed from the series (7.31). When 

 g is positive I have not been able to find a practicable method of ob- 

 taining ^(g) other than the numerical integration of (2.11). The results 

 are shown in Table 2.1. Since ^(g) is an integral function its Taylor's 

 series about, say, g = — .5 converges for all values of g. The coefficients 

 in this series may be computed from (7.31). However, I was unable to 



* Reference 11, page 424. 



