1194 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 



In our problem ?? is (negative) imaginary and m/v may be an integer or a 

 rational fraction. For any given ?? and mjv the conditions (565) to- 

 gether with the limiting form (566) determine an eigenvalue X, and 

 hence by (563) determine A; but only a small amount of work has been 

 published on the eigenvalues of Mathieu functions with imaginary 

 parameter or of fractional order. We shall look at some special cases. 



V = \. The function /(^) is one-half cycle of a cosine curve which 

 varies from — 1 to + 1 ; we expect results similar to those found for the 

 symmetric discontinuity of case (i) and the linear variation of case 

 (iii). The eigenfunctions of the first two modes (w = 1 and m = 2) are 

 se2(T, ^) and se4(r, d-). The eigenvalues of these two functions for purely 

 imaginary t? have been computed by Mulholland and Goldstein out to 

 a point which corresponds to C = 8x , and an asymptotic formula is 

 given for larger values of C. The values of Ai/tt" and A^/tt are plotted 

 forO -^ C S. 100 in Fig. 28; the corresponding eigenfunctions resemble 

 those shown in Fig. 23 for the stack with a symmetric discontinuity. 

 Again we find that Ai and A2 are real for small positive C, equal for a 

 particular value of C, and conjugate complex for larger C. The leading 

 terms of the asymptotic formula are, in our notation, 



Ai 



A* 

 A2 



[4.7124C"^ - 3.0842 - 1.0901C"* - • • • ] 

 + i{C - ^:jY1^& - 1.0901C"- - 



(568) 



J/ = 1. Here/(^) is one full cycle of a cosine function, varying from — 1 



12 



20 



30 



50 

 C 



Fig. 28 — Real and imaginarj^ parts of Aj/tt^ and Aa/Tr^ for a nonuniform stack 

 whose average properties vary as one-half cycle of a cosine function across the 

 stack. 



29 H. P. Mulholland and S. Goldstein, Phil. Mag. (7), 8, 834 (1929). In this 

 reference \a or 4/5 corresponds to our X and 89 to our »?. 



