144 BELL SYSTEM TECHNICAL JOURNAL 



gml<P — X/^mn^nl " +2gmO^ + gmO(p. (3-24) 



The vanishing of the determinant of the coefficients of the g„o's in (3-23) 

 determines .V values of (p-, the /th being (p( . Once <p- is selected the g„o's 

 are determined to within a common multiplying factor (which may depend 

 upon v). This factor is then fixed to within a multiplying constant by the 

 necessar>^ and sufficient condition that (3-24) be consistent^*, namely, 



^ hm{2gmO<P + gmO<p) = (3-25) 



m 



where hm is any solution of the transposed system 



hm<P — z2 Anmhn = 0. 

 n 



Because A„m = Amn we may take hn to be gmo • Equation (3-25) may 

 then be integrated and leads to the second of equations (3-20) when we set 

 the constant of integration equal to unity and identify <p and gmo with (p( 

 and Sm( , respectively. The first equation in (3-20) follows directly from 

 (3-23). 



Since equations (3-20) do not completely satisfy (3-24) (gmi remains to be 

 determined) our WKB solution for a set of equations is not, in a sense, as 

 good an approximation as it is for a single equation. Nevertheless it still 

 represents, just as in the case A^ = 1, the leading part of the asymptotic 

 form approached as the ^mn's vary more and more slowly with v. 



In matrix form, the WKB approximation to the solution of 



y = Ay (3-26) 



is 



y = Se-r + Se'-d^ (3-27) 



where y and (f^ are column matrices, A and S square matrices, and exp 

 (±H) a diagonal matrLx having exp (±^/) as the ^th term in its principal 

 diagonal. The element in the mth row and fth column of S is Smc whence, 

 from (3-20), 



S^ = AS 



S'S^ = S^S' = I (3-28) 



H = $ 



where the primes denote transposition of elements, I is the unit matrLx of 

 order N and $ is the diagonal matrix having sr^ as the h\\ term in its principal 

 diagonal. That the non-diagonal terms of S'S are zero follows from the 

 first of equations (3-20) and from (ft 9^ (fk \i ( 9^ k. 



