142 BELL SYSTEM TECHNICAL JOURNAL 



integral equation may be obtained from the differential equation (3-1) and 

 the boundary conditions (3-2) and ii-?>) by the one-dimensional analogue 

 of the method used in Section 3 of the companion paper^ If we multiply 

 both sides of 



^ - Jh = s(v) (3-16) 



(where s(v) has been added for generality) by Ga(i'u , ^0, integrate twice by 

 parts over the intervals (vi , Vq — e), (vo + e, Vo) with 6 > and Vi < 

 I'o < Vo , and finally let e ^ we obtain 



Kfo) = f Ga{vo,v) s{v) - y{v)Jf ^, ir 



+ Ga(To,n)ly' - e-yU,; - Ga(T,,vd{y' + d-'yUr,. 



dv 



(3-17) 



Equation (3-15) follows when we put s(v) = and let vi -^ — x , vo -^ ^ . 

 It will be recognized that (3-17) and (3-15) are closely related to integral 

 equations occurring in the work of R. E. Langer^ and E. C. Titchmarsh^. 

 When // has, for example, one or more simple zeros in — x < v < x 

 the integral in (3-15) contains a factor which becomes infinite and the 

 integral equation fails. However, we shall not concern ourselves with this 

 case beyond remarking that it involves results obtained by H. Jeffreys'", 

 Langer^, Furr>''' and others. 



3. So far we have been considering the solution of only one equation 

 whereas we really require the solution of a set of equations. If it is apparent 

 that most of the disturbance is given by the first equation of the set it may 

 be possible to proceed by successive approximations, each of the remaining 

 equations being of the form (3-16) with s{v) determined by the solution of 

 the first equation. 



Another method of dealing with a system of -V equations is that of numeri- 

 cal integration. As a contribution towards obtaining the boundary condi- 

 tions at large positive and negative values of v we shall state a generalized 

 form of the WKB solution. Although this solution is related to the general 

 results obtained by Birkhoff'-, Langer^ and XewelF^ concerning the asymp- 

 totic forms assumed by the solutions of a system of ordinary linear differen- 

 tial equations of the first order, it is worth mentioning explicitly. 



Let the wth equation of the set be 



A' 



A'm = S Amnyn , W = 1 , 2 , • • • , iV (3-18) 



n=l 



where the .lm«'s are relatively slowly varying functions of v (see equations 



