222 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1953 



value by picking a sequence of trial functions for ho(w) with the ap- 

 propriate number of parameters and imposing successively (79), (80) and 

 (81) upon this sequence; this leads us to a sequence of values for (w) 

 which can then be examined. In such a procedure careful consideration 

 of the trial functions is an important element. The following information 

 is available. It was proved in Section IID that ho(w) is logarithmically 

 infinite at the origin. At infinity, on the other hand, ho(w) falls as e""" 

 times some power of w. One way to check this is to drop the terms con- 

 taining 1/c as factor in (46) ; the solution of the recursion system becomes 

 then 



K(w) ^ {2v + l)e"V 



where k is some unknown exponent. Armed with this fore-knowledge, 

 we shall use the following sequence of trial function for l%{w) 



ho(w) = pEi{w) + qKo{w) + re~^ + swKi{w) (82) 



where 



Ei{w) = / — 



Jw U 



du 



and Kq(w), Ki{w) are the modified Hankel functions of order zero 

 and 1.^^ 



We find in zeroth approximation from normalization only 



(83a) 



(83b) 



2' This definition, which is in accord with the tables of Jahnke-Emde, differs 

 from the usual one by a factor 2/7r. This change is suggested by Watson, Bessel 

 Functions, p. 79, and proves convenient in the following. 



