226 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1953 



start out with a relation between (1,1) and (3, 2). By the use of regular 

 recursion formulas we can successively transform this into a relation 

 between (3, 0) and (3, 2), then (3, 0) and (5, 1) and finally between 

 (3, 0) and (7, 0). Here we have for the first time the normal situation in 

 which we do not get the actual value of a moment of ho(c) but only a 

 relation between two or more of such moments; the reason for this is 

 that the system fails to connect up with (0, 0) which equals unity a 

 priori. A similar situation prevails for the next truncated relation; it is 

 originally a relation between (2, 2 ) and (4, 3 ) and is finally reduced to one 

 between (2, 0), (6, 0) and (10, 0). Similarly, the next truncated relation 

 reduces to a relation between (5, 0), (9, 0) and (13, 0) and so forth. The 

 first three of these reduced relations come out to be 



{w') = 10 (87) 



S{w') = 112(^') (88) 



fV) = 27V> + ^^(.'») (89) 



These formulas will now be imposed upon a sequence of trial functions 

 for ho(w) suitably chosen. Again, we may make use of the information 

 of Section IID, according to which hoiw) is logarithmically singular at 

 the origin. For large w we proceed as previously from (46) leaving off the 

 terms of 1/c. We get then 



K(w) ^ (2v + l)e-^ " 



2 k 



This suggests the following trial function for }h{w) 



h{w) = vEiiW) + qKoihiv") + re~" "' + svl'Kiihw') (90) 

 The best zero order approximation is actually obtained by the function 

 K,{W)' We find 



