358 



BELL SYSTEM TECHNICAL JOURNAL 

 TABLE I 



to 



15 



20 



uV2 



0.1 

 0.5 

 1 

 2 



3 

 4 



0.1 



0.5 



1 



2 



3 



4 



0.1 



0.5 



1 



2 



3 



4 



0.1 

 0.5 

 I 



2 

 3 



4 



of the Laplacian expansion as applied to the evaluation of An(x), 

 X = wV2, for values of x ranging from 0.1 to 4 and for A^ = 9, 15, 10 

 and 20, respectively. 



Designating by sAwix) the approximate value obtained for Ax{x) 

 when one takes into account the first 5 terms in each of the two series 

 S, S', the last three columns of the table show the proportional errors 

 incurred when 5 = 1,2 and 3, respectively. It will be noted that for 

 iV = 10 and x — 4 the second approximation is closer than the third; 

 this situation will occasion no surprise if it is recalled that to obtain 

 the best results the natural order of the terms may have to be altered 

 when, for example, one expresses a term of the binomial expansion in 

 a series of Hermite functions. 



For the convenience of one who wishes to calculate ^iv(^) for values 

 of A^ other than 9, 10, 15 and 20, there are given in Table II the values 

 of M~('"+^'iiC„ for m = 0, 1, 2, 3, 4, 5 and those values of u covered by 

 Table I. 



I am indebted to Miss E. V. Wyckoff of Bell Telephone Labora- 



