562 



BELL SYSTEM TECHNICAL JOURNAL 



We thus obtain 



Zbh = 



It] w=o n\ 



where ^ „ is defined as 





(89) 



(90) 



In spite of the complicated appearance of (90) the A 's are in reality 

 ver}' simple functions of aa, as the accompanying list (91) will show.^^ 



An = 



1 



aa 



At = h , 



^1 = 0, A2 

 A,= - 



3/7 



aa 



2_ _ 

 2^2 



A,= - 



r2/,2 ' 



a'a 



12 



a^a" 



(91) 



. _ 1 , 9 ,60 



Aq — - H r— ; + c = 



0-a a%^ a^a° 



The formula (90) can be made more rapidly convergent by partially 

 summing the numerator and the denominator by means of hyperbolic 

 functions. Thus, the numerator becomes 



a 1 , , smh at 



-r cosh at -\ ;r 



laa 



1 



3_i + 

 4a ^ 



_ 3^2 



a2 ^ 



and the denominator 



T sinh at + 



3(o-/ cosh at — sinh cr^) 



+ 



The reader will readily see that there would be no difficulty in using 

 this method to obtain other expansions somewhat similar to (89). 

 For example, we might write a = b — t in (75) and express our 

 results in terms of the outer radius. In this respect the method that 

 we have used has greater flexibility than Dwight's ; but there seems to 

 be little advantage gained from it, since the simple formulae (82) are 

 sufficient for most practical purposes. 



-^ The values given in (91) are exact, not approximate. One of them, namely, 



A, 



h'{cra) loicra) 



1 



Ko'icra) Koicra) 



is one of the fundamental identities found in all books on Bessel functions. The 

 rest are consequences of analogous, though ess familiar, identities. The general 

 expressions for the coefficients An were obtained by H. Pleijel [20 J. 



