

Expansions of Bessel Functions, 245 



But 



\ tank j3 (tanh £ - l)d/3 = — log — V-g + tanh £ + log 2 — 1 ; 



and thus 



;,-,dog.:-ilog^ )+ n + «log. co J i/3 -nta„b^ 



= - ^j" rf/30, - % cotlr + * cot* f}- . . .) 



- * (*° <i/3 fe cosech* /3 - j£ cosectf /S coth- /8 + . . . \ 

 by the use of the identical relation, and finally 



= (>i_ A + * - . . . )- (* coth £_ £cotb* /8...). 

 But by Stirling's series, n being large, 

 n-n log „ - . l 0g *. =} log 2 + T-&- - At + «-?*- 5 -■ ■ • (69) 



where the B's are Bernoullian numbers*. 

 And finally 

 ', = »(tanh/3-/3)-i log 2- \^- \ %-■■■) 



It appears at once that J„(:) is ultimately rapidly evan- 

 escent when n greatly exceeds z. Accordingly, T x and T 

 become identical, as also t x and t. 



Thus when n is not integral, and the coefficients X, fi in 

 the expansion are identical with those of expansion A, 

 then if 



n > z = z cos /3, 

 we have 



J_„(~)~eos /<7rJ„(~) = sm ><7r . I-—) 6> ~' 

 where 



2Tsmh£=l-^coth*/3 + ^coth r £-...j . (71) 



* 7-:. <i. ride Whittaker, ; Modern Analysis,' p. 194. 



