Expansion of Bessel Functions of High Order. 705 



The function (p. 

 By (5) ^_l_^( 1+ h + h + V 1 



where t 'c 2 = 4c 2 -?r? 2 . 



Ttas |* = f(l+^ + ^+..A . • • (31) 



^~ 2~\ or or / 



where Ah= — ^3 = i? j 



^ 3 = -X 5 + V=i(2^ 2 -25), l'(32) 



Ps 



o= _X 7 + 2\ 3 X 3 -X 3 3 = t 1 6 (40m 4 - 1112m 2 - 1073). ^ 



and so on. 

 Thus by (6) 



= f ' ^cot 2 ( 1 -=™%° - ^tan 2 6-!**** 6 . . . W 



J 2 \ cos ??r m 4 / ' 



where sina=5-, (3d) 



or 



nir m . . m ( 1 ra 2 \> 



= -T^f 6*i+— 2 tan 2 0+^tan 4 0+ ...\*0 



2"U \ m ~ m ' 



and finally 



_iJ^l t an«-^(tan«-^tan 3 a U^(tana- 

 2m\m 2 m 4 \ 3 / m 6 \ 



han 3 «+|tan 5 *)- ...J. . . (34) 



Final Results. 



Collecting the results, it appears that when n and z are 

 both large, and z — n is of order not too small in comparison 



