96 Mr Watson, Bessel functions of large order 



* 



Bessel functions of large order. By G. N. Watson, M.A., 

 Trinity College. 



\^Received 14 June 1917.] 



1. When the order of a Bessel function is large, the asymp- 

 totic expansion of the function assumes various forms depending 

 on the values of the ratio of the argument to the order of the 

 function. The dominant terms of the asymptotic expansions are 

 given by the formulae : 



(i) When n is large, x is fixed and ^x< 1, then 

 Jn {nx) ~ (27rw)"* (1 - a^y^x"" [1 + V(l - a?)]"' exp [n V(l - x% 

 (ii) When n is large, x is fixed and x>l, then 



Jn{nx)<^{^'jrny^{x'^- l)~^cos [?i V(*'^ - l)-n^e(r^x- lir]. 



_ 2. 

 (iii) When n is large and € = 0{n ^), then 



J^{n + ne)^T(i)/{7r2is^n^]. 



The corresponding complete asymptotic expansions, valid for 

 general complex values of n and x, have been given by Debye*. 

 Accounts of the history of the approximate formulae are to be 

 found in Debye's memoirs and also in two papers f which I have 

 published recently. 



It is evident that there are transition stages between the 

 domains of validity of the three formulae quoted ; and not much 

 is known about the behaviour of J„ (jix) in these transition stages. 

 Consequently I propose to establish approximate formulae (involv- 

 ing Bessel functions of orders'] + ^) which exhibit the behaviour 

 of the Bessel function right through the transition stages. These 

 formulae are more exact forms of some approximations which 

 Nicholson § obtained some years ago without estimating the 

 margin of error or the precise ranges in which the results were 

 valid. 



* Math. Ann., lxvii. (1909), pp. 535—558. Milnchen. Sitzungsberichte [5], 1910. 



t Proceedings, xis. (1916), pp. 42—48. Proc. London Math. Soc. (2), xvi. (1917), 

 pp. 150—174. 



J These functions have been tabulated by Dinnik, Archiv der Math, und Phys., 

 XVIII. (1911), p. 337. 



§ Phil. Mag., Feb. 1910, pp. 228—249. 



