234 Dr. J. W. Nicholson on the Asymptotic 



When n and z are both great, a first approximation to R, 

 is obviously 



T> 1 . 1 n * 1 . 3 7l 4 . 



B=l+i. ? +__ + ..., 



or £ 



which leads to one of Lorenz's expansions, when n is half an 

 odd integer. But the error involved would be very doubtful. 

 Now writing as in usual notation for Bessel functions of 

 imaginary argument, 



K (\)=l g-xcosh*^ . . . . (29) 



then it may be shown * that when the series terminates 



4~ f °° 

 Wl ==R= — 1 K (2s sinh *) cosh 2nt dt, (30) 



or, as a reversible double integral, 



R = — 1 J e -2s sinh < cosh * cos h 2n* rf* efyr. (31) 



But this expression remains finite and determinate when 

 n is not half an odd integer, and it may, moreover, be proved 

 by direct substitution that it is still a solution of the same 

 differential equation. Accordingly, it is still the value of R, 

 as it also takes the correct value when ,c = a© . Thus for all 

 real values of n and z, 



4~r°° 



R= — I K (2zsmht)cosh2ntdt, . . (32) 

 v Jo 



and when 2n is not an odd integer, the series u l9 though 

 divergent ultimately, may be used for the computation of 

 the integral. 



Expressing the value of R in terms of the Bessel functions, 

 we deduce, when n is not an integer, 



»(*) + J-n 2 (~) —2J n (z)J_ n {z)cosn7r= — 2 sin 2 ?i7r j K (2 < 2rsinh t) cosh 2titdf f 



Jo 



.... (33) 



and when n is an integer, 



(.« 

 K [2z sinh t) cosh 2ntdt. (34) 



* Phil. Mag. Dec. 1907. 



