Okr — Extensions of Fourier^ s and the Bessel-Foiwier Theorems. 211 



on 0, in addition to those required when ?2 + 1 is positive, as the integral 

 with respect to p might not converge. 



It will first be supposed that a is not zero. 



It is supposed, also, in the first instance, that n is not an integer ; if ?i is 

 an integer, the limiting form of this equation is to be taken. 



It is not supposed that ii is necessarily real ; but, to simplify the verbiage, 

 only the case of a real n is discussed. 



I proceed, in fact, to give a proof of equation (14), or perhaps I ought 

 more strictly to say, to indicate a mode of discussing it, which applies to all 

 values of n. I use, however, the K functions which, as usual, are defined by 

 the equations 



K,,[x) = ^!ll_{/.„(x') -/„(*)! = ,-^^— {i"J_„(ix)-i-"J„(ix)}, (15) 

 Asmmr 2 sin mr 



where the argument of each power of x in /+,((;X') is zero when x is real and 

 positive. I proceed to establish equations involving K functions which are 

 together equivalent to (14), and its analogue obtained by changing the sign 

 of n. One of these equations is 

 Lt. 



cj-h 



■h 



XK„ (- iXr) d\ K,, {iXp) p(f> {p) dp, 



L/t r^ c^ 



or, A=» \Kn(iXr)dX\ Kn{- iXp)p(l>{p)dp, 



cJ-h J a 



/_„(Ar)Jl„(Ap) + Jn{Xr)Jn(Xp)]p(p{p)dp* 



4 sin^WTT 



XdX 



-ji>{r-.), (16) 



provided r > a, the path of A being as above, and the arguments of iXr, iXp 

 thus passing from 37r/2 to 7r/2, those of - tAr, - iXp from + 7r/2 to - 7r/2. 



This contour is to be deformed into one at a great distance from the 

 origin ; and therefore we consider the asymptotic expansions of the Bessel 

 functions for large values of the variable. The fundamental equation is 



27r~^ sin mrKn(x) = I-n(x) - I,^{x) = (2/7r«)s sin mr . e~'-, (17) 



w^here arg. re ^ - tt and ^ + tt. (It really holds if arg. x> - ZttI'2 and < 37r/2, 

 but not at these limits.) And the fractional error in the right-hand member 

 is, when x is sufficiently great, of order x~^. 



By changing x into ye-''' and ye"^" in succession, we deduce from (17) 



{/.„ [y) + In{y) \ sin im - i [I-n{y) - In{y)] cos m; -= {^juy)^ sin mr . ey ; (18) 



- {I_n{y) - Iniy) } COS 2mr - i [ Ln(y) + In [y] ] sin 2mr ^- {^jny)^ sin uk . e-y ; (19) 



* Terms involving the inoducts J^J-n disii]>pear, as positive and negative values of A annul each 

 other, on deforming, for these terms, the contour into a line. 



