Okr — Extensions of Fourier'' s and the Bessel- Fourier Theorems. 219 



ratio of tiie left-hand member to the coefficient of A on the right, it is finite 

 wlien \p is zero, as appears from a consideration of the order of magnitude of 

 the terms ; and it is zero when \p is infinite, from the form of the asymptotic 

 equation ; it cannot be infinite for any finite Kf> ; lience it has some finite 

 maximum value. 



If now we substitute for JnQ^o) in (51) from this inequality, and replace 



Kn{-i\r) by ;7r/(-2ar)j4e'^ 



the limit of the former of the two terms in the integral is zero, as follows 

 from Fourier's theorem. It remains to consider 



a[ Ir^ai r |e'M'-p)pi,/,(p)f/p|. (53) 



C J -ll j 



Interchanging the order of integration, and writing A = Ae'^, the integral in A 

 is seen to be less than 7r/(r - e), and thus the double integral is less than 



7rr--2 (r - £)-> \ p^(l>{p)dp\, 







which, under the condition stated, diminishes indefinitely with £. 



When Jn is replaced by J.n, this proof holds if n :j> |. 



If n > ^, we might proceed to reconsider equations (16), (27). It appears, 

 however, more simple in the light of what precedes to keep to the discussion 

 of (40) with the sign of n changed in J; for, when this is shown to have the 

 value |^(^'-£)' if we express Km terms of Jn, J-n, the products JnJ-n will 

 disappear. 



Thus, we return to the consideration of (51), with Jn changed into '/-„, 

 i.e. of 



Lt r'* r^ 



A=a^ \dxKn{- i\r) J.ni'^p)p({>(p)dp. (53a) 



c J -^' - 



Suppose n < m + |, but not < m - |, where m is a positive integer. We now 

 have an inequality of the form : — 



I J^nQ^p) minus m terms of its asymptotic expansion | < ^ | (\p)~"c''^<' \ , (53b) 



which follows by an argument similar to that used in establishing (52). 



On substituting in (53a) from this inequality, and replacing ICn{-iXr) 



by {7r/(- 2'iX?'))ie*^'", if \ p^""(j)(p)dp \ converges all the integrals in p 



J 



converge, and do so uniformly for all A's in the range. And the argument 

 used of (53) applies to the first of these integrals, and, a fortiori, to the 

 others. 



If r is zero, the integral in (14) is, of course, zero when n is positive 

 infinite when n is negative. 



