On^— Extensions of Fourier'' s and the Bessel-Fonrier Theorems. 243 



Fp denoting a polynomial ; and n is not necessarily positive. And the 

 solution obtained here becomes 



TT (2 sm mr)'^ ^B j p ^_n 



"( ■ ^Fp{X){d/db/>J„[Xh) 



= |{^(r-.) + <^(r + £;i. (52) 



The proof may follow the same general plan as before, but, as in the 

 corresponding case in which a is zero in Hankel's integral, some modification 

 is required. This may be done on the lines of Part I., Art. 10. 



And, as expansions of the type given there, (52), (53b), are to be used 

 for both t7„(Ap) and /-^(Ap), (or else for Z'„(± *Ap)), a sufficient condition 

 to be satisfied by ^ in neighbourhood of p = is that jo\p^'^'(j){p) dp\ 

 should converge, where ^j is the greater of the numbers n, ^. 



When n is an integer, or indeed in any case, the numerator, in so 

 far as it involves Xh, may be expressed in terms of the F's. 



Art. 5. Nature of the Convergence, Order of Magnitude of the Terms. 

 Term hy Term Differentiation. 



The nature of the convergence of the series and the order of magnitude 

 of the terms are the same as those of the Fourier series. 



I consider the Bessel series : like considerations apply to the trigono- 

 metrical, which, however, might be discussed independently. In (49) the 

 fraction whose residue has to be obtained reduces asymptotically, when A 

 is large, to the form 



sin {A (r - &) - /3] sin [\ fp - a) - a] /sin [X{h - a) + ^ - a], (53) 



where a, /3 are constants which may be complex. Omitting a factor 

 [l - «)~\ the corresponding residue assumes asymptotically the form 



( - a ) I - a \ 



where m is an integer and 7, 8 are constants.* Thus the dominant portion of 

 the m*^ term in the series is obtained, save as to a constant factor, by 

 multiplying (54) by {p/r)i(p(p) dp, and integrating from a to h, the fractional 

 error being ultimately of order m~\ When this is divided into two terms 

 by expressing the product of the sines as the difference of two cosines, one 

 differs from the term in a Fourier series only by having (mir + 7) instead 



* It must be borne in mind that the angles themselves in (54) are only asymptotically coneet ; 

 it suffices that the eiTor in eacb dirninislies indefinitely with increasing ni ; it is, in fact, of order wr'. 



