Orr — Extensions of Fourier'' s and the Bessel- Fourier Theorems. 227 



Moreover, even when (f ) is not zero, if S is of lower dimensions than G 

 the first term in (71), when integrated with respect to A, is seen when we 

 keep in mind conditions (ii), (iii) of Art. 1, to disappear from (70), unless x is 

 zero. Thus, in this case, i.e. the case in which the integral is continuous at 

 X = 0, and does give the value of there, the first differentiation is legitimate, 

 save when x = Q, even if ^(f) is not zero. 



Art. 16. Nature of the Convergence of the Integrals in Art. 1. 



A reference to the proof shows that the nature of the convergence of the 

 integrals in Art. 1 depends on that of two ordinary Fourier integrals, and on 

 that of (6) which converges (uniformly and absolutely) to zero. Thus, as in 

 the Fourier case, the convergence is uniform, except in the neighbourhood of 

 points of discontinuity, or (integrable) infinity, and the point x = 0. And the 

 same is true when, as in Art. 2, x is made negative. 



The order of magnitude of the coefficients of dk is also usually the same 

 as in the Fourier integral. By repeated use of equations of type (70), (71), 

 as in the corresponding Fourier case, it appears that a discontinuity at Xi in 

 (/»(P)ic usually gives rise to terms in the coefficient of d\ in (12) asymptotically 



of the form 



sin I 



\'P-' \(xy ±x) \ib^PHx,) 



COS / I r \ / 



Art. 17. Nature of the Convergence of the Integrals m Arts. 13, 14. 

 Diferentiation under Sign of Integration, 



In considering the nature of the convergence, and the order of magnitude 

 of coefficients of the integrals in Arts. 13, 14, we may evidently substitute 

 the asymptotic expansions of the functions.* I take the more general integral 

 in (69), and consider the first form of the left-hand member. Since F{±i\a) 

 is equal to the product of e+*^'' and an asymptotic series in descending powers 

 of A, the expression for the denominator is simply another series in descending 

 powers. If we replace the terms of the numerator by their asymptotic expan- 

 sions, we then see that the dominant term in the coefficient of d\ tends to 

 equality with a numerical multiple of 



sm 

 cos 



Cb 



sm 



\{r-a) X(p-cc)(p/r)mp)dp, (72) 



J a eos 



both factors being sines, or both cosines. The nature of the convergence is 

 thus the same as in Fourier's integral ; so, too, is the order of magnitude of 

 the coefficients. 



^' Compare Aits. 11, 12. 



