Ork — Extensions of Fourier^ s and the Bessel- Fourier Theorems. 221 



asymptotic expansions of tlie functions;* thus, the dominant term of the 

 coefficient of d\ tends to equality with a numerical multiple of 



■■b 



cos {(2/1 + l)Trl^ - \r] cos [(2n + l)nl-i - \p] {plr)i<j>(p)dp ; (55) 



. a 



when the product of the two cosines is expressed as a sum, one portion of 

 this is a Fourier coefficient ; and arguments, which it is unnecessary to give 

 in detail, apply to the other, very similar to those used in the Fourier case. 



If (55) is replaced by the complete expression, on integrating by parts it 

 appears that a discontinuity or boundary value in 0('')(?') at ;' = i\ gives rise 

 to a part of the coefficient in the development of ^(?-) whose dominant term, 

 when X is large, is a multiple of 



\'P-' cos {(2?i + IJttM - \r} cos {{2n + Ijtt/J- - An + Qj + l)7r/2! 



x(n/r)il<^('^)(n)|;, (56) 



while, if ^ and its derivatives up to the i^'^ are continuous and vanish at the 

 boundaries, the coefficient contains no part of order so high, (p''''^ being sup- 

 posed finite.f (The first part of this statement refers to the simplest mode of 

 representing the coefficient rather than embodies a physical fact ; for in the 

 equation J udv = uv - / vdu any constant may be added to v ; while it is 

 impossible to create or destroy a discontinuity in a function without affecting 

 values elsewhere.) 



Aet. 12. Differentiation of the Fourier- Bessel Fquation under Sign of 



Integration. 



The conditions laid down as sufficient to render differentiation under the 

 sign of integration legitimate in the Fourier case+ equally suffice here. 



For, when (p is finite and continuous and vanishes at the boundaries, the 

 dominant part of the expression obtained by differentiating with respect to r 

 the coefficient of d\ may be written in the form 



TT-'i cosX(r-p).(plr)hp'(p)dp~Tr-' [ cos {{2n+l)Tr/-2 -\(r + p)l(pl>')^\p)dp. 



(57) 

 If (ji'(p) is a Dirichlet function, the former of these is the coefficient in a 

 (Fourier) integral, uniformly convergent except near discontinuities and 

 (integrable) infinities in ^', and the latter is the coefficient in an integral 

 uniformly convergent everywhere in the range. And, this being so, 



* It is legitimate to integrate, and in tlie present case to differentiate, asymptotic power series 

 term by term. It is legitimate, also, to multiply and divide according to ordinary rules. See 

 Whittaker, I.e., pp. 167, 168; Brorawich, '■ Infinite Series, pp. 331, 340. 



f Or rather ^(p^') a Dirichlet function. 



+ See foot-note to Art. 11. And in Arts. 11, 12, I suppose that a is not zero. 

 R. I. A. PROC. VOL. XXVII., SECT. A. - [32] 



