Orr — Extensions of Fourier's and the Bessel- Fourier Theorems. 215 



Hankel's original equation for + n, the following, in which it has been 

 deemed unnecessary to divide the range of p into two : — 



\clX [ Y„(Ar) Yn{\p) Pi>(p)dp = 2{<p (r - e) + (r + e) }. (36) 



The only case in which the path of A is reducible to a line is that for 

 which n is zero, and then we deduce 



[ dx\ \ {r,{Xr)Jo(Xp) + Jo(Xr)ro{Xp)]dp = 0. (37) 



Jo J « 



As this method of obtaining Hankel's original equation when n is integral 

 involves the differentiation of (31) with respect to n, I indicate a slight 

 variation, applicable to any case in which ?i + 1 is positive. 



We have 



— Uni - TT J niri ni 



iriJn [x] = e -i Kn (xe ~^) - e^~ K,i {xe ~^) . (38) 



In the integral 



[ Xdx[ Jn{Xr)Jn(Xp)p^{p)dp (39) 



Jo J'» 



make this substitution for JniXr), and change the term involving 



^«(Ar6?) 

 into a similar integral from to - co, involving 



-niri \J^ f/i -iri p' 



{TTiyc-^n-^\ XdXKn{Xre~) Jn[Xp)p<pip)dp, (40) 



c J -ft J a 



Kn (Xre ~^ 

 We thus express (39) in the form 

 ■"'^» Lt. d' 



c]-h 



the path passing above the origin. Deforming the path into a contour 

 everywhere at infinity, and using the asymptotic values as before, we 

 evaluate (39), and obtain |^(r - t). 



For the range from r to h the substitution (38) is to be applied to JniXp), 

 instead of to Jn[Xr]. 



Art. 8. An alternative Discussion: SommerfekVs Investigation extended to an 



unrestricted n. 



It may be of interest to point out another method by which equation (14) 

 may be established when n + 1 is negative. In one of the known proofs* 



* Sommerfeld, Die Willkiiilicheu Fimctionen in der Mathcmatiscbeu Pliysik, Iiiaug. JJiss., 

 Konigsberg, '91 (or '01). I have no first-haiid knowledge of this paper; probably my heading of 

 this Art. is too ambitious in that Sommerfeld's investigation might apply to more arbitrary functions 

 than does that given here. See Hardy, " Further Ilesearclies in the Theory of Diveigi-nt Series and 

 Integrals," Camb. Phil. Trans., xxi., p. 44. 



[31^j 



