Orr — Extensions of Fourier^ s and the Bessel- Fourier Theorems. 241 



interchanging r and p in the multiplier of p(p(p)dp; and this is true, what- 

 ever be the range of values of p. For the difference of the two integrands 

 thus considered is 



^t IKMICXfire-^) - K„{fxr)K,ipp&'')] pcf> {p)dpd^ ; (41) 



and as this is a multiple of 



in { I„(fip)I-n ifir) - /„ [fxr] Ln ifip) \ p(}> (p) dp dfi, (42) 



which has no finite singularities, it integrates to zero when fj. describes any 

 closed contour. 



On thus interchanging r and p, it is seen by reasoning similar to that 

 which precedes, that, if h > r > a, the value of the integral is 



-7r=0(r + £). (43) 



If r = a or h, its value is, of course, zero. 



Adding these results, using Cauchy's theorem of residues, and dividing by 

 - 27r^ there results the equation 



-b 



{K„ (/x«) /C Wc-^) - K„ {fxae^^)K„(inp) ] p<i> (p) dp 



a 



= 7r(2sin%7r)-^Si^ 



\[J,,{\T)J.,.{\h) - J_n{\r)J,X\h)\ 



Jn{\p)J.n{\ct) - J.n {\p)J,, (A«) } p./. ip) dp 



= 2 1 ('' - e) + (■'' + ^) } ' a <T <h 



= 0, /• = a or r = h; (44) 



and, in the left-hand member, fxa, fxb, or \a, \h may be interchanged in the 

 numerator. 



Akt, 3. Generalization of Preceding Result. 



Equation (44j admits of considerable generalization. In it t/'„(A«), J.n{\a) 

 may be replaced respectively by Fi(Xa), F^{\a), where 



F,{\a) - ^fp(X){dlda)Pj,X\a), (45) 



FfXa) = :^fp{\){d/dayj-,.{\a), 



(46) 



