Orr — Extensions of Fourier'' s and the Bess el- Fourier Theorems. 225 



Consider, next, the range from r to l. If we now take the second form, 

 (64) it is similarly seen that the first term now gives 



and that the second term is zero when r > a, but 



— (p {r + e) when r = a, 



so that the complete integral is zero in the latter case, as is evident d priori. 



Thus the equation expressed in either of the forms (60), (63), (64) s 

 established. 



Art. 14.-4 Generalization of the Preceding Result. 



Equations (60), (63), (64) are examples of a general type which can be 

 established in a similar manner. 



Suppose F{ix) denotes a function of the form ':^pfp{x)(dldxyKn(ix), 

 where, for each value of j;, fp (x) is a polynomial which is an even function 

 of o: when ^9 is even, and an odd when ^j is odd, or else vice versa: (by using 

 the differential equation satisfied by K, such a function F(ix) may be expressed 

 in a variety of forms, and includes, for example, Kmiix), and all of its deriva- 

 tives with respect to x, where m '- ■?i is an integer) ; then, provided the equation 

 F{x) = has no roots whose arg. lies between ±7r/2, the following hold: — 



f^ { K,liX.r)F^-i\ayK,l-ikr)F{i\a)] [K,li\p)F{-iXa)-K„{-iXp)F{iXa)] 



XdX 



F{iXa) F{- iXa) 



-p4>{p)dp 



Lt. 



h 



XdX 



-k 



Kn{-iXp) 



F{- iXa) 

 X \K„(iXr)F{-i\a) ~ Kn{- i\r)F{i\a)) p^{p)dp* 



rh 



^^- ' XdX 



rb 



-h 



K n{-i\r) 

 a F(-iXa) 



X {K,{iXp)F(- iXa) - K„{- iXp)F(iXa)] p<l>{p)dp* 



= - ^ {-^O'-O + <^0' + OJ' cL<r<h. (69) 



TT" 



If r = b, the right-hand member is to be replaced by ~:y.<j>{^^ ~ f ) 5 



if r=^a, by - ^ ^j^; k-ie^i'^Fi_ih)IF{-ih)U{a+,), 



the limit of the expression in brackets being either zero or 2, according to 



* In these two forms of the integral, the F's may be replaced bj"- functions of a still more genei-al 

 character. 



