Orr — Extensions of Fourier'' s and the Bcssel- Fourier Theorems. 239 



Suppose it is desired to express, for values of ;• between a and I, an 

 arbitrary function 0(?') in the form' 



SM'J"„(Ar)-^^^/_,(Ar)|, (28) 



where A', B' and the vahies of A are determined by the equations 



A'Jn{\ct)^B'J.,,{\a) = 0, (29) 



A'Jn{U) + B'J_„{\h) = 0. (30) 



The form of the theorem is, of course, well known ; though, save in the case 

 of a = 0, B' = 0,* I cannot give a specific reference. The coefficients are 

 generally obtained by assuming the possibility of such an expression. 



Using, as more convenient, the K functions, and writing A = in, the 

 admissible values of ji are given by the equation 



Kn{ixa)K,,{f.the--^) - K,{f,ae--^)K„(^ib) = 0, (31) 



while each term in the sum is a multiple of 



^n{^ia)J^n(f^re-') - K„{fxae-^)K„{^,r\ (32) 



and also of 



K,^xh)K„(,cre^^) - IQ{jjJje-^)K.{fjir). (33) 



It is to be noted that (32), (33), and the left-hand member of (31) are 

 uniform functions of n, aud consist solely of terms of even degree. 



Equation (31) has an infinity of roots, (known to be real), and those whose 

 value is large ultimately tend to the form 



IX = nTri/(h - a), 

 where n is an integer, positive or negative. 



Consider the integral 



X [K^ [fxa] Kn {^pe^^) - K„ (fxae^j [{„ {fxp) \p((>(p) dp 



\K,Xfxa)K4ixhe-') - K,Xfxae^') K,,{fxh)\ 



(34) 



where the path of p is as in (11). To make the meaning of each K definite, 

 suppose that the initial and final arguments of p are - 37r/2, 7r/2, and that 

 when arg. fx is zero, each power of p has its principal value. 

 First, suppose that r is neither a nor I. 

 Supposing CO to be real and positive, the approximate equations 



^,.(iu^) = (WV^O^^-'^^ (35) 



K„(nxe-') = - i{ir/2px)ie'^- (36) 



* For this case, see Gray and Matliews, " liessel Funclions," chap. vi. 



[34*] 



