112 BELL SYSTEM TECHNICAL JOURNAL 



g > 10 



Jo' {qy/ - i) = i Jo (gV- i) 



Ko' (g V^ ^ - iKo {qV^^i) 



Note II — Transformation of Fourier-Bessel Expansion 



In problems involving Fourier-Bessel expansions it is sometimes 

 necessary to transform quantities of the form 



cos scbj sin sd)] , 

 —-7—, — ;r-. log Pi. 

 Pj Pj 



from the system of coordinates pj, (pj to the systems p, 4> or r, d which 

 are related as shown in Fig. 5. 



The necessary formula may be derived as follows. We have 



cos S(f)j + i sin S(j)j e^'^j /e'^^j 



'Pj V Pj ) ^j, 



Pj 



where Zj is the conjugate of the vector Z'j = p^t**^-'- Similarly 

 we may write 



Cj = c.-e^'^ + '^i) 



The vectors Zj, Z and C, as may be seen from Fig. 5, have the 

 lengths Pj, p and c, respectively, and the directions indicated by the 

 arrows. 



By vector addition, 



Z'j = Z -^ Cj 



whence Zj = Z' -\- C'j, 



where Z' and C'j are the conjugates of Z and Cj respectively. 

 By expansion 



J_= 1 ^ JLHl - -i- ^ -u ^ (^ + 1) ^ 



z] iz' -\- c'y c:^L 1 c'i 1.2 c'2 



5 (5 + 1) (5 + 2) Z'' 

 1.2.3 Cp 



-] 



