DIFFERENTIAL EQUATIONS 



201 



EXPANSIONS IN BESSEL'S FUNCTIONS 



9.170 Schlomilch's Expansion. Any function f(x) which has a continuous 

 differential coefficient for all values of x in the closed range (o, IT) may be expanded 

 in the series: 



f(x) = a Q + 2 <*iJo(kx)> 



k=i 



a Q = /(o) + - C u f*f (u sin 0)d6du, 



^ Jo Jo 



a k = I u cos ku I 2 }'(u sin 6) dOdu. 



Jo J o 



f(x) = a x n 



/j 



a Q 



= > 



= 2(n+ i) / f(x)x n+l dx, 



xf(x)J n (a k x)dx. 



(Bridgman, Phil. Mag. 16, p. 947, 1908) 



where: 



J 



and 



Ak = 2 W(M*&) - 



(Stephenson, Phil. Mag. 14, p. 547, I97> 



f 



SPECIAL EXPANSIONS IN BESSEL'S FUNCTIONS 



9.180 



CO 



^V X U r /S 



T ci T\ -V O J i T 1 / O J. I 1 I OC I - 



CO 



2. COS 3C = JoW + 2/^ (-l) A /2/;(^). 



