38 BELL SYSTEM TECHNICAL JOURNAL 



where x = w m t; p = w/2co m ; and the f s (x) functions are denned and 

 determined by the integral identities, 



r /,Mri * c -(i&ir (3 - 24) 



for all integral values of s. 



For 5 = 0, the solution of this equation is known; it is 



fo(x) =COS X. 



The solutions for 5>0 are gotten from the recurrence formulas 29 



fs(x)= I cos (* — X)/,_i(X)dX. 



Jo 



Repeated applications of this formula give 



Mx) =^(P 2s (x)cos x + Q 2 s(x)sm x) 



where P 2s and Q 2s are polynomials in x of the 2s th and (25 — 1)'* 

 orders respectively. Thus : 



^-"WL + HW^. + tW 



and 



2jt! ' ry ' (25-2)! ' ,w (2j-4)! ' * ' 



(terminating in term in * 2 /2!), 



x 2s ~ 1 x 2i_3 x 25-5 



0to = fl(5) (27^17! + b{s) (25^31! +C(5) (27^5)1 + ' ' 



(terminating in term in x/l\). 



The a, /3, 7 . . . a, b, c, . . . coefficients are functions of the order 5; 

 the first few coefficients are: 



a(s) = l, 5 = 0, 



«(*) = -g— • * = 1. 



, (s) ,_ (25-2)^25 + 1) ^^ 



^. N _ (2f-2) (25 + 1) (25-3) (25 + 1) (25 + 2) ^ 

 °W~" 8 346 ,5 ~ 



If the foregoing expressions for the / s functions are substituted in 

 the series solution (3.23) for A n (t) and if the series are rearranged as 

 explained below, we get writing wt/2 = px = y, 



A' n (t) = ^ J Jiniy) cos x + pRi(y) sin x + p 2 R 2 (y) cos x . . I • 



29 See equation 10, The Heaviside Operational Calculus, 5. 5. T. J., Nov., 1922. 



