54 BELL SYSTEM TECHNICAL JOURNAL 



We assume that 1/H(p) can be formally expanded in the series 





We shall here introduce a necessary restriction on the function 1/H{p). 

 It must include no function which is represented asymptotically by 

 a series all of whose terms are zero; that is a function <p{p) such that 

 the limit, as p approaches oo, of p"(l>ip) is zero for every value of n. 

 The function e~^ is such a function. (See Whittaker & Watson, 

 p. 154.) 



With this restriction understood, start with the integral (9) and 

 integrate by parts; we get 



L^ = h{o) + n e-f^m\t)dt 

 (P) Ja 



H{p) 



where h^''\t) = dydt"h{t). 



Now let p approach infinity; in the limit the integral vanishes and 



h{o) = 1/H{cc) = ao 



from the asymptotic expansion. 

 Integrate again by parts; we get 



p(l/H(p) - ao) = h^'Ko) + n e-P'h^^Ki)dt. 



Now let p again approach infinity; in the limit the integral vanishes, 

 and the right hand side, by virtue of the asymptotic expansion, 

 approaches the limit Ui, whence h''^^ (o) = ai. Proceeding in this 

 manner, repeated integrations by parts establish the relation 

 h^"^ (o) — a„. But provided the series is absolutely convergent, then 



h{t) = VaW(o)/V«! 



2 



= "^ anf'/n ! 



which establishes the formula. 



The power series solution is applicable to a large class of physical 

 problems and has been rigorously established under certain restric- 

 tions by other methods than that employed above (see papers by 

 Bromwich, Phil. Mag, May 1920, p. 407; Fry, Phys. Rev. Aug. 

 1919, p. 115; and the writer, Trans. A. I. E. E. 1919, p. 345). 



