CIRCL'IT THEORY AND OPFMATIONAL CALCULUS 709 



straightfon\'ard, is actualK- in practice extrenicis- l.iliorioiis and (om- 

 plicatcd. We note tliese |)oints in passing; a more complete estimate 

 of the value of the power series solution will be made later. 



To summarize the preceding: Heaviside, generalizing from specific 

 examples otherwise solvable, arri\'ed at the following rule : — 



Expand the right hand side of the operational equation 



h = \/ll{p) 

 in inverse powers of p: thus 



h^ao+ai/p+at/p-+ . . . +«„'/?"+ .... 



and then replace t; by /"/"'• ^^^ operational equation is thereby con- 



P 

 verted into the explicit power series solution : — 



/»=ao+ai//l!+aj/V2!+ • . +a„r/nl+ . . . (35) 



As stated above, this rule was arrived at by pure induction and 

 generalizatit)n from the known solution of specific problems. It can- 

 not, therefore, theoretfcally be regarded as satisfactorily established. 

 The rule can, however, be directly deduced from the integral equation 



\-^=fhit)e-^'dt. 



pmp) 



To its derivation from this equation we shall now proceed. 



First suppose we assume that h{t) admits of the power series ex- 

 pansion 



//„ + /ii//l!+/;:/-, 2! + 



Substitute this assumed expansion in the integral, and integrate 

 term by term. The right hand side of the integral equation becomes 

 formally 



ho/p + h,'p^ + h-./p'+ 



by virtue of the formula 



/'°°iy<" = -Lfor/>>o. 

 Jo nl p"^^ 



.\ow expand the left hand side of the integral equation asymptotically 

 in inverse process of />: it becomes 



a<,//)+a,//>=+a,//)'+ 



where 



ao+ai/p-\-ai/pr+ .... 



