CIRCUIT THEORY AND OPERATIONAL CALCULUS 711 



AK.iiii U't p approaih iiillnity: in the limit the left hand side of the 

 (.■((uation l)fci)nit's Oi and wc li.i\«.' 



/j<'>(o)=a,. 



PrDCi'cdinv; by successive partial inirur.itioiis we tluis establish the 

 general relation 



Hut by TaNlor's theorem, the power series expansion of /;(/) is simply 



/i(0=/»(o)+/'<'Ho)/ l!+//<2)(o)/r2!+ 



whence, assuming the lonvergence of this iw pans ion, we ^et 



//(/)=ao+rt,/ l! + a«/=;2!+ ...= Vrt„r w! (35) 



which establishes the power series solution. It should be carefully 

 noteil, however, that it does not establish the convergence of the 

 power series solution. As a matter of fact, however, I know of no 

 physical problem in which Hip) satisfies the conditions for an asymp- 

 totic e.vpansion, where the power series solution is not convergent. 

 On the other hand many physical problems exist, including those 

 relating to transmission lines, where a power series solution is not 

 derivable and does not exist. 



The process of expanding the operational equation in such a form 

 as to pertnit of its being converted into the explicit solution is what 

 Heaviside calls "algebrizing" the equation. In the case of the power 

 series solution the process of algebrizing consists in expanding the 

 reciprocal of the impedance function in an asymptotic series, thus 



1 //(/)) ~rt„ + a, p+a-2 p-+ 



Regarded as an expansion in the variable />, instead of as a purely 

 symbolic expansion, this series has usually only a limited region of 

 convergence. This fact need not bother us, however, as the series 

 we are really concerned with is 



ao+a,//i!+a2/V2!+ 



It is interesting to note in passing that the latter series is what Borel, 

 the French mathematician, calls the associated function of the former, 

 and is extensively employed by him in his researches on the sunima- 

 bility of divergent series. 



The process of "algebrizing," as in the examples discussed abo\e, 

 may often be effected by a straight forward binomial expansion. 



