CIRCUIT THEORY AND OPERATIONAL CALCULUS 745 



niarkably enough the expansion is everywhere converKent, while 

 in yet others its application leads to a series which is meaningless 

 for a certain range of values of /. 



Heaviside himself gives no inforniation which would serve us as a 

 guide in informing us when the rule is applicable and when it is not. 

 Consequently it becomes a matter of practical importance, not only 

 to investigate the underKing mathematical philosophy of the rule 

 and to establish it on the basis of orthodox mathematics, but also to 

 de\elop if possible a criterion of its applicability. In this investiga- 

 tit>n we shall have recourse to the integral equation of the problem. 



We shall take up first tlie type of problem (Class I) in which the 

 operational equation is 



and assume that /•"(/)) admits of the formal power series expansion 



/•■(/>) =*o+fri/>+6i/>'+6a/''+ . . . (Ill) 



The corresponding integral equation is 



Vp 



= f h(t\e-t'iU. (112) 



We now assume the existence of an auxiliary function kit), defined 

 and determined by the auxiliary integral equation 



\ow since 



F{p)= n k{t)e-t"dl. (113) 



4==/%- 4= (114) 



it follows from (112), (113), and (114) and Bf)rers Theorem, or 

 Theorem I\', that 



h(l)=A=f'-^}^dT. (115) 



Vt''o y/l-T 



Now if we differentiate (113) repeatedly with respect to p and put 

 p = o, it follows from the expansion (III) that 



b, = {-\)' r-Mt)dt. (116) 



This equation presupposes, it should be noted, the convergence of 

 the infinite integrals for all values of n, and therefore imposes severe 



