CIRCUIT THEORY ASH OPERATIONAL CALCULUS 757 



If, on the other hand, the real part of p„ is nonativc, we \vrit(< f I 11) as 



\ \ >.'«' v/-r / 



The term e'-' ultimately dies away, and the delinile inte^jnil ran lie 

 expanded asymptotically in accordance with the theory discussed 

 under Rule 1, again leading to an asymptotic scries identic.d with that 

 given by direct application of the Ueavisidc Rule to the operational 

 equation. 



Consequently since the operational equation in //„ can be asymptotic- 

 ally expanded by means of the Heaviside Rule, the operational equa- 

 tion in h = '^lim is similarly asymptotically expandible, and the 

 Heaviside Rule is verified for equation (1;^:?). 



We have now covered, more or less completeK', the tiieoretical rules 

 and principles of the operational calculus in so far as they can be 

 formulated in general terms. We shall now apply these principles 

 and rules to the solution of important technical problems relating to 

 the propagation of current and Noltage along lines. In doing, so, while 

 we shall take advantage of our table of integrals with the corresponding 

 solutions of the operational equation, we shall also sketch Heavi- 

 side's own methods of solution. 



We shall close this discussion of divergent and asymptotic expan- 

 sions with a general expansion solution of considerable theoretical 

 and practical importance in the problem of the building-up of alter- 

 nating currents. It will be recalled from Theorem III that the response 

 of a network of generalized operational impedance II(p) to an e.m.f. 

 E(t) impressed at time t = o '\s given by the operational formula 



^_V{p) 



, mp) 



where E= V(p) is the operational equation of the applied e.m.f.: that 

 is, analytically 



--V(p)= rE{t)e-P'dt. 



P ^0 



\ow suppose that the impressed e.m.f. is sin uit: then liy formula (/i) 

 f)f the table of integrals 



