iiRirir 77//.()A"i" .;.\7) ori:R.irn>x.ii. i.ii.cri.rs 70s 



\W virtue of this tlieoriMU, we have the advantage, at the outset, 

 of a ke\' to the ineauiiiK of Heavisiile's ojierational e(|uations, aiul a 

 means of chcekini; and deducing liis rules of sohitioii. Tliis will 

 serve us as a guitle throu^ihoul our further stud\-. 



Returning now to Heaviside's own point of view and method of. 

 attack, his reast>ning may l)e described somewhat as follows:-^ 

 The operational e(|ualion 



/; = ! //(/>) 



is the full e(iui\alent of the (iitferential e(|uations of liie problem and 

 must therefore contain the information necessary to the solution 

 proviiled we can determine the significance of the symbolic operator 

 p. The only way of doing this, when starting with the operational 

 equation, is one of induction : that is, we must compare the operational 

 equation with known solutions of specific prol)lems and thus attempt 

 to infer by induction general rules for interpreting the operational 

 ecjuation and con\erting it into the re(iiiired explicit solution. 



The Power Series Solution 



Let us start with the simplest possible problem: the current in 

 response to a "unit e.m.f." in a circuit consisting of an inductance L 

 in series with a resistance R. 



The differential equation of the problem is 



L^A+IL\=\, t^o, 

 at 



where A is the indicial admittance. Consequently replacing d dt by 

 p, the operational equation is 



A = ~^- 

 pL + R- 



The explicit s )luti<)n is easily deri\ed : it is 

 A =~^(\ -€-"•) 



where a = R L. Note that this makes the current initially zero, so 

 that the e(|uilibrium boundary condition at t = o is satisfied. 



Now suppose that we expand the operational equation in inverse 

 powers of p: we get, formally, 



pLl+ap Rpi+a p Rip ^p' ^'p P "f 



by the Binomial Theorem. 



