CIRCUIT THEORY .INP Ol'ERATIONAL CAl.Cl'l.VS 713 



solutions which actually conceal rather than reveal the significance 

 of the original clitTerential efiu.itions of the problem. On the other 

 hand, the following remark of his indicates to me that Heavisidc 

 has a <iuite exaggerate*.! idea of the value and fundamental character 

 of i)t)wer series in general: "I regret that the result should he so 

 complicated. But the only alternatives are other ec|uivalent infinite 

 series, or else a definite integral which is of no use until it is evalu- 

 ate<l, when the result must be the series (135), or an equivalent one." 

 As a matter of fact the properties of most of the important functions 

 of mathematical ph\sics have been investigated and their \alues 

 computed by methods other than series expansions. I may add that 

 in technical work the power series solution has proved to be of re- 

 stricted utility, while definite integrals, which He.iviside * i)articu- 

 larly despisetl, have proved quite useful. 



The Expansion Theorem Solution ' 



We pass now to the consideration of another extremely important 

 foim of solution. Heaviside gives this solution without proof: we 

 shall therefore merely statethc solution and tlicii fl('ri\T it from the 

 integral equation. 



(iiven the operational equation 



li = l/H(p) 



which has the significance discussed above: i.e., the response of the 

 network to a "unit e.m.f.". The explicit solution may be written as 



^=m+XpjF{p;) ^^^^ 



where p\. p« ■ ■ ■ pn are the n roots of the equation 



II{p)=o 

 and 



//W = [^//(/»]^^^^. (39) 



.As remarked above, this solution, referred to by him as The Ex- 

 pansion Theorem, was stated by Heaviside without proof; how he 

 arrived at it will probably always remain a matter of conjecture. 

 Its derivation from the integral equation is, however, a relatively 

 simple matter, though in special cases troublesome questions arise. 



* Vide a remark of his to the effect that some mathematicians took refuge in a 

 definite integral and called that a solution. 



* This terminology is due to Heaviside. A more appropriate and physically 

 significant expression would l>e "The Solution in terms of normal or characteristic 

 vibrations." 



