A Generalization of Heaviside's Expansion Theorem 



By W. O. PEN NELL 



The expansion theorem is one of the most frequently used methods of 

 evaluating operational forms arising from the operational calculus de- 

 veloped by Heaviside. The original theorem, however, is applicable, in 

 general, only to expressions containing integral powers of the operator didt. 

 This paper describes an extension to, or a generalization of the original 

 expansion theorem whereby, in general, operational forms with either 

 fractional or integral powers of the operator can be evaluated. A number 

 of operational equivalents are given to be used with the theorem, one of 

 which is the equivalent used by Heaviside. Examples of the application 

 of the theorem to electric circuit problems are shown. 



THE well known expansion theorem given by Heaviside in Vol. H 

 of his "Electromagnetic Theory" may be stated as follows: 

 An operational equation of the form h = Y{p)/Z{p), may under 

 certain well known restrictions on the functions Y and Z, have as its 

 solution 



A = |[^+Et^^^^"', n=l,2,3---. (1) 



Z(0) „ pnZ {pn) 



p is the differential operator d/dt, and pi, p2 - - • are the roots of 

 Z{p) = 0. Z'ipn) is the result of substituting pn for p in d(Z(p))/dp. 

 The theorem is true only when no root is zero and all roots are unequal. 

 Y(p) and Z(p) must contain p to positive integral powers only. Various 

 proofs of this theorem have been given and perhaps the simplest de- 

 pends upon the expansion of Y(p)/Z(p) by partial fractions. 



The expansion theorem is valuable in the solution by operational 

 methods, of problems in mathematical physics, and especially electric 

 circuit theory problems. 



Generalization of the Expansion Theorem 



The generalization of this theorem may be stated as follows : Under 

 certain circumstances it may be possible to write the operational 

 equation 



^ Z(p) ^' ^' D{q)' 



where g is a function of the operator p. With suitable restrictions on 

 the functional forms of iV and D the solution of the operational equa- 

 tion is given by 



7V(0) iV(g„) 



where \p{t, qn) is the equivalent of the operational expression q/(q — q„) 



482 



