A Generalization of Heaviside's Expansion Theorem

HE well known expansion theorem given by Heaviside in Vol. II 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 * = + »= 1 . 2 . 3 - a) p is the differential operator d/dt, and pi, pn · · · are the roots of Z{p) = 0. Z'{pn ) 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 depends 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. G E N E R A L I Z A T I O N O F 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 h = h m Z(P ) as aS h h = m D{q) ' where q is a function of the operator p. W i t h suitable restrictions on the functional forms of N and D the solution of the operational equation is given by . N(0) , ^ N(qn ) f / , . t . = + (2)