A construction theorem for evaluating operational expressions having a finite number of different roots

Abstract

In the course of the past 25 years Heaviside's direct operational method of solving differential equations has, to a large extent, been replaced by solutions involving functional transformations. While mathematically sound this trend has resulted in placing a powerful analytical tool far above the mathematical equipment of the average engineer. With the exception of the celebrated expansion theorem or the method of partial fractions no direct method appears to have been discovered by which the average engineer or engineering student could solve even relatively simple differential equations by Heaviside's calculus. Several expansion theorems have been developed <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1,2,3,4</sup> to cover cases where repeated roots are present but none of these have been set up in a form which could be used by undergraduates. It does not appear that any direct general procedure has ever been suggested to cover the usual type of asymptotic solutions which appear in problems in electrical engineering.