An Extension of Operational Calculus

HE Heaviside operational calculus postulates at the outset that the initial (boundary) conditions at reference time / = 0 are those of equilibrium; that is to say, the system is at rest when suddenly energized at time t = 0 by a " u n i t " impressed force. By unit impressed force is to be understood a force which is zero before, unity after, time / = 0. In a paper published in Volume 7, 1929, of the Philosophical Magazine, Van der Pol briefly indicated the appropriate procedure for extending the operational calculus to cover arbitrary initial conditions. The present paper is an exposition of this generalization for a system of a finite number of degrees of freedom, followed by an application to the differential equations of the transmission line. While stated in the language of electric circuit theory, it is to be understood that the processes are generally applicable to a wide variety of problems. We start with the canonical equations for a network of n degrees of freedom Zllll + 212^2 + ' ' ' + Zl nln = -Ei (1) Znl I +Z712/2 + · · · + Znn In = En where Now multiply the equations (1) by e~ pt throughout and integrate from 0 to infinity. Also let Jm and Fm denote the Laplace transforms of Im and Em ; thus Jm = ° Im e~ rt dt, ( 3 ) Fm = I Jo Em e-»' dt.