An Extension of Operational Calculus 



By JOHN R. CARSON 



THE Heaviside operational calculus postulates at the outset that 

 the initial (boundary) conditions at reference time / = are 

 those of equilibrium; that is to say, the system is at rest when suddenly 

 energized at time / = by a "unit" impressed force. By unit im- 

 pressed 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 Maga- 

 zine, Van der Pol briefly indicated the appropriate procedure for ex- 

 tending 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 



Znll + Z12/2 + • • • + Zlnln — El 



(1) 



Znlll -\-Zn^h + • • • + Znn^n = En 



where 



^'■' = (^4 + ^"+i/_/') 



(2) 



Now multiply the equations (1) by e~p^ throughout and integrate 

 from to infinity'. Also let /„ and F„, denote the Laplace transforms 

 of Im and Em', thus 



/,„ = r /„,e-"' dt, 



° (3) 



F,n - I Er„e-p' dt. 

 Jo 



Now let /m° and Qm^ denote the initial values (at time / = 0) of 

 /„ and the charge Qm in the mth mesh; also let us replace Zjk- of (2) by 



Zi, = pLj, + Rik + 1//>C,A-. (4) 



340 



