156 BELL SYSTEM TECHNICAL JOURNAL 



The foregoing results can undoubtedly be derived by integration 

 of the Bromwich integral (2) along the contour suggested by March 

 {I.e.). Wiener in his paper on "The Operational Calculus" {Math. 

 Ajinalen, Bd. 95, 1925) gives an entirely different treatment of the 

 problem. The operational calculus he deals with, however, differs 

 under some circumstances from that of Heaviside, as Wiener himself 

 remarks. A paper by Tibor v. Stacho on "Operatoren Kakiil von 

 Heaviside und Laplaceshe Transformation" (publication 1927 VI 15 

 by the Hungarian University, Francis Joseph) may also be consulted. 



n 



Subject to certain well known restrictions a function f{t) can be 

 expressed as the Fourier integral 



/W = T"; HP)e^''dp. (28) 



the path of integration being along the imaginary axis. We assume 

 for the moment that this equation is valid. 



Now suppose that f{t) represents a force applied to an electrical or 

 dynamic system whose "steady state" or forced response to an applied 

 force F{p)eP^ is 



H{p)' • 



Then the forced response g{t) of the system to the applied force /(/) 

 is given by 



However, in applying the foregoing to the Heaviside problem we 

 encounter an initial difficulty. This is that if /(/) is taken as the unit 

 function (zero before unity after, / = 0) it does not admit of formu- 

 lation as the Fourier integral (28). The unit function, however, when 

 multiplied by e~'' when c is a positive real constant, does admit of such 

 formulation, and it is easy to show that the unit function itself is 

 given by 



c-fioo , 



— dp c > 0. (30) 



c-i« P 





Consequently, if the unit function is the force impressed on the sys- 

 tem, the forced response is 



iTTiJc-io, pn{p) 



