NOTES ON THE HEAVISIDE OPERATIONAL CALCULUS 157 



If now all the singularities of the integrand lie to the left of the imag- 

 inary axis, then k{t) = /?(/) and (31) is the formulation of the 

 Heaviside problem. Suppose, however, that the electrical or dynamic 

 system specified by II{p) is "unstable"; that is, it contains some 

 internal source of energy which makes its transient oscillations in- 

 crease with time / instead of dying away. In such a case H{p) 

 will have zeros to the right of the imaginary axis, and in order that 

 (31) shall be the solution of the Heaviside problem, c must be taken 

 so large that all the singularities of the integrand lie to the left of 

 the path of integration. Consequently 



h{t) -^ I -^TTTT^P' 



J_ r+'" g^' ^^ (2) 



iiriX-i^ pn{p) 



pro\ided c is so chosen that all the singularities lie to the left of the 

 path of integration in the complex plane. This is Bromwich's formu- 

 lation of the Heaviside problem. ^° 



From the foregoing it follows that the Fourier integral 



-f 





is, in general, the formulation of the Heaviside problem if and only 

 if, all the singularities of the integrand lie to the left of the imaginary 

 axis. If there are singularities on the imaginary axis, the integral 

 is ambiguous, while if there are singularities to the right of the im- 

 aginary axis, the integral gives an incorrect solution of the Heaviside 

 problem. ^^ 



As a simple example consider the operational equation 



h = \/H{p) = ^ 



P-is' 



where the real part /Sr of /3 is positive. The correct solution as given 

 by either (2) or (3) is 



// = / < 



= e^' t > 0, 



^° The appropriate mathematical methods of solving the infinite integral (2) are 

 dealt with in great detail by Jeffreys in his "Operational Methods in Mathematical 

 Physics" (Cambridge University Tracts). 



" To prevent misunderstanding it should be stated that the application, when 

 permissible, of the classical Fourier integral (2a) to the Heaviside problem, was 

 known long prior to the work of Bromwich. Bromwich's essential and important 

 contribution lay in showing that the path of integration must be shifted to the 

 right of all the singularities, together with a verification of an important form of 

 solution, first given by Heaviside, of the operational equation. 



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