NOTES ON THE HE AVI SIDE OPERATIONAL CALCULUS 159 

 provided c is sufficiently large to make the transient term 



/^C-|-lQO 



T„{t)dp 



^ C—iaa 



negligibly small. Analytically this requires that c be so large that 

 the zeros of pllip) shall all lie to the left of the axis pji = c. 



Ill 



The foregoing discussion tacitly assumes the existence of an unique 

 solution of the operational equation. On the part of the physicist 

 this assumption is entirely proper because if the operational equation 

 is the symbolic formulation of a correctly set physical problem an 

 unique solution must and does exist. When approached from the 

 purely mathematical standpoint, however, the case is different and 

 there is no assurance of the existence of a solution. As an example 

 consider the operational equation 



h = e" 



The corresponding integral equation 



f = r 



P Jo 



h{t)e-p'dt pR > 



has no solution, while Bromwich's formula 



c-ioo P 



1 r'+"^ ov 



gives h = t < — 1 



- 1 / > - 1 



which is obviously incorrect. As a matter of fact the operational 

 equation itself has no solution. 



To formulate the necessary and sufficient conditions for the exis- 

 tence of a solution we may proceed as follows: If a solution exists 

 it is given by either of the equations 



h{t) = ^ r j\P)e"'dp, (2) 



KP) = f 



00 



hit)e-p'dt pR > c, (3) 



. 



