THE HEAVISIDE OPERATIONAL CALCULUS 47 



Now by the conditions of the problem Vq is zero before, unity after 

 time /■ = 0; hence the foregoing equations are operational formulas 

 and by (9) 



e-p'I,{t)dt, 



Ig-T = re-P'VAt)dL 

 p ^0 



The solutions of these equations are obviously 



/;<: = for t<x/v, 



— \j J for t>x/v, 



V^ = ior t<x/v, 



= 1 for t>x/v, 



which are, of course, the well known solutions of the problem. The 

 directness and simplicity of the solution from the definite integrals is, 

 however, noteworthy. 



By virtue of the foregoing analysis the Heaviside operational cal- 

 culus becomes identical with the methods and rules for the solution 

 of integral equations of the type 



l/pH{p) =f\-p'h{t)dt (9) 



to which brief consideration will now be given. 



An integral equation is, of course, one in which the unknown func- 

 tion appears under the sign of integration; the process of determining 

 the unknown function is the solution of the equation. Integral 

 equations of the form of (9) were first employed by Laplace and may 

 be referred to as equations of the Laplace type. More recently they 

 have become of importance in the modern theories of divergent series 

 and summability. The solution of a large number of integral equa- 

 tions of the Laplace type has been worked out; however the procedure 

 is usually peculiar to the particular problem in hand. In this con- 

 nection it is noteworthy that, from a purely mathematical stand- 

 point, Heaviside's operational calculus is a valuable contribution to 

 the systematic solution of this type of integral equations. That is to 

 say, methods which he developed for the solution of his operational 

 equation suggest systematic procedure in the solution of the integral 

 equation (9), as might be expected from the relationship pointed out 

 in the present paper. 



