CIRCUIT THEORY /INP OPERATIONA!. CALCULUS 719 



reasoninij. In this chapter we sliall employ this eciuivalenre ti) eslal>- 

 lisli iTrt.iiii Keneral forimil.is aiul llieorems for the solution of oper- 

 ational ctiuations. That is to s.iy, wc shall make use of the principles 

 that (1) any method applicable to the solution of the integral e<iualion 

 supplies us with a corresponding mcthwl for the solution of the 

 operational equation, and (2) a solution of any specific integral e(|ua- 

 tion gives at once the solution of the corresponding operational 

 equation. We turn therefore to a brief discussion of the appmpriate 

 nielh(xls for solving the integral eciuation. 



It may he said at the outset, that the solution of the integral e(|ua- 

 tion, like the evaluation of integrals, is a matter of considerable art 

 and experience; in other words there is not, in general, a straight- 

 forward proce<lure corresponding to the process of ditTerentiation. 



On the other hand, as a purely mathematical question, it is always 

 possible to invert the integral equation and write down hil) as an 

 explicit fimction in the form of an infinite integral. F"or example 

 it may lie shown from the I-"ourier Integral that 



IT J O) 



where a{tS) is defined b\' 



1 



//(/oj) 



= a(co) + i^(a)). 



Later on we shall briefly consider the Fourier Integral; for the 

 present the preceding formula will not be considered further. In 

 certain problems it is of value; for the explicit derivation of h(t), 

 however, it is usually too complicated to be of any use except in the 

 hands of professional mathematicians. As a matter of fact, a direct 

 attack on this formula would be equivalent to abandoning the unique 

 simplicity and advantages of the whole Operational Calculus. 



It has been noted above that any solution of the integral etiuation 

 supplies a solution of the corresponding operational equation. This 

 principle enables us to take advantage of the fact that a very large 

 number of infinite integrals of the type 



£ 



f(l)e-P'dt 



have been evaluated. The evaluation of every infinite integral of this 



type supplies us, therefore, with the solution of an operational equation. 



Of course, not all the operational equations so solvable have physical 



significance. Many, however, do. Below is a list of infinite integrals 



