722 BELL SYSTEM TECHNICAL JOURNAL 



In formulas (1), (m), (n), J„{x) denotes the Bessel function of order 

 n and argument x. In formula (p), Io{x) denotes the Bessel function 

 Joiix) where i = V — 1 • 



This list might be greatly extended. As it is, we are in possession 

 of a set of solutions of operational equations which occur in important 

 tfchniral problems and which will be employed later. 



The foregoing emphasize the practical and theoretical importance 

 of recognizing the equivalence of the integral and operational equa- 

 tions. With this equivalence in mind, the solution of an operational 

 equation is often reduced to a mere reference to a table of .infinite 

 integrals. Heaviside did not recognize this equivalence. As a 

 consequence many of his solutions of transmission line problems are 

 extremely laborious and involved and in the end unsatisfactory 

 because expressed in involved power series. 



.\ot all the infinite integrals corresponding to the operational 

 etiuations of physical problems have been evaluated or can be recog- 

 nized without transformation. This statement corresponds exactly 

 with the fact that a table of integrals is not always sufficient but 

 must be supplemented by general methods of integration. We turn, 

 therefore, to stating and discussing some general Theorems applicable 

 to the solution of Operational Equations. 



In the derivation of the operational thcortnis, which constitute the 

 general rules of the Operational Calculus, the following proposi- 

 tion, due to Borcl aiui known as Horel's theorem, will be frequently 

 employed.* 



If the functions f{t) , flit) , and f«(t) are defined by the integral equations 



F{p) = f me-^'dt 

 »'o 



Fx{p)= rf,{t)e-'"dl 



F.(p)= rf.{l)e->"dl 



and if the functions /•', Ft and F-i satisfy the relation 

 F{p) = Fi{p).F,{p) 



• For a prcKjf of this important theorem the reader is referred to Rorel, " Leioiis 

 sur les Series OiverRentcs" (1901), p. 104; to Bromwieh, "Theory of Infinite Series," 

 pp. 28(^-281; or to I-'orcl, "Studies on Divergent Series and Summaliility," pp. 93-94 

 (l>cinK Vol. II of the Michigan University Science Series, published liy Macmillan). 

 The proof depends on Jacobi's transformation of a double integral: see Kdward's 

 "Integral Calculus," 1922, Vol. II, pp. 14-l.S. 



