52 BELL SYSTEM TECHNICAL JOURNAL 



which agrees with the foregoing and is the actual asymptotic ex- 

 pansion.^ 



The foregoing discussion is sufficient, it is hoped, to show the place 

 of the integral formula (9) in relation to the Heaviside operational 

 calculus. It is believed to be particularly applicable in connection 

 with a number of questions relating to divergent series and solutions 

 which Heaviside's work has raised and which have received too little 

 attention from mathematicians. 



APPENDIX I 



A proof of the integral formula 



\/pH{p) = n e-i"h{t)dt ' (9) 



can be made to depend very simply on the formula 



d 



' '0 



^(0 =j^fF{t-y)h{y)dy. (5) 



This equation may be regarded as well established and can in fact 

 be deduced in a quite general manner by synthetic arguments. It is 

 derived and employed in papers by the writer (Trans. A. I. E. E., 

 1911, pp. 345-427, and Phys. Rev. Feb. 1921, pp. 116-134) and is 

 deducible at once from the work of Fry (Phys. Rev. Aug. 1919, pp. 

 115-136). 



On the basis of equation (5) the deduction of formula (9), in which, 

 however, no pretense to rigor is made, proceeds as follows; 



If the function F{t) in equations (3) is set equal to e^', the complete 

 solution (5) includes the particular solution ^ 



eP^Hip) 



which irfvolves / only through the exponential term. The complete 

 solution must, therefore, admit of reduction to the form 



x{t) = e^^/Hip) +>'(/) (a) 



where y{t) is the complementary solution. 



' The procedure by which Heaviside arrived at the foregoing asymptotic solution 

 is not, however, always so fortunate For example if a terminal inductance is sub- 

 stituted for the terminal condenser of the preceding problem, precisely the same 

 procedure gives an incomplete result. Heaviside recognized this and added an 

 extra term without explanation (Elm. Th. Vol. H, p. 42) but his solution appears 

 to be doubtful in the light of some recent work by the writer in applying the formula 

 of the present paper to the same problem. 



* Provided H {p) '^ o. This restriction is of no consequence in physical prob- 

 lems, where the roots of H (p) are in general complex with real part negative. 



