748 BELL SYSTEM TECHNICAL JOURNAL 



Tlie infinite integrals are known and have been evaluated. Sul)- 

 stituling their values this series becomes: — 



V2^/ ( ^8X/'^2!(8X/)^'^3!(8X<)''^ • " | 



which is in fact the well known asymptotic expansion of ihc fuiulion 

 e-^7„(X0. 



A second example may be worth wliile. Consider the case of an 

 e.m.f. e~^' impressed at time t = o on a cable of distributed resistance 

 R and capacity C: required the current entering the cable. The 

 required formula is * 



T- l~C d pe-xc-^) ^ 



by ob\ious transformations. 



Asymptotic expansion of the definite integral as in the preceding 

 example gives the asymptotic formula 



\ TrRt I 2X<^(2X/)2^(2X/)'^ ■ ■ ) 

 The operational formula of the problem is 



HI: 



P+X 



.•\l)l)l\ing the Heaviside Rule, we get the asymptotic exijansion 



\ tRI I 2X^"'" (2X/)- (2X0' ■ ■ ■ * 



which agrees with the jireceding formula, deri\eil from the definite 

 integral. 



We shall now discuss a specific problem in which the Hea\ iside Rule 

 breaks down. For example let us take the preceding problem, and 



•The derivation of the formulas in this problem is left as an exercise for the 

 reader. 



