740 BELL SYSTEM TECHNICAL JOURNAL 



in the example discussed before, the asymptotic expansion results 

 from repeated partial integrations; thus 



aV7 2aJ, tVt 



2-a-Jt 



O-UT 



^±?l -^—^dr 



and finally 



aV t 2a'tVT 2-a-J, t^\/t 



e-"' j ,_ J_ , Jj3_ _ 1.3.5 , / 

 a^/7 ( 2at'^ {2at)^ (2a0'"*" ' ' • T (90) 



The series (90) is divergent just as is (82) of a preceding problem 

 and the error committed by stopping with the smallest term, is of 

 the same character and subject to the same di.scussit)n. Willi this 

 understanding we write the solution (89) as 



For large \alues of t {at>5) this series is accurately and rui)idly 

 computable. Furthermore it shows by mere inspection the be- 

 havior of V{t) for large values of /, and that it ultimately approaches 

 zero as l/y/wat. 



Let us now see how Heaviside attacked this problem ami how he 

 arrived at a divergent solution from the operational formula. Re- 

 turning to the operational equation (85), it can be written as 



r= ^/L - (92) 



l+vA/« 



Now expand the denominator by the binomial theorem: we gel 

 fnrmallv 



