736 BULL SYSTEM TECHNICAL JOURNAL 



The straijjiitforward Hea\iside soliilion is obtained by expanding 

 the operational equation as follows : 



Identif>'ing \/p\\\th \l\/rl (from known solutions of allied problems) 

 and substituting for !//>" multiple integrations of the Hth order we get 



l-SJ- i 1 I (2^^^ (2X/)-^ 1 .3(2X0^ _ _ ) _ 

 ^~\^l^+ 2 2.3.4+2.3.4.5.0 * ^''^ 



It can be verified that this solution is ronvergent and equivalent 

 to (77). 



This problem, while simple and of minor technical interest, will serve 

 to introduce us to the very important and interesting question of 

 asymptotic series solutions. 



An asymptotic series, for our purposes, may be defined as a series 

 expansion of a function, which, while divergent, may be used for 

 numerical computation, and which exhibits the behavior of the func- 

 tion for sufficiently large values of the argument. 



Let us return to equation (77). We observe that the series solu- 

 tion (78) of the definite integral becomes increasingly laborious to 

 compute as the value of / increases. This remark applies with even 

 greater force to the Heaviside solution (79) on account of the alter- 

 nating character of the series. Right here we have an excellent 

 example of what I regard as Heaviside's exaggerated sense of the 

 importance of series solutions as compared with definite integrals. 

 Consider the solution in the form of (77) as compared with Heavi- 

 side's scries solution (79). The former is incomparably easier to 

 interpret and to compute, eitlicr by numerical integration or by 

 means of an integraph or planimcler. In fact the series (79) is prac- 

 tically unmanageable except for small values of /. 



Returning to the question of an asymptotic expansion of the solu- 

 tion (77), we observe that the definite integral_^appearing in that 

 c(iuation can be written as, 



J/»ig-xi /'°°e-X' /""e"^' 



y/t Jo y/T Ji y/T 



(80) 



