746 BELL SYSTEM TECHNICAL JOURNAL 



restrictions on k{l) and hence on F{p). We shall suppose that these 

 restrictions are satisfied, and discuss them later. 

 Now (115) can be written as: — 



h{t) = ^ f'dT.kir) (l-r/0-'''2. (117) 



It can he shown that, if k{t) satisfies the restrictions underhing 

 (IIG), the integral (117) has an asymptotic solution obtained as 

 follows: — Expand the factor (1 — t//)~'' by the binomial theorem, 

 replace the upper limit of integration b>' -x. , and integrate term l)y 

 term : thus 



'^'^~^U"'^'^'^'4/rr!^('^^' 



^ (2/) 



1 'A f'^ t- 



{\m 



Finali\' from (lUJj we get 



which agrees exaclK- witli llie Heaviside.rulc for this case. 



The foregoing says notiiing regarding the asymptotic character 

 of the solution. It is easy to see (iualitati\ely, however, that (118) 

 and therefore (119) does represent the behavior of the definite in- 

 tegral (117) for large values of /, provided k{t) converges with suffi- 

 cient rapidity. 



The foregoing analysis ma\' now he sumniarizetl in liie following 

 proposition : 



If the operational equation h = \/II{p) is re<liicH)le to the form 



h = F(p)Vp 

 and if F(p) admits of power series expansion in p: thus 



F(p)=bo + lhp + hip'^+ . . . +b„p"+ 



50 that, formally, 



h = {bo + b,p + b,p-+ . . . +b„p"+ . . . )y/p 



an explicit series solution, usually asymptotic, is obtained by replacing 

 y/p by l/y/irl and p"{n integral) by d''/dt", whence 



■:^^{bo-K^i+b.~-b.]^+ . . .) 



