752 BELL SYSTEM TECHNICAL JOURNAL 



Successive substitutions in the integral equation (123b) and repeated 

 differentiations with respect to p, lead to the set of formulas, 



f g{t)e-t"dt^ -^ as p-^0 

 ''o Vp 



fJtUi)e-''dt ^-l^^asp-^0 



e-'"dt<^ — ^as /)-»o 

 2Vp 



(123d) 



t.gi{l)e-t"dt^^ -^ as p-^0 



Now assuming that h{t) satisfies the restrictions stated in the pre- 

 ceding proposition, it follows from that proposition, that 



g(0"«i/V'T/ as t—^x 

 ^'^'^~-2^Vr/^^'-^" 



,,, 1.3.5 a; . 



From the set equations (123d) and (123e) it follows by successive 

 sulistitutions that 



HD' 



, 1 / 1 . 1.3 1.3.5 ^ V 



which agrees with the series gotten by applying the Heaviside Rule. 



The defect of this derivation, which, however, appears to be in- 

 herent, is that it requires us to know or assume at the outset that h{i) 

 satisfies the required restrictions. Consequently an automatic ap- 

 plication of the Heaviside Rule may or may not give correct results. 

 On the other hand if we know that an expansion solution in inverse 

 fractional powers of / exists, the Heaviside Rule gives the series with 

 extraordinary directness and simplicity. 



The tyfje of expansion solution just discussed will now be illustrated 

 by some specific problems. The first problem is that of the propagated 



