ciHCUir iiii-ony .ixn iiri.n.uioiWii. c.iixii.us 7Ji 



Theorem IX 



If the operational e<iii(ilioii 



y; = l IKp) 



is reducible to the form 



h ^1^L_ 



\+\K(p) 



uhere X «5 a real parameter, and if the auxiliary functions f=f(t) and 

 k = k(t) are defined by the auxiliary operational equations 



J=F(p) 



k = K{p) 



then /i(/) is determined by the Poissan Integral equation 



hit)=f{t)-\f'h{r)kil-T)dT. 



This theorem is of considerable practical importance in connection 

 with the approximate and numerical solution of operational equations 

 when the operational equation and the equivalent Laplace integral 

 equation prove refractory. In such cases, as will be shown later, 

 the numerical solution of the Poissan integral equations can often 

 be rapidly and accurately effected, and in many cases the quali- 

 tative properties of //(/) can be deduced from it without detailed 

 numerical solution. 



The proof of this theorem proceeds as follows: 



By \irtue of the relation h = \ 'll(p) the operational equation 



can be written as 



1 + XA'(/)) 



.A direct application of Borel's theorem or Theorem IV gives at once 

 the explicit equivalent 



h{t)=f{t)-\£h(r)k{t-T)dT 



