URCl'ir llll.OKV .tXI) Ol'l-fx.llloX.II. C.llxriA'S 7.?.i 



Tlif i()rri->|>(iii(liiii; inti'Kr.il iM|iiatii)n i^, of cniirsc, 



= f l(t)e P'dt. (72) 



4 



c vp+ \ 



R p 



W'f shall i;i\c (wo solutions of this |)roi)lrin; lirst liic solution (jf 

 the iiitei-ral equation, anil secoml the t\|)iral Hea\isi(U' solution 

 directly from the operational equation. 



luiuation (72) nia\' l)e written as 





P+\ 



(73) 



Now supix)se that /(/) is the solution of the ec|uation 



-j=.^=fj,,e.„ (74) 



it follows at once from Theorems (I) and (II) of the preceding chapter 

 that 



m = yj^{i+xp,)m. 



(75) 



Also from formula (c) of the table of integrals and Theorem (V'a) the 

 solution of (74) is 



J(t)=- 



V-f 



whence 



'^•H^.\9h^f:vi< 



(76) 



(77) 



The integral appearing in (77) can not be e\aliiated in finite terms; 

 it is easily expressible as a series, however, by repeated integration 

 by parts. Thus 



Proceeding in this way by repeated partial integration we get for the 

 integral term of (77) 



2v^e-X'jl+?^'4-(^>... [. (78) 



' 1.3 1.3.0 i 



