ELECTRIC CIRCUIT THEORY 67 



It follows at oiUT froni our oprrational nilrs, and (2().S) and f20l), that 



I = v[c^G j' dt'^F, (206) 



/•' ?)F 

 V=-v / %-dt. (207) 



Our problem is thus reduced to evaluating the function F, from the 

 operational equation (205). This equation can be solved by aid of 

 the operational rules and formulas already given. The process is 

 rather complicated, and there is less chance of error if we deal instead 

 with the integral equation of the problem 



6 V 



-= I F{t)e-t"dt. (208) 



2 t/0 



Now let us search through our table of definite integrals. We do 

 not find this integral equation as it stands, but we do observe that 

 formula (m) resembles it, and this resemblance suggests that formula 

 (m) can be suitably transformed to give the solution of (208). We 

 therefore start with the formula 



g-XV/)=+l 



-1 /•<» 



1 Jx 



(m) 



This, regarded as an integral equation, defines a function which is 

 zero for /<X and has the value Jo{\/t'^ — \^) for f^X, Jo being the Bessel 

 function of order zero. We now transform (m) as follows : 

 (1) Let X/> = g and t/\=ti. Substituting in (m) we get 



-V52+A2 />oo 



Now% in order to keep our original notation in p and /, replace q h\ p 

 and /i by /; we get 



Vp"" +X2 



= I e-f"Jo{\Vf-l) dt. (m.l) 



(2) In (m.l) make the substitution p = q-\-}x and then in the final 

 expression replace qhy p\ we get 



I 



e-'''.e~^''J„{Wt--l)dt= - , (m.2) 



V(^+/x)^+x2 



