CIRCUIT TIII.ORY ASn OI'l-KAIIONAI. CALCULUS 72i 



thrn 



/(/)= f'Mr)Ml-r)dr 



= f'Mr)W-T)dr. 

 Jo 



The operational theorems will now he stated and hrittly proved 

 from the integral equation identit>-. 



Theorem I 



If in the Operational Equation 



h = l,II{p) 



the generalized impedance function f[(p) can he expanded in a sum of 

 terms, thus 



//(/)) ih(pyih(p) ■ ■ ■ ^ihip) 



and if the auxiliary operational equations 

 1 





H,{p) 

 1 



Jhip) 



can be solved, then 



h=hi+h-, + . . . +//„. 



This theorem is too ob\ious to require detailed proof: in fact it is 

 self evident. The power series and expansion theorem solutions are 

 examples of its application. In general, however, the appropriate 

 form of expansion of \'H(p) will depend on the particular problem 

 in hand. The theorem, as it stands is a formal statement of the fact 

 that solutions can often be obtained by an appropriate expansion 

 whereas the equation cannot be solved as it stands. 



Theorem II 

 If h = h{l) and g=g{l) are defined by the operational equations 

 h = \/mp) 

 g = l/pH(p) 



the 



g{t)= f'h{T)d. 

 Jo 



