716 BELL SYSTEM TECHNICAL JOURNAL 



Now substitute this known form of solution in tlie integral equation 

 of the problem and carry out the integration term by term. We get 



Setting p = o, we ha\e at once 



Co = \/II{o). (50) 



To determine Q let p = pj+q where g is a small quantil\' ultimately 

 to be set equal to zero, and write the equation as 



Comp)+Xfff,cj=i. (51) 



If now p = pj-\-q and q ai^proaches zero, this becomus in the limit 



PjII'(Pj)Cj=\ (52) 



or 



PjH {pj}' 



whence //(/) =y^^ + V _£^ (54) 



11(0) ^PjII'(Pj) 



which is the Expansion Theorem Solution. 



We shall not attempt to discuss here cases where the expansion 

 solution breaks down though such cases exist. In every such case, 

 however, the breakdown is due to the failure of the impedance func- 

 tion IKp) to satisf\' the conditions necessary for the partial fraction 

 expansion (40), and correlati\"ely the non-existence of a solution in 

 normal vibrations. Furthermore, it is usually possible by simple 

 modification to deduce a modified expansion solution. It may be 

 added here, that while the proof given above is also limited implic- 

 itly to finite networks, the expansion solution is \-alid in most trans- 

 mission line problems. 



Let us now illustrate how the expansion solution works 1)\' applying 

 it to a few simple examples. Take first the case considered in the 

 preceding chapter in connection with the power series solution. Re- 

 quired the charge Q on a condenser C in series with an inductance L 

 and resistance R in response to a "unit e.m.f." The operational 

 equation is 



= . \ 



^ Lp'^+Rp+l/C 



„_ 1 1 



