CIRCUIT THEORY AND OPERATIONAL CALCULUS 715 



proviiltti the real part of X is not positive (a condition s^itisfieil in all 

 network problems), we see at once that e(iuations (42) and {A'A) are 

 Siitisfied by taking 



I'.m-K.-l^y (47) 



ronsei|uentl\- from (43) and (47) it follows that 



which establishes the Kxpansion Theorem Solution. 



As implied above, the partial fraction expansion (40), on which 

 the expansion theorem solution depends, imposes certain restrictions 

 on the impedance function //(/>). Among these are that II{p) must 

 have no zero rf>ot, no repeated roots, and !///(/>) must be a proper 

 fraction. In all finite networks these conditions are satisfied, or by 

 a slight modification, the operational equation can be reduced to 

 the required form. The case of repeated roots, which may occur 

 where the network involves a unilateral source of energy- such as an 

 amplifier, can be dealt with by assuming unequal roots and then 

 letting the roots approach equality as a limit. Without entering 

 upon these questions in detail, however, we can \ery simply and 

 directly establish the proposition that the expansion theorem gives 

 the solution whenever a solution in terms of normal or characteristic 

 vibrations exists. The proof of this proposition proceeds as follows. 



It is known from the elementary' theory of linear differential equa- 

 tions that the general solution of the set of differential equations, 

 of which the operational equation is h = \/II(p), is of the form 



h{l)=^Co+^Cj^'^ 



where pj is the jth root of //(/>) =o, and Co, Cx . . . Cn are constants of 

 integration which must be so chosen as to satisfy the system of dif- 

 ferential equations and the imposed boundary conditions. The 

 summation is extended over all the roots of //(/>) which is supposed 

 not to have a zero root or repeated roots. 



