592 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 



Proof: Let qo = T~\po) 9^ =° . For 5.26 we have 



where the eciuahties are by definition and the inequaUty is 5.2G apphed 

 to matrices rational in q, since 



Fr(q) = F(T(q)) 



is by hypothesis finite at qo . The remaining conclusion of 5.26 follows 

 similarly. The proofs of 5.27 and 5.29 are equally simple. 



5.35 Theorem: If we extend 5.3 to aS* by defining 



Sl{Z, po) = S*{Z, , T-\po)), 



then 5.31 holds for S* ^^ith no restrictions on po or T" (po). 

 Proof: By their definitions, 



S*r(Z, Po) = 5*(Zi , T-\po)) = SciZ, , T-\po)), (4) 



Avhere V is such that V^ {T^^ {po)) is finite. But 



S,.{Z, , T'\po)) = S{Z, , U-\T'\po))) (5) 



where 



Z,(r) = Z^ixT) = Z{T{U{:r))). 



Let y{r) = T{U(r)). Then, by definitions, 



S{Z, , U-\T-\po))) = Sy(Z, Po) = S*(Z, Po), (6) 



since V~^ipo) = U'^\T^\po)) is finite. The theorem follows from (4), 

 (5), and (6). 



5.4 Definition: Let 



S*(Z, Po) = [Ti , 72 , • • • , Tn]. 



Define 



8{Z, Po) = 7i + 72 + • • • + 7n , 



5(Z) = j:hz,po), 



where the latter summation is over all poles po of Z(p), including po = °° • 

 This 8(Z) is the degree of Z for which we must establish the properties 

 claimed in 2.11 through 2.17. These properties will be demonstrated in 

 5.41 through 5.45, in numerical order, saving 2.13, which is deferred to 

 5.46. 



