264 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 



16.13 Conversely to 16.12, if Z{p) is symmetric and Jk(p) is PR for 

 every real k, then Z(p) is PR, and J kip) is PR for all k. 



Proof: J kip) is rational so (i) of 16.0 holds in any frame by 14.13. 

 Clearly (iv) of 16.0 holds. 



Now for real k, by (10) of 10.33 and 14.02 



Jkip) = Jkip) = iZip)k, k). 



Hence Zip) = Zip) by 14.13. This is (ii) of 16.0, and (iii) there holds 

 by hypothesis. 



16.2 Proof of 16.01: By 15.11 and 16.12, J,(p) has no poles in r+ . 

 This is 16.01 by the definition 14.3 of pole. 



16.21 Corollary: Any PR Zip) can be considered as defined throughout 

 r+ : for any k, Jkip) is defined throughout r+ by 16.2. For each p, 

 as a function of A', Jkip) defines Zip) (14.13). 



16.3 Pz-oo/o/ 16.03: In any frame [Z"'(p)] = [Zip)]'' = [r(p)] consists 

 of rational elements, by direct calculation of the inverse matrix. In a 

 real frame [F(p)] = [Z'^p)] is symmetric and real for real p by the same 

 argument (both facts are also deducible geometrically). Hence we have 

 the duals of (i), (ii) and (iii) of 16.0 for Yip). Clearly Yip) is defined 

 throughout r+ . 



Now suppose that for some veW and some po€T+ we have 



Re(y, Yipo)v) < 0. 

 Then there is a keK such that v = Zipo)k. Therefore 



Re[Zipo)k, k) = l\eiZipo)k, k) < 0. 



Since this is impossible, we have the dual of (iv) of 16.0 for Yip) and 

 Yip) is PR. 



16.4 Proof of 16.04: This is immediate from 15.14, 14.3, and 16.12. 



16.5 Proofs of 16.05 and 16.06: Suppose Zip) has a pole at p = io:o . 

 Then iZip)k, k) does and that pole is simple by 15.15 and 16.12. Then 

 by 14.3 we can write 



Zip) = L^ 7?o + Zoip) 



P — tOii) 



where Zoip) is regular at p = iwo . Now Zoip) has a pole at p = —iojo 

 by 16.5, so a similar argument gives 



Zip) - ~ Ro + ^ . /?! + Zrip), (1) 



p — iwo p -\- two 



