FORMAL REALIZABILITY THEORY — II 571 



(X(p)0\ + ^0,^1 + k)-h (G-'(p)0\ + i.),ii + J2) 



2) 

 + {L(v)(j-2-k),j,-k) 



obtained from (IG). By the definitions of X(p), L(p), and G~ (p) this is 

 a rational function taking real values for real p. Hence we need only show 

 of (2) that its real part is non-negative when Re(p) > to show that it 

 and M(p) are Pll. 



Referring to (G) and (11) of paragraph 4.4 and (11) of 4.5G for the 

 definitions, we see that (2) can be written 



- \- Uijx + k),j, + /.•) + "^ iG'\j, + J2), Jl + J2) 

 VL 2 



+ mj,-k),j2-k)] 



(3) 



+ P [- (Bij^ + /v), Jl -\-l^)+l (G-'Ul + J2),ji + J2) 



+ {FU2- /v),i2-fc)j. 



That is, the quadratic form in question has poles, simple ones, only 

 at and 00 , and has no constant term. If we can show that the residues 

 at these poles are non-negative, then it will follow not only that M(p) 

 is PR but that M{p) is of the form 



- Mo + pM„ 

 V 



where each of these summands is realizable by 3.1. 



Unfortunately, there still remains some computation to verify that 

 the residues of (3) are non-negative. 



4.62 We first recapitulate some relations obtained earlier; 



r(2)/ N ^/ N . 1 



Z''\p) = Z{p) + -A + pB; (4) 



V 



this is (5) of 4.42. 



'(2)/ \ 2p (3) 





this is (7) and (8) of 4.45. 



Z''\p) = -H+pF-^Z''\p)', (6) 



V 



this is (11) and (12) of 4.491. 



