FORMAL REALIZABILITY THEORY — II i)iO 



". HGA = H - A, 



^0 (22) 



2FGB = F - B. 



These are fundamental to the evaluation of the residues of (3). Before 

 calculating these residues, we draw a further important conclusion from 

 the formulas just developed. 



Relation (20) exhibits .1 as a product of II and a possibly singular 

 matrix (viz., Fi''(0).l). Hence 



rank (.1) < rank {H). 



But relation (21) shows // as a product of .1 b}" 



4 HG + //IT'(O). 



Wo 



Hence 



rank (//) < rank (A). 



That is, 



rank (.4) = rank {H), 



(23) 

 rank (B) = rank (F), 



the latter being established in the same way. 



4.65 The formulas developed in 4.64 are all quite symmetric as between 

 relations obtained at p = °° and those at p = 0. We shall now con- 

 tinue to the evaluation of the residue of (3) at p = oo . The evaluation 

 at p = proceeds in an exactly similar manner. 

 The residue in question is, from (3), 



- {B{J, + A-), h + A-) + hiG~\h + J2), ii + j^ ^ ^ 



24 



Here ;i and j-i are any elements of J and k any element of Ki . The range 

 of G is J and the operator G~^ operates from J to U = J*, representing 

 the inverse to the operation G from U to J. 

 Let us define h and eliminate j-i by the relation 



j, = -III + 2GB(j, + k) - J, . (25) 



Since the range of (i is J, h e J. 



The definition analogous to (25) for the other pole of (3) is 



i2 = 4/1+ 4g.4(Ji+ a-) -,/i. 

 COo OJo 



