FORMAT. HIOALIZAHILITY TIIKOIJV II .)/ / 



for some Vi e Vi . liut /i e J = (Vi)" ((I) of 4.5). Heuco the kocoiuI term 

 here \aiiishes and term 3 cancels term 7. By an exactly similar arj^u- 

 inciit, since GB(J\ + />') e J, we lind that lei'in ."> cancels term <). 

 Consider tei'in I, and wiite it in the form 



•2{a 'A, /,-,) = 2(((7-')*/m , h) 



= 2{(a ■)*A^, ,A) =2(r; 'I-^j,). 



This follows hy (i, 7. '2, 14.0) and the fact that (/ ' is symmetric. Fnt- 

 ting in the definition of /,i , and using the fact that G and B are real, 

 we get 



2((r% , h) = 2{G-'GB{J^ + k). Ti) 



= 2{G^'GBin + k), h). 



Xow J is ival (4.42) so /i e J. Therefore the reasoning used on term 3 

 yi(4ds finally 



2(7^0, + k), h) 

 as the value of term 4. 

 We now write term 8 as 



2{Fh, k.) 

 and transform it to 



2(Fh,h), 

 !)>■ the reasoning just used on 4. Putting in what k-^ is, this is 



2(2FGB0, ^k) - Fj, - Fk, h). 

 Using the reality of G and B, and (22), this is 



- 2(B0i + k), Ti). 



This cancels term 4 and all terms save 2 and (> are accounted for. Fi- 

 nally, then, the residue of (3) at p = » is 



2(G-'li, h) + MFh, h). {:21) 



Since G~ is definite in J and F is semidefinite, this residue is non-neg- 

 ative, and indeed not zero if /i ?^ and /i e J. 



4.7 We have established the non-negativity of the residue of (3) at 

 /J = CO. A similar argument (exactly parallel, in fact) wdll establish the 

 same for the residue at p = 0. Each term in the representation 



Miv) = - J/o + pM^ 

 V 



