FOUMAL KKALIZAIULITV THKORV I 2(il 



exists for every keK and is not identically zero. By lo.lS, tliis limit 

 ((k) defines an operator Ro , tlie residue* of Z(p) at po , by 



(Rok, k) - t{k) for keK. 



The changes in (2) re(iuired lo define a poh' at p = co are obvious. 

 14.21 A pole Po of order //; of Z{p) is a pole of some matrix element of 

 [Z{p)], of order m, in any frame, and no element of [Z(p)] has a pole at 

 /;o of order exceeding m. For the elements of [Z(p)] are defined by the 

 values of (Z{p)k, k), by 14.11 and llaimos'' loc. cit. 



XV. POSITIVE REAL FUNCTIONS 



15.0 Let /(p) be a scalar function of the complex variable p. Following 

 Brune" we define f(p) to be positive real if 



(i) f(p) is a rational function of p, 



(ii) /(p) = m, 



(iii) Re(p) > implies Re(/(p)) > 0. 



The property (i) of being rational is of course on a quite different 

 level of ideas from the other properties, but it saves words later to in- 

 clude it specifically in the meaning of positive real. 



We abbreviate the words positive real to PR. 



15.01 The open region of the complex plane consisting of all finite p 

 such that Re(p) > — the right half plane — we denote by T+ . 



15.1 Brune, loc. cit., established a number of properties of PR functions 

 f(p) which will be useful to us here: 



15.11 f{p) has no poles in r+ . 



15.12 If Re(/(p)) = for some peT+ , then/(p) = for all p. 



15.13 If it exists, -~ is PR. 



f(p) _ _ 



15.14 If /(p) has a pole at p = pn ,it has one at p = po . 



15.15 If /(p) has a pole at p = icco , that pole is simple and 



lip) = -^^2 r + hip), 



where r > 0, and fi{p) is PR. 



* Properly, T^o is a residue only when m = 1. There is no convenient name 

 available for general m. 



