FORMAL HKALIZABILITY THEORY — II r)<)8 



5.41 Clearly 8(Z) is an iutegor and non-nogativo. If 8{Z) = 0, thou every 

 7 at every p^ is zero. Tlenee no fh , not even oo , is a pole of Z. Hence 

 (>acli el(Mnent of Z(p) is a constant. This estal)lisjies 2.1 1 and 2.12. 



."). 42 Snppose 



Z(p) = Z,(p) + Z,(p) 



where each Zi(p) is finite at e\'ery pole of the other. The poles of Z(p) 

 are then exactly the poles pi of Zi and those po'^ of Zo . At each pole, 

 5.27 api)lies in the enIarti;(Ml sense of 5.34, so 



Hreakinii the sum defining 5(Z) into sums over the /Jo" and po"^ pi'oves 

 that 



8{Z) = 6(Zi) + 5(Z2). 



This is 2.14. 



5.43 If 



Zip) = f(p)R, 



where R is a constant matrix, then the normal form of Z(p) is f(p) 

 times a diagonal matrix of the same rank as R (5.15). 2.15 then follows 

 at once. 



5.44 If 



Zi(p) = AZ(p)B, 



where A and B are constant and non-singular, the poles of Zi(p) and 

 Z(p) are the same. At each, 5.26 applies in the enlarged .sense of 5.34. 

 Therefore 5(Zi) = 5(Z). This is 2.16. 



5.45 If Zi(p) is Zip) bordered by zeros, they have the same poles. One 

 verifies at once from 5.11 that the normal form of Ziip) is that of Zip) 

 bordered by zeros. Since also ZiiTiq)) is ZiTiq)) bordered by zeros, it 

 follows that 



S*iZr , po) = S*iZ, po) 



at every pole, whence 6(Zi) = 5(Z). This is 2.17. 



5.46 We must prove that if Zip) is non-singular, then 



8iZ) = 8iZ~') 

 Proof: Choose a bi-rational transformation p = Tiq) such that at 



