You are here : Control System Design  Index  Book Contents  Appendix C  Section C.3 C. Results from Analytic Function Theory

(C.3.1) 
where is the region bounded by .
We first consider a simple case in which is representable in both of the forms:
(C.3.2)  
(C.3.3) 
Then
(C.3.4) 
One can now integrate to achieve
(C.3.5)  
(C.3.6)  
(C.3.7) 
By a similar argument,
(C.3.8) 
For more complex regions, we decompose into simple regions as above. The result then follows.
We then have the following converse to Theorem C.3.
Theorem C.5 Let and have continuous derivatives in and let be simply connected. If , then is independent of path in .
Suppose that
Then, by Green's Theorem (Theorem C.4),
(C.3.10) 