You are here : Control System Design  Index  Book Contents  Appendix C  Section C.8 C. Results from Analytic Function TheoryC.8.3 Poisson's Integral for the Unit Disk
Theorem C.10 Let be analytic inside the unit disk. Then, if , with ,
where is the Poisson kernel defined by
ProofConsider the unit circle . Then, using Theorem C.8, we have that
Define
Because is outside the region encircled by , the application of Theorem C.8 yields
Subtracting (C.8.21) from (C.8.19) and changing the variable of integration, we obtain
from which the result follows.
Consider now a function which is analytic outside the unit disk. We can then define a function such that
Assume that one is interested in obtaining an expression for , where , . The problem is then to obtain an expression for . Thus, if we define , we have, on applying Theorem C.10, that
where
If, finally, we make the change in the integration variable , the following result is obtained.
Thus, Poisson's integral for the unit disk can also be applied to functions of a complex variable which are analytic outside the unit circle.
