C. Results from Analytic Function Theory
where is the Poisson kernel defined by
Consider the unit circle . Then, using Theorem C.8, we have that
Because is outside the region encircled by , the application of Theorem C.8 yields
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
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.