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## C. Results from Analytic Function Theory

#### C.8.3 Poisson's Integral for the Unit Disk

Theorem C.10  Let be analytic inside the unit disk. Then, if , with ,

 (C.8.17)

where is the Poisson kernel defined by

 (C.8.18)

##### Proof

Consider the unit circle . Then, using Theorem C.8, we have that

 (C.8.19)

Define

 (C.8.20)

Because is outside the region encircled by , the application of Theorem C.8 yields

 (C.8.21)

Subtracting (C.8.21) from (C.8.19) and changing the variable of integration, we obtain

 (C.8.22)

from which the result follows.

Consider now a function which is analytic outside the unit disk. We can then define a function such that

 (C.8.23)

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

 (C.8.24)

where

 (C.8.25)

If, finally, we make the change in the integration variable , the following result is obtained.

 (C.8.26)

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.

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