Theorem 12.8 (Poisson Integral Formula for the Unit Disk). Let be a function that is harmonic in a simply connected domain that contains the closed unit disk .  If takes on the boundary values  

            ,  

then   has the integral representation  

            ,   

which is valid for  .

Proof.

Proof of Theorem 12.8.

Proof of Theorem 12.8 is in the book.

Complex Analysis for Mathematics and Engineering

(c) 2006 John H. Mathews, Russell W. Howell