This section follows Section 12.1 where we introduced the Complex Fourier Series expansion. The Dirichlet problem for the closed unit disk is to find a real-valued function u(x,y) that is harmonic in the unit disk D and that takes on the boundary values  

    ,  

at points    on the unit circle  .  

Theorem 12.7 (The Dirichlet problem in  D). If has period and has the Fourier series representation

    ,  

then the solution to the Dirichlet problem in  the unit disk D  is  

    ,  

where denotes a complex number in the closed disk .  

Proof of Theorem 12.7.

Proof of Theorem 12.7 is in the book.
Complex Analysis for Mathematics and Engineering

    An important method for solving this problem is our next result which is attributed to the French mathematician Siméon Poisson.

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 of Theorem 12.8.

Proof of Theorem 12.8 is in the book.
Complex Analysis for Mathematics and Engineering

Example 12.3.  Find the function u(x,y) that is harmonic in the unit disk and takes on the boundary values  .

Solution 12.3.