Theorem 12.7 (The Dirichlet problem in  D). If [Graphics:Images/DirichletProblemDiskMod_gr_6.gif] has period [Graphics:Images/DirichletProblemDiskMod_gr_7.gif] and has the Fourier series representation

            [Graphics:Images/DirichletProblemDiskMod_gr_8.gif],  

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

(12.11)            [Graphics:Images/DirichletProblemDiskMod_gr_9.gif][Graphics:Images/DirichletProblemDiskMod_gr_10.gif],  

where [Graphics:Images/DirichletProblemDiskMod_gr_11.gif] denotes a complex number in the closed disk [Graphics:Images/DirichletProblemDiskMod_gr_12.gif] .  

Proof.

Proof of Theorem 12.7.

Proof of Theorem 12.7 is in the book.

Complex Analysis for Mathematics and Engineering

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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