Example 11.8.  Find a function    that is harmonic in the unit disk  ,  which takes on the boundary values

(11-8)              

Figure 11.8  The Dirichlet problems for and .  

Explore Solution 11.8.

Construct the solution via a known conformal mapping.  This example is similar to Example 11.21.

The above solution is theoretical, it can be used for computations.  If the trigonometric identity    is used we get a version that plots the boundary values better.  Then use Mathematica to make a contour plot of the solution.

Next, use Mathematica to make a 3D plot of the solution.

The function    is harmonic in the unit disk  ,  and takes on the desired boundary values.

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