Originally Posted by

**ComplexXavier** I am trying to find harmonic functions with set boundary conditions for closed regions by using conformal mapping.

For example, I am looking for a harmonic function u(x,y) that is defined and continuous on D − {0}, where D {0 <= argz <= 3pi/2}, (3/4 plane) such that u(x, 0) = 1 for x > 0, and u(0, y) = 0 for y < 0.

So, I decided to perform a conformal mapping z--> 2/3log(z), which will map the 3/4 plane to an infinite strip. How can I utilize this information to find the desired harmonic function? I thought that if I could somehow apply the boundary conditions given and find them on the infinite strip, I could then find a harmonic function and then push back the harmonic function to the 3/4 plane.