# Thread: Harmonic Functions and Conformal Mappings

1. ## Harmonic Functions and Conformal Mappings

Hello there,

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 <=arg z <=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. However, this is all talk and I am not sure how to go about accomplishing this.

I appreciate any help that can be provided. Thanks!

2. 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.
You are thinking along the right lines there. Suppose that $\displaystyle z\mapsto f(z)$ is a conformal map taking $\displaystyle D\setminus\{0\}$ to a strip $\displaystyle 0\leqslant \text{Im }z\leqslant k$. Next, you want to find a harmonic function taking the value 0 everywhere on the lower boundary of the strip, and the value 1 everywhere on the upper boundary. The easiest way to do this is to use the fact that the real part of an analytic function is harmonic. The analytic function $\displaystyle g(z) = -iz/k$ has exactly the properties that you want. So define $\displaystyle u(x,y) = \text{Re }g(f(x+iy))$.