# Solving Laplace's equation using conformal mapping

1. May 8, 2012

### lonewolf5999

I'm trying to use conformal mapping to solve for a function u(x,y) satisfying Laplace's equation ∇2u = 0 on the outside of the unit circle (i.e. the complement of the unit disk), with boundary conditions:

u = 1 on the unit circle in the first quadrant,
u = 0 on the rest of the unit circle.

To start with, I'm trying to simplify the boundary conditions by conformal mapping, so I first tried mapping the outside of the unit circle to the upper half plane with a linear fractional transformation w = -i(z-1)/(z+1), and then sending the upper half plane to the rectangle using an elliptic integral transformation W(w) = ∫ dt/sqrt[(1-t2)(1-k2t2)], integrated from 0 to w.

I did this because I was trying to send the unit circle in the first quadrant to one full edge of the rectangle, and the same for the part of the unit circle in the other quadrants, but that doesn't seem to work, because after doing the elliptic integral transformation, it turns out the unit circle in the first quadrant gets sent to only half of the bottom edge of the rectangle, so that doesn't simplify my boundary conditions much. Rotating the unit circle before transforming it doesn't seem to help either, since none of the quarter-circles are sent to a full edge of the rectangle.

Any suggestions on how I can proceed, or what other sorts of transformations would help me simplify the boundary conditions, would be helpful!