Finding a Function H in C to Satisfy Conditions

[libelec]

Homework Statement

Find a function H in C such that $${\nabla ^2}H = 0$$ for y>0, H(0,y) = 1 for y<-$$/pi$$, H(0,y) = 0 for y>$$/pi$$ and H(0,y) = -1 for -$$/pi$$<y<$$/pi$$.

The Attempt at a Solution

I haven't been able to came up with anything. All the conform transformations that I know allow me to solve the Dirichlet problem with only 2 conditions, or 3 but with two of them with the same value. I was told that I could just leave the geometry of the problem like it is (that is, not make any transformation) and propose the solution A$$\theta$$₁ + B$$\theta$$₂ + C, being $$\theta$$₁ the argument of [z - (0 -i*Pi)] and $$\theta$$₂ the argument of [z - (0 +i*Pi)], but the solution I find doesn't satisfy the border conditions.

Any ideas?

 

[libelec]

Nobody knows this?

 

[libelec]

I leave a graphic of the problem.
http://img692.imageshack.us/img692/333/asdasdxk.png

Does anybody know the answer? Because I kind of need it urgently...

Thanks.