In our current Complex Variables homework assignment, we are given the following problem:
Let $\Omega = \lbrace z \in \Bbb C : |z|<1, Im(z)>0\rbrace $ and $f \in C(\partial\Omega,\Bbb R)$ such that $f(t) = 0$ for $-1 < t < 1$. We are supposed to find a harmonic function $u \in C(\overline\Omega)$ such that $u = f$ on $\partial\Omega$.
When we solved the general Dirichlet problem for the whole disk $B_1$ in class, we used the poisson integral formula, i.e. $u(z) = \frac{1}{2\pi}\int_{|\xi|=1}f(\xi )\frac{1-|z|^2}{|\xi-z|^2}d\xi$ was the solution to $u|_{\partial B_1}=f$ and $\Delta u = 0$. I have tried to modify this integral so that it works for the homework question, without success. I'd be glad for any ideas.
Hint: Define $g$ on the unit circle by setting
$$\begin {cases} g(e^{ix}) = f(e^{ix}),& 0\le x \le \pi \\g(e^{ix}) = -f(e^{-ix}), &-\pi \le x\le 0 \end {cases}$$ Let $u$ be the Poisson integral of $g$ on the closed unit disc.