Green's function for Dirichlet problem on a half disk

Question

Let $D=\{z=(x,y):x^2+y^2<R^2, y>0\}$ be the half disk with radius R. Then if we consider the Dirichlet problem on this domain, i.e., we want to find $$ \Delta u=0, ~~z\in D,\\ u=f,~~z\in\partial D, $$ where $f$ is continuous.Then what's the expression of solution $u(z)$ in terms of it's boundary value?

Answer 1

\begin{align} u(z) &= \frac{1}{2\pi i} \int\limits_{|\zeta| = 1 \atop \operatorname{Im} z > 0} f(\zeta) \Big( \frac{\zeta}{\zeta-z} + \frac{\bar\zeta}{\overline{\zeta-z}} - \frac{\zeta}{\zeta-\bar z} - \frac{\bar \zeta}{\bar\zeta-z} \Big)\,\frac{d\zeta}{\zeta} \\ &\qquad + \frac{1}{2\pi i} \int_{-1}^1 f(t) \Big( \frac{z-\bar z}{|t-z|^2} - \frac{z-\bar z}{|1-tz|^2} \Big)\,dt \end{align}

See this paper by Begehr and Vaitekhovich for details. (Assuming $R=1$. General case by scaling.)