Past exam of the perturbation methods and applied complex variables course of the University of Oxford / 2010 / B / 4 / iii / Solution

Let $g=f^{-1}$, so $g(z)=\sqrt{1-z^{-2}}$, choosing the analytic branch with $\mathrm{Im}g>0$; for example $g(1/2)=i\sqrt{3}$. Let $\zeta \in \Omega$ be the source point. For the convention $-\Delta G={\delta }_{\zeta }$, the Dirichlet Green function in the upper half-plane is obtained by an opposite-sign reflected source:It vanishes for real $Z$, has the logarithmic singularity $-(2\pi {)}^{-1}\log |Z-W|$, and is harmonic elsewhere. Conformal invariance of two-dimensional harmonic functions now givesThe branch specified above maps every boundary bank to the real axis, so the Green function vanishes on both slit banks and the vertical wall. Local conformality preserves the coefficient of the logarithmic singularity. If the convention is instead $\Delta G={\delta }_{\zeta }$, negate this expression.