I would like to compute the Green function for the Laplacian on the complex plane, with mixed Neumann and Dirichlet boundary conditions (see equation 4.150 of Blumenhagen, Lust, Theisen "Basic Concepts of String Theory".)
Let us start from a simpler case, to set the notation: consider Neumann conditions on both end points of the open string.
Using the method of the image charge, I know that the solution should be in the form
\begin{align}
G(z,w)=-\frac12 {\rm ln}|z-w|^2+a\ {\rm ln}|z-z'|^2,\quad (1)
\end{align}
where $a$ and $z'$ are to be determined by imposing the boundary conditions.
Neumann conditions on both end points correspond to $z\partial/\partial z=\bar z\partial/\partial \bar z$ when $z=\bar z$: this implies that \begin{align} -\frac12\frac{1}{z-w}+a\frac{1}{z-z'}=-\frac12\frac{1}{z-\bar w}+a\frac{1}{z-\bar z'}. \end{align} Since in general $w\neq\bar w$ and since the choice $a=+\frac12, \ z'=w$ leads to the trivial solution $G=0$, the only interesting solution is $a=-\frac12,\ z'=\bar w.$ This means that \begin{align} G(z,w)=-\frac12 \left({\rm ln}|z-w|^2+ {\rm ln}|z-\bar w|^2\right), \end{align} which is the correct solution (see formula 4.149.)
How do I proceed for mixed boundary conditions?
This time I want to impose Neumann for $\sqrt z=\sqrt{\bar z}$ (positive real line) and Dirichlet for $\sqrt z=-\sqrt{\bar z}$ (negative real line).
However, the ansatz $(1)$ does not have square roots or fractions in the argument of the logarithm, which instead appear in the final solution.