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I am trying to solve the 2d version of Laplace's equation ($\frac{\partial^2 V}{\partial x^2}+\frac{\partial^2 V}{\partial y^2}=0$) for the semi-infinite bar shown below in the shaded region.

enter image description here

That is for the region bounded by $x= \pm L$ and $y=0,\infty$ given the boundary conditions that $V=0$ for $x=\pm L$ and $V=V_0$ for $y=L$. I know this can be done via separation of variables, but I want to do it via complex analysis and conformal mappings. I know that we can get to:

enter image description here

Via the mapping $z'=\sin(z)$. But I don't know what further mappings to do to make the problem more simple to solve, any suggestions?

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Once you reduced the problem to half-plane, the argument function can be used to produce a harmonic function with any piecewise constant boundary values you want. Specifically, to have $v_0$ on $[a,b]$ and zero value elsewhere, you would use $$ h(z) = \frac{v_0}{\pi}(\arg(z-b)-\arg(z-a)) $$ where the argument is understood as taking values in $[0,\pi]$ in the upper half-plane.

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