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It is actually an electrostatic problem, I have to find the potential in the region descripted above using the laplace differential equation with the next boundary conditions: $V$ is $0$ in the straight sides and $V_0$ in the curve side.

The region with the boundary conditions

The problem asks for me to solve the laplace equation in a semi-infinite strip first, where the finite side has a potential V and both infinite sides have a 0 potential, which is an standard variable separation problem and can be easily solved.

The first thing that confuses me is that it doesn't seem like a conformal mapping because any function that transform the first region into the second can't preserve angles. My question is if I do an approximation an convert my region into a triangle Does exist a function that can transform my triangle in an semi infinite strip?

Thanks in advance.

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The probably source of your confusion is that conformal map doesn't need to preserve angle at boundary. For example, the map $z\mapsto z^{\pi/\alpha}$ is a conformal map that maps your sector to a half-disc, but it stretches angles at the vertex.

The semi-infinite strip $\{z=x+iy\mid x<\log R, y\in (0,\alpha)\}$ is mapped, by the exponential, to your sector conformally.

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