Schwarz-Christoffel transformations

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I would like to visualize the Schwarz-Christoffel transformations. Do you have an idea how to implement those? For example a polygon (as a domain) in the complex plane and subsequently also a map (decomposed into several steps) into the upper half-plane of the complex plane should be drawn. I am not quite sure how to do this. The formulas for these transformations are well-known.