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As an elementary result in complex analysis, one can use the argument principle to show that the Schwarz-Christoffel transform is injective on the interior of the polygon to which it maps. Could this one-to-one correspondence be used to show that such an inverse mapping exists for at least a subset of the upper half-plane? In my search for an answer, I found page 24 of the following paper that mentions the existence of a trigonometric inverse for a mapping to a bar-shaped region. Does a generalization of this result hold? More specifically, has existing literature produced an inverse formula for any polygonal geometry?

I should also hope to refer anyone curious about the existence and Holder continuity of an inverse to the following post.

Thank you all.

Inverse Schwarz-Christoffel Mapping

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    $\begingroup$ Schwarz-Christoffel map is not always injective. When it is injective, the inverse map (from its image to the upper half-plane) of course exists. So what is your question exactly? This inverse map is sometimes an elementary function, as in your example. In most cases it is not. $\endgroup$ Apr 7, 2021 at 0:45
  • $\begingroup$ Is there a general formula that could work for all polygonal geometries? Do you know of one? $\endgroup$
    – Talmsmen
    Apr 7, 2021 at 1:00
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    $\begingroup$ @JPwin, no, there's in general no explicit formula in terms of elementary functions. For example, for rectangles you can write a formula using elliptic functions, for triangles, the Schwarz-Cristoffel map gives you a hypergeometric function (and the inverse is rather obscure function) $\endgroup$
    – Kostya_I
    Apr 7, 2021 at 6:57
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    $\begingroup$ What do you mean by a "general formula" and formula for what? Schwarz-Christoffel map IS a formula. Are you asking whether the map or its inverse is an elementary function or what? Or conditions under which it is injective? Can you state your question precisely? $\endgroup$ Apr 7, 2021 at 12:00

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I don't know a general answer for your question, but I would be very excited to learn more about this topic. Even having more examples would be very interesting. Here is what I do know.

In the forward direction - from the upper half plane $\mathbb{H}$ to the polygon $P$. If $P$ is a triangle, then the SC map is basically given by the Beta function. If $P$ is a regular $n$-gon, the SC map can given in terms of a hypergeometric function. See Exercise 5.19 of the paper "Mappings to polygonal domains" in the book "Explorations in complex analysis".

In the backwards direction - from $P$ to $\mathbb{H}$. If $P$ is a "rational" triangle (all angles are rational multiples of $\pi$) then it may be unfolded to give a quadratic differential $q_P$ on a Riemann surface $X_P$. (The unfolding procedure comes from the theory of billiards.) The inverse to the SC map is then, by the Schwarz reflection principle, a branched covering from $X_P$ to the Riemann sphere. In the presence of symmetries this covering map can have pretty expressions.

I carefully understood this covering map in two cases beyond the trigonometric example. When $P$ is a square or a hexagon we treat it as the unfolding of a rational triangle $Q$. We deduce that $X_Q$ is a square or hexagonal torus. Thus the covering map is the Weierstrass $\wp$-function for the square and hexagonal lattices, respectively. We can express these (as usual) in terms of theta functions

Remarks:

When $P$ is a rectangle then a similar discussion holds - this case is discussed in the BSc thesis you link to.

When $P$ is rational, the covering map lifts to give an automorphic map from $\tilde{X}_P$ (the universal cover) to $\mathbb{H}$. This will have a Poincaré series. But Poincaré series have very poor convergence properties, so I don't really regard this as an "answer".

When $P$ is not rational, I don't see how to proceed... It would be nice to have "non-existence results", but I don't know what those would look like.

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