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.

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.

As an elementary result in Complex Analysis, one can use the Argument Principal 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.

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.

As an elementary result in Complex Analysis, one can use the Argument Principal 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?

Thank you all.

As an elementary result in Complex Analysis, one can use the Argument Principal 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.