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The Riemann's mapping theorem allows to map any simply-connected domain in the complex plane $\mathbb{C}$ conformal onto the open unit disk. Conformal is meaning preserving angles. This is a complete existence statement. However constructive methods do exist. Im currently working on how to construct an explicit conformal map from a given domain onto the unit circle.

As Mathematica seems to have strong capabilities for visualizing things I wondered how to make use of it for my work.

First of all I need to define a domain for Mathematica. Im not interesting in those standard domains like $\lbrace z \in \mathbb{C} : \Re(z) > 0 \rbrace$. I rather would like something else which allows me to make a real custom domain. I though about defining some vertices then let Mathematica connect them with lines and define the inner component as my domain. So this would work for me but is kind of limiting as I cant enter like $50$ points. But thats a compromise for the beginning.

Next I need Mathematica to construct a conformal map onto the unit circle. Im not interested in some expression. I just want to see the result. This should be a plot of the unit circle. Of course the best part is the angle preserving so I need to apply a texture onto my domain which gets mapped onto the plot.

This is an image which I would like to recreate:

enter image description here

On the left you can see the domain which I would have defined of the boundary points. On the right you can see a conformal mapping of the domain on the left. The colors let you check that it is indeed angle preserving. This is what I would like to achieve with custom domains.

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    $\begingroup$ You will have to explain what you need Mathematica to do in A LOT more detail. Right now, it is quite unclear to me what you need. Specifically what calculations would you like to carry out? What have you tried yourself? $\endgroup$ – MarcoB Dec 3 '19 at 17:05
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    $\begingroup$ I rewrote the whole question. Thanks for the feedback $\endgroup$ – Arjihad Dec 3 '19 at 17:42
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    $\begingroup$ The following mathoverflow.net/questions/314189/… may be useful. $\endgroup$ – user64494 Dec 3 '19 at 18:03
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    $\begingroup$ Here is an implementation for the case when the map can be found analytically mathematica.stackexchange.com/questions/111479/… $\endgroup$ – yarchik Dec 3 '19 at 19:18