# Approximation of conformal mapping as a sum of elementary conformal mappings

Hi, I would like to approximate any 2d conformal mapping, as a sum of elementary conformal mappings. So I would have some basis, a conformal mapping with some parameters, and by adding several ones of those, I would mimic any conformal mapping. I don't know a-prioris the map I'm going to approximate, I just want to be sure that I can approximate all possible conformal maps.

I wonder if such problem have been studied, and I'm not even aware of the keywords that might help me to find the answer. May I have some names, keywords to help me to find what I'm searching for ?

[edit following Omer's answer] the mappings I wish to approximate are isomorphism of a circle, which are bounded. Typically a polygon with straight edges, curved edges...

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That depends a lot on the metric in which you wish to approximate.

For entire functions you have e.g. Taylor series. For holomorphic functions you can add simple poles.

If the function can be defined only on some arbitrary domain you'll need many more basis elements.

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Hum... I want to approximate maps that transform a circle, in say, a polygon, with straight edges or curved edges, a wing profile etc... Always bounded shape. –  Marmakoide Apr 12 '11 at 14:10

Have you tried looking into the theory of Schwarz-Christoffel mappings? These are conformal maps of the upper half plane to a polygon (bounded or unbounded), so you could approximate your region by a polygon, and then apply SC maps. I know that this is different from the original question, but depending on what application you have in mind, it might work as well.

SC mappings have many interesting properties. First of all, for simple geometries (say, a triangle mapping to a disc) they can be explicitly written down, and for more complex geometries they involve a simple contour integral ("simple" in the sense of being easy). Secondly, there is a beautiful book by Driscoll and Trefethen which has many important examples, and for those examples that aren't in the book there is a very fast Matlab package written by Driscoll. I'm an applied mathematician, but ever since learning about SC maps and the tools available to deal with them, I almost never have to resort to using other conformal maps.

Last, this might be somewhat conjectural, but if you're interested in complex approximation theory, you could also look into Faber polynomials and how they are defined in terms of conformal maps. Basically, the $m$th Faber polynomial $f_m(z)$ is the polynomial part of the $m$th power of some uniformizing map of your domain, and the interesting fact is that the level sets $|f_m(z)| = 1$ provide "optimal" approximations (in some sense) to the boundary of the original domain. This is also explained in the book by Driscoll and Trefethen.

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