# 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
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.