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I'd like to write a little program that transforms a fractal generated in the square $(-1,1)^2\subset\mathbb C$ conformally to the unit disk $|z|<1$. I know that conformal mappings from the unit disk to polygons can be described by Schwarz-Christoffel-Transformations, and with the help of several articles I finally came up with the hypergeometric function $e^{\pi/4 i}z\; _2F_1(1/2,1/4,5/4,z^4)$ that maps - as said - the unit disk conformally to the square $(-1,1)^2$. I used Mathematica to plot this function, and the result is perfectly fine.

My question is now: What is the inverse of this function?

There is a wikipedia article that gives a similar definition of the inverse in terms of Jacobi's Elliptic Function $cn$, but this doesn't work with my plotting. I think I have to adjust this only a bit to get what I want, but I was not able to so far.

Thx for your help in advance!

Community
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sranthrop
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1 Answers1

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The inverse can be expressed in terms of the Weierstrass $\wp$--function for the lattice with periods $1$ and $i$. See the discussion here.

If I may ask - which articles were you reading? I really like that expression for the SC function in terms of the hypergeometric function.

Sam Nead
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