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I want to derive a conformal mapping $f\!:\!A\!\to\! B$ where $A=\{ (x,y)\ |\ x\!\in\![0,1]\ \text{ and }\ 0 \leq y \leq x \}$ and $B=\{ (x,y)\ |\ x\!\in\![0,1]\ \text{ and }\ 0 \leq y \leq \frac{1}{\sqrt{3}}x \}$. The regions $A$ and $B$ are the right-angled triangles with angles $\{\frac{\pi}{4},\frac{\pi}{2},\frac{\pi}{4}\}$ and $\{\frac{\pi}{6},\frac{\pi}{2},\frac{\pi}{3}\}$, respectively.

Can anyone help me derive an explicit equation for $f(x,y)$? I am vaguely familiar with the Schwartz-Christoffel mapping and the Schwartz triangle mapping, but I do not have a rigorous enough understanding of complex analysis to apply these to the above case. Any advice would be much appreciated!

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There are two expressions depending on what you prefer: hypergeometric functions or elliptic functions.

  1. Let $f$ be the Schwarz-Christoffel map of the upper half-plane onto $A$, and $g$ the Schwarz-Christoffel map onto $B$ (both sending $(0,1)$ to $(0,1)$). Then your map $A\to B$ is $g\circ f^{-1}$. Explicit formulas: $$f(z)=\frac{\int_0^z\zeta^{-3/4}(\zeta-1)^{-1/2}d\zeta}{\int_0^1\zeta^{-3/4}(\zeta-1)^{-1/2}d\zeta},$$ $$g(z)=\frac{\int_0^z\zeta^{-5/6}(\zeta-1)^{-1/2}d\zeta}{\int_0^1\zeta^{-5/6}(\zeta-1)^{-1/2}d\zeta}.$$ All integrals can be expressed in terms of hypergeometric functions.

  2. Alternative method, using elliptic integrals and their inverses. Reflect both triangles with respect to the long side. In the first case you obtain a square, and the mapping function $$f^{-1}(z)=1/\wp^2(z,1,0).$$ Similarly, the map $g$ can be expressed in terms of a standard elliptic integral of the first kind corresponding to hexagonal lattice. The second triangle has to be reflected with respect to the side $(0,1)$, to obtain an equilateral triangle.

Remark on computation. Elliptic integrals can be expressed in terms of theta-functions (see Whittaker-Watson, for example). Theta-series are converging so fast they they are even suitable for computation by hand, without a computer. The little book by N. I. Akhiezer, Elements of the theory of elliptic functions, contains less material but is more user-friendly than Whittaker-Watson.

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  • $\begingroup$ Thanks so much! With regards to your first solution, unfortunately I have no experience with hyper-geometric functions and evaluating complex integrals. Is the evaluation of these integrals straightforward? If not, can you suggest how I can solve these numerically? (I am unsure how to 'numerically' represent an integral over the entire upper-half plane) $\endgroup$
    – niran90
    Jun 13, 2020 at 12:42
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    $\begingroup$ @niran: the integrals are not over the entire half-plane, they are line integrals. If you are not familiar with hypergeometric functions, look to some complex analysis textbook, for example, Whittaker-Watson. Also Maple and other computer systems easily compute these integrals. $\endgroup$ Jun 13, 2020 at 12:46
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    $\begingroup$ Inversion of $f$ in the first approach can be a problem, but in the second approach I wrote $f^{-1}$ for you in the form convenient for computation. $\wp$ function is called WeierstrassP in Maple. $\endgroup$ Jun 13, 2020 at 12:50
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    $\begingroup$ The best thing is to use $g$ from the first solution and $f^{-1}$ from second. Then you don't need to invert anything. The second approach does not give you a ready formula for $g$ but rather for $g^{-1}$. $\endgroup$ Jun 13, 2020 at 13:01
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    $\begingroup$ $g_1=1,g_3=0$. Periods are $1$ and $i$ $\endgroup$ Jun 13, 2020 at 20:07

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