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!



Conformal mapping between two right-angled triangles
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.


