Conformal mapping between two right-angled triangles 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!
 A: There are two expressions depending on what you prefer: hypergeometric functions or elliptic functions.


*

*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.

*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.
