The mapping function is a solution of the Schwarz differential equation
$$\frac{f'''}{f'}-\frac{3}{2}\left(\frac{f''}{f'}\right)^2=R(z),$$
where $R$ is a rational function with poles at the preimages of the vertices. The poles of $f$ are of second order, and the coefficients at the second order terms are determined by the angles. Schwarz equation can be reduced to a linear differential equation of the form
$y''+Ry/2=0$. In the case of a triangle, this is a hypergeometric equation.
In the case of a quadrilateral, this is a Heun equation.

Literature: Courant, Geometrische Funktionentheorie,
Caratheodory, Funktionentheorie, II, (There is an English translation),
Golubev, Vorlesungen über Differentialgleichungen im Komplexen, (transled from Russian).

The case of a triangle is completely understood (see any book on hypergeometric function)
The case of a quadrilateral is already quite complicated, and there are many
unsolved questions about quadrilaterals and Heun's equation. See for example arXiv:1409.1529 for some recent results about this, and literature there.

EDIT. More concrete examples can be found here arXiv:1111.2296 and here arXiv:1110.2696
and in references to these papers.