Schwarz-Christoffel transformation

Let

w=f(z)=c\\int\\frac{dz}{(z-a_{1})^{k_{1}}(z-a_{2})^{k_{2}}\\ldots(z-a_{n})^{k_{n}%}}+C,

where the $a_{j}$ ’s are real numbers satisfying $a_{1}<a_{2}<\\ldots<a_{n}$ , the $k_{j}$ ’s are real numbers satisfying $|k_{j}|\\leqq 1$ ; the integral expression means a complex antiderivative, $c$ and $C$ are complex constants.

The transformation $z\\mapsto w$ maps the real axis and the upper half-plane conformally (http://planetmath.org/ConformalMapping) onto the closed area bounded by a broken line. Some vertices of this line may be in the infinity (the corresponding angles are = 0). When $z$ moves on the real axis from $-\\infty$ to $\\infty$ , $w$ moves along the broken line so that the direction turns the amount $k_{j}\\pi$ anticlockwise every $z$ passes a point $a_{j}$ . If the broken line closes to a polygon, then $k_{1}\\!+\\!k_{2}\\!+\\!\\ldots\\!+\\!k_{n}=2$ .

This transformation is used in solving two-dimensional potential problems. The parameters $a_{j}$ and $k_{j}$ are chosen such that the given polygonal domain in the complex $w$ -plane can be obtained.

A half-trivial example of the transformation is

w=\\frac{1}{2}\\int\\frac{dz}{(z-0)^{\\frac{1}{2}}}=\\sqrt{z},

which maps the upper half-plane onto the first quadrant of the complex plane.