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 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 time $z$ 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.