Let

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

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