Let

where the a_j’s are real numbers satisfying a₁ < a₂ < … < a_n, the k_j’s are real numbers satisfying |k_j|≦ 1; the integral expression means a complex antiderivative, c and C are complex constants.

The transformation zw 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 −∞ to ∞, w moves along the broken line so that the direction turns the amount k_jπ anticlockwise every time z passes a point a_j. If the broken line closes to a polygon, then k₁+k₂+…+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.