Let
where the
The transformation
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
moves on the real axis from
to
,
moves along the broken line so that the direction turns the amount
anticlockwise every time
passes a point
. If the broken line closes to a polygon, then
.
This transformation is used in solving two-dimensional potential problems. The parameters
and
are chosen such that the given polygonal domain in the complex
-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.
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