PlanetPhysics/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 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.