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.