PlanetPhysics/Schwarz Christoffel Transformation

Let $w=f(z)=c\int \frac{dz}{(z-a_1{)}^{k_1}(z-a_2{)}^{k_2}\dots (z-a_n{)}^{k_n}}+C,$ where the $a_j$'s are real numbers satisfying\, $a_1<a_2<\dots <a_n$, 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 $-\mathrm{\infty }$ to $\mathrm{\infty }$, $w$ 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\, $k_1+k_2+\dots +k_n=2$.

This transformation is used in solving [[../CoriolisEffect/|two-dimensional]] potential problems.\, The [[../Parameter/|parameters]] $a_j$ and $k_j$ are chosen such that the given polygonal [[../Bijective/|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.