I found two different equations for the Schwarz-Christoffel-mapping of a unit disk to a rectangle (which are the general form of the SC-mapping, I guess). The first, e.g. from Link, page 20, is

\begin{align*} z=f(\zeta)=&A+C\int\limits^{z}\prod\limits_{k=1}^n\left(\zeta-z_k\right)^{\alpha‌​_k-1}d\zeta\\ =&A+\tilde{C}\int\limits^{z}\prod\limits_{k=1}^n\left(1-\frac{\zeta}{z_k}\right)‌​^{\alpha_k-1}d\zeta\\ =&A+\tilde{C}\int\limits^{z}\prod\limits_{k=1}^n\left(1-\frac{\zeta}{z_k}\right)‌​^{-\beta_k}d\zeta \end{align*}

with $\tilde{C}=C\prod (-z^k)^{\alpha_k-1}$, which is a constant. Therein

  • $A$ and $C$ are complex constants,
  • $z_k$ are the coordinates of point $k$ on the unit circle corresponding to the vertex $k$ of the rectangle,
  • $\alpha_k$ are the interior angles of the vertices by means of multiples of $\pi$
  • $\beta_k$ are the exterior angles of the vertices by means of multiples of $\pi$
  • and $\zeta$ is a point outside the unit circle such that $|\zeta|>1$.

However in Savins book (Link, page 14) I found the equation

\begin{align*} z=f(\zeta)=&A+C\int\limits^{\xi} \prod\limits_{k=1}^n \left(1-\frac{z_k}{\zeta}\right)^{\gamma_k-1} d\zeta\tag{I.47} \end{align*}

where $\gamma_k$ are real positive constants showing what part of $\pi$ is comprised by the outside angles of the polygon, such that $2\pi-\alpha_k\pi=\gamma_k\pi$ and $\gamma_k=\frac{3}{2}$ for each of the four vertices of a rectangle. The latter equation is taken from Smirnov V.: Course in Higher Mathematics, Part 3-2, Moscow, 1933. Unfortunately, I don't have access to that book right now.

Could someone please tell me, if the two equations are identical and why ?


It seems that one of them is for mapping the inside of the circle to the inside of the rectangle, and the other is for mapping the outside of the circle to the outside of the rectangle.


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