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 ?
