7. Corollary 2.7.1 - Nice Schwarz-Christoffel formula

If we integrate $\frac{f^{″} (z)}{f^{'} (z)}$ twice using Schwarz-Christoffel, we get a nicer formula:

f (z) = c \int_{0}^{z} (ζ - a_{1})^{α_{1} - 1} \dots (ζ - a_{n})^{α_{n} - 1} d ζ + c^{'}

And we may further show that $| c | =$ 1;; $1$ , therefore $f$ is a euclidean motion of the integral. If $a_{n} = \infty$ , we exclude it from the integral.

Remark

Although this is a nice form, it is in general very difficult to determine what that $a_{i}$ must be. We may choose three of the points freely (since this is unique up to an automorphism of 1;; $H$ ), but determining the rest of the coefficients is called the problem of successive parameters.