# 施瓦茨-克里斯托费尔映射

## 定义

${\displaystyle \{\zeta \in \mathbb {C} :\operatorname {Im} \,\zeta >0\}}$

${\displaystyle f(\zeta )=\int ^{\zeta }{\frac {K}{(w-a)^{1-(\alpha /\pi )}(w-b)^{1-(\beta /\pi )}(w-c)^{1-(\gamma /\pi )}\cdots }}\,{\mbox{d}}w}$

## 例子

${\displaystyle f(\zeta )=\int ^{\zeta }{\frac {K}{(w-1)^{1/2}(w+1)^{1/2}}}\,{\mbox{d}}w\,}$

${\displaystyle z=f(\zeta )=C+K\operatorname {arccosh} \,\zeta ,}$

## 其它简单映射

### 三角形

${\displaystyle z=f(\zeta )=\int ^{\zeta }{\frac {dw}{(w-1)^{1-a}(w+1)^{1-b}}}}$

### 正方形

${\displaystyle z=f(\zeta )=\int ^{\zeta }{\frac {{\mbox{d}}w}{\sqrt {w(w^{2}-1)}}}={\sqrt {2}}\,F\left({\sqrt {\zeta +1}};{\frac {\sqrt {2}}{2}}\right)}$

## 参考

• Tobin A. Driscoll and Lloyd N. Trefethen, Schwarz-Christoffel Mapping, Cambridge University Press, 2002. ISBN 0-521-80726-3.
• Z. Nehari, Conformal Mapping, (1952) McGraw-Hill, New York.
• Darren Crowdy，[1]Schwarz-Christoffel mappings to unbounded multiply connected polygonal regions，Math. Proc. Camb. Phil. Soc. (2007)，142, 319.