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本帖最后由 hbghlyj 于 2023-1-10 19:53 编辑 Schwarz–Christoffel mapping
Consider a polygon in the complex plane. The Riemann mapping theorem implies that there is a biholomorphic mapping $f$ from the upper half-plane\[\{\zeta \in \mathbb {C} :\operatorname {Im} \zeta >0\}\]
to the interior of the polygon. The function $f$ maps the real axis to the edges of the polygon. If the polygon has interior angles $\alpha ,\beta ,\gamma ,\ldots$, then this mapping is given by
\[f(\zeta )=\int ^{\zeta }{\frac {K}{(w-a)^{1-(\alpha /\pi )}(w-b)^{1-(\beta /\pi )}(w-c)^{1-(\gamma /\pi )}\cdots }}\,\mathrm {d} w\]where $K$ is a constant, and $a < b < c < ⋯$ are the values, along the real axis of the $ζ$ plane, of points corresponding to the vertices of the polygon in the $z$ plane.
例子
考虑$z$平面中的半无穷带。这可以视作顶点是$P=0$, $Q=\pi i$和$R$的三角形，当$R\to\infty$的极限。极限时有$\alpha=0$和$\beta=\gamma=\pi/2$。假设我们要找映射$f$，有$f(−1) = Q$，$f(1) = P$，和$f(∞) = R$，那么$f$是
$$ f(\zeta) = \int^\zeta \frac{K}{(w-1)^{1/2}(w+1)^{1/2}} \rmd w $$
计算积分得到
$$ z = f(\zeta) = C + K \operatorname{arccosh}\zeta$$
其中$C$是个（複）积分常数。条件$f(-1) = Q$和$f(1) = P$给出$C=0$和$K=1$。因此施瓦茨—克里斯托费尔积分是$ z = \operatorname{arccosh}\zeta$。下图描绘这个映射。
| 从上半平面到半无穷带
Schwarz–Christoffel mapping of the upper half-plane to the semi-infinite strip |
施瓦茨三角形映射把上半平面映射到其边是圆弧的三角形。 |
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