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在open mapping theorem(complex analysis)的证明中:
Assume $f : U →\mathbb C$ is a non-constant holomorphic function and $U$ is a domain of the complex plane.
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Consider an arbitrary $w_0$ in $f(U)$. Then there exists a point $z_0$ in $U$ such that $w_0 = f(z_0)$. Since $U$ is open, we can find $d > 0$ such that the closed disk $B$ around $z_0$ with radius $d$ is fully contained in $U$.
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$e$ is the minimum of $|g(z)|$ for $z$ on the boundary of $B$ and $e > 0$.
Denote by $D$ the open disk around $w_0$ with radius $e$.
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This means that the disk $D$ is contained in $f(B)$.
The image of the ball $B$, $f(B)$ is a subset of the image of $U$, $f(U)$.
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$e=\min_{\partial B}\abs{g(z)}>0$,以$w_0$为中心$e$为半径作开圆盘$D$.
那么$D$是$f(\partial B)$内的以$w_0$为中心的最大的开圆盘.
最后证明了$D\subseteq f(B)$.
我有个问题:是否有$\partial f(B)=f(\partial B)$? 是否有$f(\mathring B)=\mathring{f(B)}$? (此处\mathring表示"内部") |
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Czhang271828
发表于 2023-5-24 20:00
