2 votes
Accepted

Branch cut of $\operatorname{Log}(z^2-z)$

To find branch cuts it is often easier to think of it through the question: "when is $z^2-z$ in the cut of the parent function" rather than identifying a formula for the composite function $\...
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2 votes
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Alternate Version of Schwarz Lemma

You can proceed as in the proof of the “usual” Schwarz Lemma: $$ h(z) = \frac{f(z)-f(0)}{z} $$ has a removable singularity at $z=0$, and $|h(z)| \le \frac{2M}{R}$ for $|z| =R$. Now apply the maximum ...
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2 votes
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Order of a zero and Identity Theorem

Without further context, the number $n$ here can be zero (by that I mean, $f(c)\neq0$ is possible). The point is, if the statement didn’t hold, then $f^{(n)}(c)\equiv0$ for all $n\in\Bbb N_0$ and ...
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2 votes
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biholomorphic function $f\colon \mathbb{C}\to\mathbb{C}\setminus \{0\}$

Let $\gamma$ be the image of the unit sphere $\partial B_1(0)$ under the proposed inverse $g: \mathbb C^\times \longrightarrow \mathbb C$ of your biholomorphic map $f : \mathbb C\longrightarrow \...
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  • 117k
1 vote

Is composition of analytic function with invertible function analytic?

On $\mathbb{C}$ the map $z\mapsto \bar{z}$ is invertible. If $f\colon\Omega\to\mathbb{C}$ is analytic on $\Omega$ it is holomorphic, but the composition with $z\mapsto \bar{z}$ will not be holomorphic,...
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1 vote

Is composition of analytic function with invertible function analytic?

Take the following invertible function $f_w:\Bbb R\to\Bbb R$ with $w\in\Bbb Q\!\setminus\!\{0\}$: $$f_w(x) = \begin{cases} x, & x\in\Bbb R\setminus\Bbb Q \\ x+w, & x\in\Bbb Q \end{cases}$$ ...
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1 vote

Order of a zero and Identity Theorem

Take $g=0$. The negation of $iii)$ is: $\forall c ,\exists n$ such that $f^n(c)\ne0$. Since there exists such an $n$ for all $c$ and $n$ is a natural number, we can pass to the smallest such.
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