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 $\...
- 10.4k
2
votes
Accepted
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 ...
- 87.7k
2
votes
Accepted
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 ...
- 10.4k
2
votes
Accepted
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 \...
- 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,...
- 1,566
1
vote
Is composition of analytic function with invertible function analytic?
Take the following invertible function $f_w:\Bbb R\to\Bbb R$ with $w\in\Bbb Q\!\setminus\!\{0\}$:
$$f_w(x) = \begin{cases}
x, & x\in\Bbb R\setminus\Bbb Q \\
x+w, & x\in\Bbb Q
\end{cases}$$
...
- 6,044
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.
- 184
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