Questions tagged [complex-analysis]
For questions mainly about theory of complex analytic/holomorphic functions of one complex variable. Use [tag:complex-numbers] instead for questions about complex numbers. Use [tag:several-complex-variables] instead for questions about holomorphic functions of more than one complex variables.
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Confusion about (closed) submanifolds of Stein manifolds
I'm starting to study Stein manifolds and I think I'm confused by something that is probably very elementary (which possibly highlights how little I truly understand about basic manifold theory in ...
- 1,008
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Rings of holomorphic functions (of one complex variable) is a Bezout's domain
I am trying to solve the following problem:
Show that if $U$ is an open subset of $\mathbb{C}$, every finitely generated ideal of $H(U)$ is a principal ideal.
I know that if $\{g_1,\cdots,g_n\}$ is ...
- 539
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Find the Laurent expansion of $f(z) = \sin{\frac{1}{z(z-1)}}$ in $0<|z-1|<1$
Here is my idea:
$\sin{\frac{1}{z(z-1)}} = \sin{\left( \frac{-1}{z} + \frac{1}{z-1}\right)} = \sin{\left(\frac{1}{z-1} - \frac{1}{z}\right)} = \sin{\frac{1}{z-1}}\cos{\frac{1}{z}} - \cos{\frac{1}{z-1}}...
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Domain of convergence of complex series $z\not=1,\sum_{n=1}^{\infty}\frac{1}{n^{2}3^{n}}\left(\frac{z+1}{z-1}\right)^{n}$
I need to find the domain of convergence of the following series:
$z\not=1,\sum_{n=1}^{\infty}\frac{1}{n^{2}3^{n}}\left(\frac{z+1}{z-1}\right)^{n}$
Clearly the radius of convergence is $R=3$ since:
$$\...
- 129
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Integrals of the form $\int f\left(x,\frac{\sqrt{x+m}}{\sqrt{x+n}}\right)\,dx$
Let $b-a=m$ and $b+a=n$, where $a$ and $b$ are real numbers. Then the following identities hold:
$$\frac{1-e^{\pm\text{i}\alpha}}{1+e^{\pm\text{i}\alpha}}=\tan{\left(\frac12\sec^{-1}\left(\frac{x+b}{a}...
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Great Picard theorem
The great Picard Theorem is well known.
I wonder what instead can be said about the values of the function
$$
\operatorname{F}: z \mapsto \left(\operatorname{f}\...
- 160
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1
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Calculate $\displaystyle\int_0^\infty \frac{x^2}{1+x^7} \, dx$ [duplicate]
Calculate $\displaystyle\int_0^\infty \frac{x^2}{1+x^7} \, dx$
This showed up on a complex analysis qualification exam. First, I will write
$$\displaystyle\int_0^\infty \frac{x^2}{1+x^7} \, dx = \lim_{...
- 2,756
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Montel Theorem version
Let us suppose that $F$ is a family of meromorphic functions defined on an open subset
$D$. If $z_0\in D$ is such that $F$ is not normal at $z_0$ and $z_0\in U\subseteq D$ , then $$\bigcup_{f\in F} ...
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Montel's Theorem corollary/alternative formulation
I have been exploring normal families and Montel's theorem and recently I came across this alternative formulation (or corollary?):
Let us suppose that $F$ is a family of meromorphic functions defined ...
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Is there some link between analytic continuation and potential theory?
It seems like the analytic continuation of a function has a lot in common with the process of trying to define a potential for a vector field (or a differential 1-form). In particular, an analytic ...
- 185
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2
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84
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Showing that a family of holomorphic functions is normal
Let $\mathcal{F}$ be a family of holomorphic functions on $\mathbb{D}=\{|z|<1\}$ so that for any $f \in \mathcal{F}$,
$$
\left|f^{\prime}(z)\right|\left(1-|z|^2\right)+|f(0)| \leq 1 \quad \text { ...
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Proving existence of a root in the unit disk using Rouché's Theorem
Let $a\in\mathbb{C}$, and let $n \ge 2$. Prove that the polynomial $2022+az+2023z^n$ has a root in the unit disk, $D(0,1)$.
There's an algebraic way to solve this with Vieta's formulas, by observing ...
- 728
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Why is $\frac{1}{2 \pi i}\int_{∂D}\frac{f'(z)}{f(z)-\omega_0}dz$ either $0$ or $1$ if $f$ satisfies certain requirements, including injectivity?
I've come across this problem:
Assume that $f$ is holomorphic and injective on the closed disk $\overline D = [z : |z| ≤ ρ]$, and that $f(z) \neq \omega_0$ for each $z$ on the boundary $∂D = [z : |z| =...
- 613
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Question about equivalent statements of Picard Theorem
I want to prove following (Big Picard Theorem forms):
Theorem.
The followings are equivalent:
a) If $f \in H(\mathbb{D}\setminus\{0\})$ and $f(\mathbb{D}\setminus\{0\}) \subset \mathbb{C} \setminus \{...
- 996
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Prove $p \colon \mathbb{C}^{*} \rightarrow \mathbb{C}^{*} \colon z \mapsto \bar{z}^{2}$ is a covering map
I'm stuck on this problem and generally struggle to show that some maps cover maps.
Consider $p \colon \mathbb{C}^{*} \rightarrow \mathbb{C}^{*} \colon z \mapsto \bar{z}^{2}$. Prove that $p$ is a ...
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