# Questions tagged [cv.complex-variables]

Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.

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### Can we find a holomorphic representation in these mappings

$\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\PSU{PSU}$Let $U\in \mathbb C^2$ be a given open set, $A$ be the set composed by maps (not necessarily continuous) $f:U\to \PSL(2,\mathbb C)$. We call ...
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### Upper bound for the complex Beta function

The question is almost the same as here. What is the upper bound for a complex Beta function $\displaystyle B(s,z)=\frac{\Gamma(s) \Gamma(z)}{\Gamma(s+z)}$ with $0<Re(s)<1$ and $0<Re(z)<1$;...
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### Is this an outer function

I think I have read this theorem but I cannot find it. Let us suppose that $\Theta(z)$ is an inner function in $H^\infty(\mathbb{C}^+)$. Is it true that $1-\Theta(z)$ is an outer function? It is clear ...
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### Regarding starlike sets in $\mathbb{C}^n$

A domain $S$ in $\mathbb{C}^n$ is called starlike about the point $x_0\in S$ if for every point of $S$, the segment of the straight line from the point to $x_0$ lies in $S$. Let $S$ in $\mathbb{C}^n$ ...
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### Upper bound for maximum modulus of polynomial on unit circle in term of the distribution of its roots

Let $P(z) = \prod_{i = 1}^n (z - z_i) \in \mathbb{C}[z]$ be a monic polynomial having all roots $z_1, \dots, z_n$ on the unit circle $\mathbb{T} := \{z \in \mathbb{C} : |z| = 1\}$. What is known about ...
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### Integral of $I = \int_{a}^{\infty}dx \frac{x^s}{(1+x)^{n}}$ [closed]

I have been trying to evaluate the following integral: $$I = \int_{a}^{\infty}dx \frac{x^s}{(1+x)^{n}}$$ If $a=0$, then this is the Mellin transform of $\frac{1}{(1+x)^n}$. However, suppose $a \neq 0$....
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### Solving an equation containing Laplace transform

Consider the equation \begin{equation} \frac{f(p)}{f(s_{1})}\mathcal{L}(y)(s_{1})+\frac{g(p)}{g(s_{2})}\mathcal{L}% (y)(s_{2})=\mathcal{L(}y)\mathbf{(}p), \end{equation} where $\mathcal{L}$ is the ...
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### An extension of the Carlson's theorem in complex analysis

For the statement of Carlson's theorem please see, https://en.wikipedia.org/wiki/Carlson%27s_theorem. There is an extension of Carlson's theorem that says that the condition that $f$ needs to vanish ...
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### Approximation of the product $(\bar{z} - a)^{-1} \cdot (z - b)^{-1}$

I would like to construct an approximation of the product \begin{equation} f(z) = \frac{1}{\overline{z}-a} \frac{1}{z-b}, \end{equation} where $a, b \in \mathbb{C}$, and $|{a}/{z}|, |{b}/{z}| <1$. ...
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### What is a holomorphic foliation?

For a smooth foliation $F$, there are three equivalent definitions: the leaves of $F$ are tangent to a smooth vector field; the foliation chart $\phi:U\to \mathbb R^k\times \mathbb R^{n-k}$ is ...
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### Are there extensions of Euler's infinite product for sine function? [migrated]

Euler product about sine function is $\frac{\sin(x)}{x} = \prod_{n=1}^\infty \left ( 1- \left(\frac{x}{n\pi}\right)^2 \right)$ I wonder if there is known results about slight modification of above ...
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### Coefficients for Expansions of $1-\zeta_p$

Let $\mathbb{Q}_p(\zeta_p)$ be the cyclotomic extension of the $p$-adic field $\mathbb{Q}_p$. Then $1 - \zeta_p$ is a uniformizer for this field. Recall that $$\sum_{i=1}^{p-1} \zeta_p^i = -1.$$ So ...
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### Order of vanishing of an analytic function along its vanishing locus

Suppose $F: {\mathbb C}^n \to {\mathbb C}^k$ is a holomorphic/analytic function with vanishing locus $V_F = F^{-1}(0)$. Can one prove that there are positive real numbers $C>0$ and $\delta>0$ ...
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### Ahlfors' proof of Bloch's theorem

In his pioneering paper An extension of Schwarz's lemma, Ahlfors proves the lower bound on the Bloch constant $B \geq \frac{\sqrt{3}}{4}$. The proof of this lower bound proceeds as follows: Let $W$ be ...
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### How to show the set of stable polynomials equals to the set of Lorentzian polynomials in degree 2

Give a homogenous polynomial $f\in \mathbb{R}[x_1,\dots,x_n]$ of degree $2$ in $n$ variables, we can consider $f$ as a quadratic form. We call $L_n^2:=$ the set of quadratic forms with nonnegative ...
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### Putnam 2020 inequality for complex numbers in the unit circle

The following simple-looking inequality for complex numbers in the unit disk generalizes Problem B5 on the Putnam contest 2020: Theorem 1. Let $z_1, z_2, \ldots, z_n$ be $n$ complex numbers such that ...
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### Short proof of the error bound in PNT assuming a zero-free strip?

I am looking for a short proof of the fact that $\zeta(z)\neq 0$ for $\Re z>a$ implies the prime number theorem with an error bound $O(x^{a+\varepsilon})$ for any $\varepsilon>0$, which would be ...
This is not the most precise question but rather a hope that someone has seen something like this. I am given a triangulation of the 2-sphere $S^2$ which I only know up to Moebius transformations. I ...