Questions tagged [cv.complex-variables]
Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.
3,298 questions
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Vector bundles over a Stein space are projective
It is a "well known" fact that
locally free sheaves over a Stein space $X$ are projective as $\mathcal{O}_X$-modules
(see e.g. just after Lemma 1.6 in O'Brian-Toledo-Tong's "The trace ...
- 1,437
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Description of all biholomorphic maps from annulus [duplicate]
Consider the collection $\mathcal{C}$ of all maps $f \colon B_1 -\overline{B}_\beta \to \mathbb{C}$ such that $f$ is biholomorphic onto its image. Is this collection $\mathcal{C}$ path-connected?
In ...
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Singular behavior of zeros of incomplete zeta function
I've been looking at the zeros of the incomplete zeta function
$\zeta_{lower}(s, z)$ recently.
$$
\zeta_{\mathrm{lower}}(s,z)=-\frac{{\Gamma(1-s)}}{2\pi i}\int_{z}^{\infty}\frac{{(-t)^{s-1}}}{e^{t}-1}...
1
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1
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Regarding subspace generated by the polynomial multiples of outer functions
Let $\mathbb{D}$ and $\mathbb{T}$ denote the open unit disk and unit circle in $\mathbb{C}$ respectively. We write $Hol(\mathbb{D})$ for the space of all holomorphic functions on $\mathbb{D}.$ The ...
CommunityBot
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Currents with logarithmic poles compared with those with no poles
I am learning Deligne homology via U. Jannsen, "Deligne homology, Hodge-$\mathscr{D}$-conjecture, and motives." There, the currents with logarithmic poles are given in Definition 1.4 by
$$
'\...
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0
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+100
Obstruction to the existence of global resolution of coherent sheaf
It is well known that any coherent sheaf on a complex manifold (or more generally on some complex spaces) admits locally a resolution with locally free sheaves. It is also well known that for non-...
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Complex projective space as a $U(1)$ quotient
$\DeclareMathOperator\U{U}\DeclareMathOperator\SU{SU}$As is well known, one can view $\mathbb{CP}^n$ as a quotient of the unit $(2n + 1)$-sphere in $\mathbb{C}^{n+1}$ under the action of $\U(1)$, ...
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Existence of zero-free strip of a Laplace transform (edited ..)
Problem
Let $\beta$ be a probability measure on $\mathbb{R}$, and define
$$
K = \left \{z \in \mathbb{C}: g\left(z\right)=\int_{-\infty}^{\infty}\exp\left(z x\right)\beta ( dx ) \text{ is well-...
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1
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Holomorphic functions of certain blow up at origin
Suppose that $D=\{z\in \mathbb C\,:\, |z|\leq 1\}$ and let $f$ be holomorphic on $D\setminus\{0\}$ such that $|f(z)|\leq e^{\frac{1}{|z|}}$ for all $0<|z|\leq 1$ and assume additionally that $\lim\...
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integral over the unit sphere of $\Bbb C^n$
Please, is there a way to calculate this integral
$$\int_{S_{2n-1}} \frac{e^{a \langle z, \zeta \rangle}}{|z - \zeta|^{\beta}} \, d\sigma(\zeta)$$
where $ z $ is a fixed point in the complex unit ball ...
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2
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1
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Section 3 of Atiyah's "On analytic surfaces with double points" — some questions
I have some questions about section 3 of Atiyah's "On analytic surfaces with double points," a short 9 page paper. Section 3 is all dedicated to proving lemma 4.
Near the end of section 3, ...
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0
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56
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Complexity of evaluation of analytic functions
Given an analytic function $f(x)$ (say as combination of elementary functions and operators), is it possible to compute $n$ first bits of the value of the function on the whole interval $[a, b]$ ...
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Can one explicitly define a right inverse for a convolution operator on the space of entire functions?
A result of Meise and Taylor in 1988 shows that every non-zero convolution operator on the Frechet space $H(\mathbb{C})$ of all entire functions on $\mathbb{C}$ has a continuous linear right inverse $...
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1
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Why "no wandering domain" fails in parabolic basin?
Theorem (Sullivan). Every Fatou component $U$ of $f$ rational map is eventually periodic, that is, there exist $n > m > 0$ such that $f^n(U) = f^m(U)$
I am familiar with the proof: spread around ...
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Bounding a number-theoretic integral
Find a good upper bound on $$\int_1^T\frac{\zeta'(s)}{\zeta(s)\zeta(1-s)}X^sdt,$$ where $s=c+it$ for a constant $c>1$ and $X>0$ is a parameter. If needed, we can assume RH.
My attempt here is ...
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2
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Proving algebraicity of compact Riemann surfaces without Chow's theorem
I am trying to write a report for a complex analysis class where I prove Riemann-Roch and apply it to prove algebraicity of compact Riemann surfaces. While writing this, I found that Riemann-Roch ...
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0
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79
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Alternative proof of parabolic implosion
I am working on an alternative proof of parabolic implosion from complex dynamics, but only allowing hyperbolic perturbation.
Theorem (Parabolic Implosion) Let $f(z)=z^2+z$ and $U_f$ be parabolic ...
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Is the hypothesis "uniformly equivalent" needed?
I am reading S. Shimorin's paper titled Complete Nevanlinna-Pick property of Dirichlet-type spaces. My question concerns Lemma 2.3. which is as follows:
Assume $\mathscr{H}$ is a Hilbert space of ...
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1
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342
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Integrate Faddeeva function
I came across this integration in my studies.
$\int_{-\infty}^{\infty}|F((w_\textbf{_} - \hat{w_\textbf{_}})\tau) |^2 . d\tau$
It uses the Faddeeva function which is $F(z) = e^{-z^2}erfc(-iz)$. I ...
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1
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Nonstationary phase method for oscillatory integral
I want to approximate an integral of the form $$\int_a^bf(t)e^{ig(t)}dt,$$where $f(t)$ is smooth, $g(t)$ is real-valued and smooth.
The stationary phase method says that if $t_0\in [a,b]$ is such that ...
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Need bound for absolute value of complex-valued special functions (Taylor coefficients of Faddeeva's w(z))
To guarantee accuracy for code [1] that computes Faddeeva's w(z) [2] using Taylor expansions around different centers, I would need upper bounds for the absolute values $|w_n(z)|$ of the coefficients
$...
- 128k
1
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2
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Bounds of zeta function near $\Re(s)=1$
Richert proved in
https://link.springer.com/article/10.1007/BF01399533
that $$ \zeta(s) =O\left( |\Im(s)|^{100(1-\Re(s))^{3/2}} (\log |\Im(s)|)^{2/3}\right)$$ uniformly in the region $\Re(s)\in [1/2,1]...
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2
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The Paley-Wiener theorem and exponential decay.
Consider a function whose Fourier transform is supported on a half-ray:
$$
A(t)=\int_0^\infty \omega(E) e^{-iEt}d E,
$$
where I can suppose $\omega(E)\geq 0$ and any suitable regularity conditions on $...
- 1,461
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2
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682
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On integral relating logarithm of absolute value of Zeta function
Sorry for such a direct question:
Consider the following integral:
$$I(t)=\int_{1/2}^{1} {\log|\zeta(a+it)|}da.$$
How to find the nature of $I(t)$ as $t\rightarrow\infty$?
CommunityBot
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7
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Identities and inequalities in analysis and probability
Usually, at the heart of a good limit theorem in probability theory is at least one good inequality – because, in applications, a topological neighborhood is usually defined by inequalities. Of course,...
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1
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How can one test whether a given analytic curve in the plane is algebraic or not?
Suppose $\Gamma$ is an analytic Jordan curve in the complex plane $\mathbb{C}$. What are ways to test whether $\Gamma$ is (contained in) an algebraic curve, i.e., whether there exists a real ...
- 148k
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145
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On the pointwise limit of a sequence of analytic functions
I have been confused with this problem for weeks now. Suppose I have Banach spaces $E$ and $F$ and a sequence of functions $f_{n}: U \subset E \to F$, where $U$ is open and nonempty. Let $x \in U$ be ...
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1
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Approximation of the product $(\bar{z} - a)^{-1} \cdot (z - b)^{-1}$
$\def\zbar{\smash{\overline z}\vphantom z}$I would like to construct an approximation of the product
\begin{equation}
f(z) = \frac{1}{\zbar-a} \frac{1}{z-b},
\end{equation}
where $a, b \in \mathbb{C}$,...
CommunityBot
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3
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1
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Mellin transform at $0$
Let $f$ be a smooth complex valued function with support on $[0,2]$ and with Mellin transform $\tilde{f}$. Recall that the Mellin transform is defined as $$\tilde{f}(w)=\int_0^{\infty}f(x)x^{w-1}dx.$$ ...
- 105k
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Does this sequence of Blaschke Product have rescaling limit $z-1$?
Background: The conformal conjugacy class of parabolic isometry of upper half plane $\mathbb{H}$ consists of $f(z) = z+1$ and $g(z)=z-1$.
Consider surjective proper holomorphic $F_n: \mathbb{H} \...
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1
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How to evaluate the following integral?
How to (analytically) calculate the following integral,
$$I = \int_{S_{2n-1}} \left( \langle z, \zeta \rangle \right)^a e^{ b \langle z, \zeta \rangle} \, d\sigma(\zeta),$$
where $\langle z, \zeta \...
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0
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How to calculate this integral [migrated]
Is there a formula of this integral
$$\int_{S_{2n-1}} \frac{e^{i a \langle z, \zeta \rangle}}{|z - \zeta|^{2\lambda}} \, d\sigma(\zeta)$$
and how to calculate it.
Thank you in advance
- 3,808
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1
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Conjecture on the number of roots of $z^n + P(z)$ within the unit disk
Some other people and I have noticed that the following seems to be true.
Fix an integer polynomial $P \in \mathbb{Z}[x]$. Let $a_n$ be the number of roots of $z^n + P(z) = 0$ that lie in the unit ...
- 109k
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1
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Sufficient conditions for decomposition of a bounded random variable into several small pieces
Given a random variable $X$ with $\mathsf{supp}\, X \subseteq [0,1]$ and $n$ positive numbers $h_1,\cdots,h_n$ with $\sum_{i=1}^n h_i=1$, I want to know some sufficient conditions for decomposing $X$ ...
CommunityBot
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1
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Where to find or how to establish a general formula for the improper integral $\int_{0}^{\infty}\frac{\sin t}{t}(\ln t)^k\operatorname{d\!} t$?
When I tried to establish the Maclaurin power series expansion of the reciprocal $\frac{1}{\Gamma(z)}$ of the Euler gamma function $\Gamma(z)$, I came across the improper integral $$
I_k=\int_{0}^{\...
- 1,091
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Integral formula of quantum dilogarithm
In the paper"Level N Teichmüller TQFT and Complex Chern-Simons Theory" arXiv:1612.06986, the authors study the quantum dilogarithm function:
\begin{equation}
\mathrm{D}_{\rm b}(x,n)=\prod_{...
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Constant mean curvature hypersurface
Assume that $f:\mathbb{B}^2\to \mathbb{C}$ is a holomorphic function defined in the unit ball in $\mathbb{C}^2$. Let $u(z)=|f(z)|(1-|z|^2)$ and consider $\Sigma =\{z: u(z)=c\}$. It seems to me that if ...
- 63.9k
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Does $P_xP_y+Q_xQ_y=0 \implies$ "non-existence of limit cycle" for $P\partial_x+Q\partial_y$"? (Complex dilatation and limit cycle theory)
Let $X=P\partial_x+Q\partial_y$ be a vector field on the plane $\mathbb{R}^2$. Assume that we have :$$P_xP_y+Q_xQ_y=0$$ Does this imply that the vector field $X$ is a divergence-free vector field ...
CommunityBot
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Double-periodic functions with (possible) poles
Consider the set of double-periodic function $f:\mathbb C/(\mathbb Z+i \mathbb Z) \setminus \{z_0\} \to \mathbb C$, where $z_0$ is a fixed point inside $\mathbb C/(\mathbb Z+i \mathbb Z),$ that have a ...
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0
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About Carleson measures on the Hardy space on the bidisc
I have been reading the paper "Carleson Measures in Hardy and Weighted Bergman Spaces of Polydiscs" by F. Jafari and there are a few things that going on that I am not entirely convinced of.
...
2
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163
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Bounds of modular functions on the Ford circles
Assume a holomorphic function from a product of two upper half planes $Z: \mathbb{H}_+\times \mathbb{H}_+\rightarrow \mathbb{C}$ with an expansion of the form
$$
Z(\tau,\tau') = \sum_{(h,h')\in S} a_{...
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0
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27
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Conformal mapping which is $\alpha$-holder continuous for each $\alpha<1$
Recall that the Jordan curve $J\subset \mathbb{C}$ is called asymptotically conformal if
$$\text{$\max_{z\in J(a,b)}\frac{|a-z|+|z-b|}{|a-b|}\to 1$ as $a,b\in J, |a-b|\to 0$}$$
where $J(a,b)$ is the ...
- 207
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Generalisation of Paley–Wiener type results for unbounded sets
Do you know an unbounded open set $A\subset \mathbb{R}^d$, $d\geq 2$ with the following property: if some integrable function $f$ on $\mathbb{R}^d$ has its Fourier transform vanishing on $A^c$ and all ...
- 6,357
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1
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Can doubly parabolic Blaschke product (BP) contained in another doubly parabolic BP?
Let $f:\mathbb{D}\rightarrow\mathbb{D}$ be a degree $d$ doubly parabolic Blaschke product with Denjoy-Wolff point at $z=1$. That is, $f(1) = 1$, $f'(1)=1$ and $f''(1)=0$.
Let $U \subset \mathbb{D}$ be ...
3
votes
4
answers
1k
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$Aut(\mathbb{CP}^n)$ [..especially $n=1$ and $n=2$..]
I am confused and curious about the meaning of the $Aut(\mathbb{CP}^n)$.
Is what is called the "linear automorphism group" of $\mathbb{CP}^n$ the same as $Aut(\mathbb{CP}^n)$? It somehow seems to me ...
- 2,858
29
votes
7
answers
7k
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Elementary proof of Riemann-Roch for compact Riemann surfaces
I am supposed to give a talk about the Riemann-Roch theorem to a seminar of first and second year graduate students. I want to do Riemann-Roch for compact Riemann surfaces, but I am open to perhaps ...
- 63.9k
0
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0
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66
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Uniformization and constructive analytic continuation of Taylor-Maclaurin series
Context. In their paper, "Uniformization and Constructive Analytic Continuation of Taylor Series", Costin and Dunne present a constructive method to greatly increase the accuracy of a ...
- 63.9k
3
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0
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125
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Analytic differential equations with estimates
I recently came across the following theorem of Martineau (and others). Let $P$ be a polynomial of $n$ variables with complex coefficients. If we identify naturally powers of partial derivatives $\...
- 6,357
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1
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443
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Expectation of complex random variable
I am researching frequency offset estimation and ended up reading a paper "Cramer-Rao Lower Bound on Frequency Offset Estimation Error in OFDM Systems With Timing Error Feedback Compensation"...
CommunityBot
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4
votes
0
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160
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An unusual uniqueness property for entire functions
For given $q\in (0,1),$ coefficients $|c_k|\leq Cq^{k^2/3},$ and non-negative non-decreasing convergent sequences $\{a_k\}_{k=0}^\infty$ and $\{b_k\}_{k=0}^\infty$ satisfying $a_k\geqslant b_k,\;k=0,...
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