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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Sign of real part and imaginary part zeta function at 1/2-x+iy and 1/2+x+iy [closed]

I want to know what the sign of the real part and imaginary part of $\zeta(1/2+x+iy)$ and $\zeta(1/2-x+iy)$ are ,are they the same? for example in this case they are the same zeta(0.25+I 10)=0.74513-0....
Kevin67's user avatar
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Numerical analytic continuation/asymptotics

I posted this question, quite a while ago, on math.stackexchange.com, here. I received an interesting answer but not sufficiently accurate for my purposes, so I'm trying here. I have a class of ...
lcv's user avatar
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On an asymptotic integral decay

Let $f\in C^{\infty}([-1,0])$ be real-valued and suppose that $$ \left| \int_{-1}^{0} f(t)\,e^{\lambda t} \,{\rm d} t \right| \leq e^{-\sqrt{\lambda}},$$ for all $\lambda > 0$. Does it follow that $...
Ali's user avatar
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Spectrum of 'complexified' Laplace operator

Let $(M^n,g)$ be a closed Riemannian manifold. Let $\Delta$ be the Laplace–Beltrami operator acting on scalar functions defined on $M$, and let $\lambda_1 < \lambda_2 \leq \cdots$ be its spectrum. ...
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Who introduced the notion of ringed spaces?

My question is very concise, please forgive it. Who introduced the concept of ringed space? My first try would be that they were introduced by Cartan in his study of analytic functions with sheaves. ...
user234212323's user avatar
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A generalisation of Cauchy-Stieltjes transform

For a nice function $\nu$ (say smooth and compactly supported), its Cauchy-Stieltjes transform is defined as $$\int_\mathbb R \frac{\nu(s)}{z-s}\mathrm{d}s$$ which is holomorphic in $\mathbb C\...
Jiyuan Zhang's user avatar
3 votes
1 answer
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On a compact operator in the plane

Let $\Omega \subset \mathbb R^2$ be a bounded domain with a smooth boundary. Let $$\bar{\partial}= \frac{1}{2} ( \partial_{x^1} + i \,\partial_{x^2}),$$ and let $G: L^2(\Omega)\to H^2(\Omega)$ be the ...
Ali's user avatar
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2 votes
3 answers
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Groups of conformal isomorphisms of simply connected surfaces

By the uniformization theorem every connected and simply connected surface $M$ is conformally equivalent to one of the following three surfaces: open disk $D$, complex plane $\mathbb{C}$, or $2$-...
Sergiy Maksymenko's user avatar
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Bound on integral of rational functions over [0,1]

I have a question about integrals of rational functions over $[0,1]$. Suppose $f(x),g(x)$ are two nonzero polynomials with non-negative (integer, in fact) coefficients. Under what non-trivial ...
user503585's user avatar
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2 answers
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Laurent Series $\sum_{n=-1}^\infty a_n x^n$ when $a_{-1} = \infty$

When dealing with complex functions, if $f(x)$ has a simple pole, then we can find the coefficient $a_{-1}$ in the Laurent expansion $f(x) = \sum_{n=-1}^\infty a_n x^n$ by evaluating the limit $\lim_{...
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Where does the Weierstrass expansion of $\operatorname{sn}$ come from?

In Table of Integrals, Series and Products (p. 869) by Gradshteyn and Ryzhik, I found the identity (called 'the Weierstrass expansion of $\operatorname{sn}$') $$\operatorname{sn}u=\frac{B}{A}$$ where $...
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If $f(x) = \sum_{n=0}^\infty a_n x^n$, then $\int_{-\infty}^\infty f(x^2) dx = \pi i a_{-\frac{1}{2}}$

I've noticed a curious relationship between the coefficient $a_n$ for a power series and the integral of the real line. For instance, take $f(x) = e^{-x} = \sum_{n=0}^\infty \frac{(-1)^n}{n!} x^n$. ...
Caleb Briggs's user avatar
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Formula in non-Abelian Hodge theory - Hodge-Riemann bilinear relations

I am currently reading about the non-Abelian Hodge correspondence. Let $(X,\omega)$ be a compact Kahler manifold. Given a Higgs bundle $(E, D_0)$ on $X$, we want to construct the corresponding flat ...
Will Fisher's user avatar
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proving inequality in Riemann zeta function

Recently I have made some interesting observations on the limit $$\lim_{k\rightarrow \infty}{\sum_{n=1}^{k}{\dfrac{(-1)^{n-1}\biggl( \cos \left(\beta\ln(n)\right)\biggr)}{n^{\alpha}}}}. $$ When this ...
MrPie 's user avatar
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Estimate of minimum of the Poisson integrals corresponding to a convergent Hausdorff sequence of smooth bounded domains from below

Let $\{\Omega_{j}\}_{j\in\mathbb{N}}$ be a sequence of smooth bounded domains in $\mathbb{C}^{n}$ such that $\Omega_{j}$ converges to a smooth bounded domain $\Omega$ in the sense that the defining ...
Naruto's user avatar
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Does a Borel transform uniquely determine a Borel measure?

It is a known fact that Borel measures are uniquely determined by their Fourier transforms. This is the motivation for the following question. I came across the concept of a Borel transform of a Borel ...
IamWill's user avatar
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Asymptotics of the Liouville sum at the primes

Let $\lambda$ denote the Liouville function, and let $L(x):=\sum_{n \leq x} \lambda(n)$ be the Liouville sum. Define $c$ to be the supremum of the real parts of the zeros of the Riemann zeta function. ...
user501735's user avatar
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Finite morphism to Stein manifold

Let $X$ be a complex manifold and let $Y$ be a Stein manifold. Assume that there is a proper finite holomorphic map $f:X\to Y$ (means that every fiber of $f$ is finite). Can we conclude that $X$ is a ...
Higgs-Boson's user avatar
3 votes
2 answers
700 views

Do all rigorous proofs of Euler's product for $\zeta (s)$ use infinitude of primes?

There are two proofs of $$\sum_{n=1}^\infty \frac{1}{n^s}=\prod_{p \text{ prime}}\frac{1}{1-p^{-s}},\quad \Re (s)\gt 1$$ which I'm aware of. I'll call the first one the Sieve proof and the second one ...
Vestoo's user avatar
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28 votes
1 answer
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A holomorphic function sending integers (and only integers) to $\{0,1,2,3\}$

Does there exist a function $f$, holomorphic on the whole complex plane $\mathbb{C}$, such that $f\left(\mathbb{Z}\right)=\{0,1,2,3\}$ and $\forall z\in\mathbb{C}\ (f(z)\in\{0,1,2,3\}\Rightarrow z\in\...
user1950's user avatar
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Does the real part of the cross ratio satisfy a maximum principle on a domain in any real submanifold?

Let $C(p_1, p_2; p_3, p_4)$ denote the cross-ratio of the $4$ points $p_i$, for $i = 1, \ldots, 4$, thought of as a holomorphic function on $$ \Omega = \{ (p_1, p_2, p_3, p_4) \in \mathbb{C}P^1 \times ...
Malkoun's user avatar
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Energy bounds (or the lack thereof) for a functional between almost Hermitian manifolds

Suppose that $(M,g,J_M)$ and $(N,h,J_N)$ are two almost Hermitian manifolds. For a differentiable function $f:M\to N$ define its pseudoholomorphic energy to be $E_+(f)=\frac{1}{4}\int_M |Df+J_N Df J_M|...
Jess Boling's user avatar
3 votes
1 answer
155 views

Can we define $\partial\bar{\partial}(\log|z_1|^2)\wedge \partial\bar{\partial}(\log|z_2|^2)$ as a current?

In complex analysis, by Poincare-Lelong theorem, we have $$ \frac{\sqrt{-1}}{\pi}\partial\bar{\partial}(\log|z|^2)=T_{z=0} $$ as currents, where $$ T_{z=0}(\eta)=\int_{z=0}\eta. $$ Now suppose we have ...
Zhaoting Wei's user avatar
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2 votes
1 answer
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Lower bounding the derivative of a simple zero of an analytic function

Let $f : \mathbb C \to \mathbb C$ be an entire function with a separated zero set, i.e. there is a $\delta>0$ s.t. $|z-z'| > \delta$ for every distinct zeros of $f$. Further, suppose that all ...
user975628's user avatar
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1 answer
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Lower bound for polynomials

As we know Bernstein's inequality for polynomials states that, if $P(z)$ is a polynomial of degree $n$ then $$\max_{|z|=1}|P'(z)|\leq n \max_{|z|=1}|P(z)|. $$ There are results related to the reverse ...
user159888's user avatar
1 vote
0 answers
63 views

Can we perturb a product of linear terms so that we keep the local geometry?

I am trying to understand the following phenomenon. Let me give an example first to elaborate my idea. Let $Q(x,y)=(y-x)\cdot (y+x)\in \mathbb{C}[x,y]$. If we look at the zero locus of $Q(x,y)$ in the ...
Kenneth.K's user avatar
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0 answers
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Asymptotic analysis for a double integral related to Airy functions

Let $Ai(x,y)$ be the Airy kernel which is given by \begin{equation}\label{equ2.12} Ai(x,y)= \begin{cases} \dfrac{Ai(x)Ai'(y)-Ai(y)Ai'(x)}{x-y}, & x\ne y, \\ Ai'(x)^2-xAi(x)^2 & x=y. \\ \end{...
Tomas's user avatar
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2 votes
0 answers
162 views

Duality of $H^1$ and BMO

While proving that the dual of $H^1$ is $BMO$ in Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, page 143, Stein says that we have $\left\Vert g \right\Vert_{H^1} \...
abbyJeffers's user avatar
6 votes
0 answers
167 views

Computing residues at $\infty$

As an initial note, let me show by example what I mean by the terminology 'residue at $\infty$' I use in the title. I assume there is some standard terminology for this stuff, so I'd appreciate it if ...
Caleb Briggs's user avatar
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1 vote
0 answers
67 views

Carleman approximation for functions from $\mathbb R$ to (closed convex subset of) a Lie algebra

I am looking for an approximation result dealing with continuous functions of a real parameter with values in (some subset of) the unitary algebra. However, I wouldn't be surprised if the following ...
Frederik vom Ende's user avatar
2 votes
1 answer
152 views

To which space does the derivative of a function in Fock space belong?

Let $f : \mathbb C \to \mathbb C$ be an entire function belonging to the Fock space $F_\alpha^2$, that is, $$ \int_\mathbb{C} |f(z)|^2 e^{-\alpha|z|^2} \, dA(z) $$ with $A$ the Euclidean are measure. ...
user975628's user avatar
2 votes
1 answer
86 views

Pair of positive harmonic functions with negative inner product in Drury-Arveson space

Define a reproducing kernel on the Euclidean ball in $\mathbb{C}^d$ by $$k(z,w)=\frac{1}{1-\langle z,w\rangle}+\frac{1}{1-\langle w,z\rangle}-1.$$ Call the corresponding real reproducing kernel ...
J. E. Pascoe's user avatar
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3 votes
1 answer
538 views

Derivative of the Riemann zeta function at $z=-2$

I have a question regarding the derivatives of the Riemann zeta function. It is known that $\zeta'(-1)=\frac{1}{12}-\ln A$, where $A$ is the Glaisher-Kinkelin constant (which is an elegant ...
L.L's user avatar
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0 votes
0 answers
76 views

Geometry of inner products between the unit vector and several given vectors

Let $\mathcal{S}$ denote the set of all unit complex-valued $d$-dimensional vectors, i.e., $$ \mathcal{S} \triangleq \left\{ \mathbf{s}\in \mathbb{C}^{d} \mid \mathbf{s}^{\mathrm{H}}\mathbf{s}=1 \...
RyanChan's user avatar
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2 votes
0 answers
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Proving that $u_0$ is a canonical solution

I've asked this question on stack exchange before but no one could help me so I wish I can get some help here. Let's first start with the definition of the canonical solution: Consider $\frac{\...
User-123's user avatar
4 votes
0 answers
140 views

Correct way to extend a sequence defined on the naturals into the complex plane

Preamble Sequences $a_n$ defined on the natural numbers are clearly not uniquely interpolated by only one function. In particular, given an interpolation $f(n) = a_n$, then $f(n) + \sin(2\pi n)$ is ...
Caleb Briggs's user avatar
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2 votes
0 answers
99 views

Real analytic periodic function whose critical points are fully denegerated

I have asked this question on MathStackExchange. My question: is there any non-constant real analytic function $f:\mathbb{R}^n\rightarrow\mathbb{R}$ such that, $$\nabla f(x_0)=0 \Rightarrow \nabla^2 f(...
Jianxing's user avatar
5 votes
1 answer
267 views

Implicit function theorem with singularities of any order

Let $\mathcal{U}\subset \mathbb{R}\times \mathbb{C}$ a neighborhood of $(0,0)$, and $f:\mathcal{U}\to \mathbb{C}$ differentiable in the first variable and holomorphic in the second variable, with $f(0,...
Lorenzo Q's user avatar
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8 votes
3 answers
418 views

A density claim

Suppose that $g_k\in C([1,2])$, $k\in \mathbb N$ are continuous functions such that $\|g_k\|_{C([1,2])} \leq \epsilon^k$ for some sufficiently small $\epsilon>0$. Is the following claim true: If $f\...
Ali's user avatar
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1 vote
0 answers
72 views

Expectation of moduli of roots

For a complex polynomial $\sum_0^n a_i z^i$ the sum of roots is readily given by the Vieta's formula. However, after trying to find out the sum of moduli of roots in terms of the coefficients, I feel ...
AgnostMystic's user avatar
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0 answers
102 views

Is this formula for 2D Fourier integral of diffraction kernel correct?

Well I have a function parametrized by $z$ $$g_z(x,y) = \frac{z}{i \lambda r^2} e^{i k r}, \quad r = \sqrt{x^2+y^2+z^2},$$ where $\lambda > 0$ is real constant and $k = \frac{2\pi}{\lambda}$. This ...
VojtaK's user avatar
  • 151
-7 votes
1 answer
228 views

Is the Klein group related to the Klein bottle? [closed]

Is the group of symmetries of the rectangle-not-square related to the Klein bottle mathematically? The reason I am asking is because I want to put a Klein bottle coffee cup in a joke about V_4 and ...
Erin Carmody's user avatar
0 votes
0 answers
233 views

how to derive this elliptic integral?

I am reading the article arXiv: 2207.09961, there are some interesting elliptic integrals, i.e. the formula (3.7) and (3.8). You can also see this image where $p_0(z)=\sqrt{-Q_0(z)}$ and $Q_0(z)=-\...
amon Hsu's user avatar
1 vote
0 answers
90 views

Expected Number of roots in $\mathbb D (0;r)$

In the literature about roots of random polynomials there are many results about the expected number of real roots of a complex polynomial building on Kac formula especially in the the asymptotic ...
AgnostMystic's user avatar
2 votes
0 answers
231 views

Where does this trig. identity hold?

Fix an integer $n\geq1$. QUESTION. Is it possible to find ALL pair of sequences of non-negative integers $(a_k,b_k)$, for $k=1,2,\dots,n$, such that $$\sum_{k=1}^n \sin^{2a_k}\theta\cdot \cos^{2b_k}\...
T. Amdeberhan's user avatar
8 votes
1 answer
497 views

Is anything known about the power series $\sum x^p$ for $p$ prime?

I'm interested in information about the power series $$\sum_{\text{$p$ prime}} x^p$$ and the related power series $$\sum_{n=1}^\infty (-1)^n x^{p(n)}$$ where $p(n)$ is the nth prime. Immediately, the ...
Caleb Briggs's user avatar
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5 votes
1 answer
144 views

Analytic continuation for disjoint domains

This question is a question about nomenclature more than anything. I have shown all the math, but I don't know what to search for for similar results. In such a sense, it is more so a reference ...
Richard Diagram's user avatar
0 votes
0 answers
101 views

The upper bound of hyperbolic cosine function in complex plane

I want to find the upper bound of the following function : \begin{equation} M(\lambda)=\max _{E \in \mathbb{C}}\left|L_{+}-L_{-}\right| \end{equation} where \begin{equation} \begin{aligned} & 4 \...
HERMIT_WELL's user avatar
0 votes
1 answer
297 views

An identity for Weierstrass elliptic functions evaluation

Let $\wp(z), \zeta(z)$ and $\sigma(z)$ be the Weierstrass $\wp$, zeta and sigma functions associated to the ODE: $$\wp'(z)^2 = 4(\wp(z)-e_1)(\wp(z)-e_2)(\wp(z)-e_3)$$ and we assume $e_1=\frac{2-c}3>...
T. Amdeberhan's user avatar
2 votes
2 answers
260 views

Most general reverse Hölder inequality for polynomials

Theorem. Let $m$ be an integer and $P_m$ the vector space of degree $m$ polynomials in one real variable. There is a constant $C$ such that, for all $a<b$ and $p \in P_m$, $$\|p\|_{L^\infty(a,b)} \...
Sébastien Loisel's user avatar

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