Questions tagged [cv.complex-variables]
Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.
3,146
questions
-6
votes
1
answer
96
views
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....
5
votes
0
answers
164
views
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 ...
6
votes
1
answer
404
views
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 $...
2
votes
0
answers
86
views
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.
...
8
votes
0
answers
320
views
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. ...
3
votes
0
answers
95
views
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\...
3
votes
1
answer
156
views
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 ...
2
votes
3
answers
455
views
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 $2$-...
0
votes
0
answers
80
views
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 ...
2
votes
2
answers
215
views
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_{...
3
votes
1
answer
249
views
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 $...
7
votes
1
answer
302
views
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 $f(x) = e^{-x} = \sum_{n=0}^\infty \frac{(-1)^n}{n!} x^n$. ...
5
votes
1
answer
252
views
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 ...
0
votes
1
answer
124
views
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 ...
1
vote
1
answer
58
views
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 ...
1
vote
0
answers
110
views
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 ...
5
votes
1
answer
265
views
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. ...
0
votes
0
answers
67
views
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 ...
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 ...
28
votes
1
answer
809
views
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\...
1
vote
0
answers
62
views
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
$$ \Omega = \{ (p_1, p_2, p_3, p_4) \in \mathbb{C}P^1 \times ...
5
votes
0
answers
208
views
Energy bounds (or the lack thereof) for a functional between almost Hermitian manifolds
Suppose that $(M,g,J_M)$ and $(N,h,J_N)$ are two almost Hermitian manifolds. For a differentiable function $f:M\to N$ define its pseudoholomorphic energy to be
$E_+(f)=\frac{1}{4}\int_M |Df+J_N Df J_M|...
3
votes
1
answer
155
views
Can we define $\partial\bar{\partial}(\log|z_1|^2)\wedge \partial\bar{\partial}(\log|z_2|^2)$ as a current?
In complex analysis, by Poincare-Lelong theorem, we have
$$
\frac{\sqrt{-1}}{\pi}\partial\bar{\partial}(\log|z|^2)=T_{z=0}
$$
as currents, where
$$
T_{z=0}(\eta)=\int_{z=0}\eta.
$$
Now suppose we have ...
2
votes
1
answer
135
views
Lower bounding the derivative of a simple zero of an analytic function
Let $f : \mathbb C \to \mathbb C$ be an entire function with a separated zero set, i.e. there is a $\delta>0$ s.t. $|z-z'| > \delta$ for every distinct zeros of $f$. Further, suppose that all ...
1
vote
1
answer
136
views
Lower bound for polynomials
As we know Bernstein's inequality for polynomials states that, if $P(z)$ is a polynomial of degree $n$ then
$$\max_{|z|=1}|P'(z)|\leq n \max_{|z|=1}|P(z)|. $$
There are results related to the reverse ...
1
vote
0
answers
63
views
Can we perturb a product of linear terms so that we keep the local geometry?
I am trying to understand the following phenomenon. Let me give an example first to elaborate my idea.
Let $Q(x,y)=(y-x)\cdot (y+x)\in \mathbb{C}[x,y]$. If we look at the zero locus of $Q(x,y)$ in the ...
4
votes
0
answers
151
views
Asymptotic analysis for a double integral related to Airy functions
Let $Ai(x,y)$ be the Airy kernel which is given by
\begin{equation}\label{equ2.12}
Ai(x,y)=
\begin{cases}
\dfrac{Ai(x)Ai'(y)-Ai(y)Ai'(x)}{x-y}, & x\ne y, \\
Ai'(x)^2-xAi(x)^2 & x=y. \\
\end{...
2
votes
0
answers
162
views
Duality of $H^1$ and BMO
While proving that the dual of $H^1$ is $BMO$ in Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, page 143, Stein says that we have $\left\Vert g \right\Vert_{H^1} \...
6
votes
0
answers
167
views
Computing residues at $\infty$
As an initial note, let me show by example what I mean by the terminology 'residue at $\infty$' I use in the title. I assume there is some standard terminology for this stuff, so I'd appreciate it if ...
1
vote
0
answers
67
views
Carleman approximation for functions from $\mathbb R$ to (closed convex subset of) a Lie algebra
I am looking for an approximation result dealing with continuous functions of a real parameter with values in (some subset of) the unitary algebra. However, I wouldn't be surprised if the following ...
2
votes
1
answer
152
views
To which space does the derivative of a function in Fock space belong?
Let $f : \mathbb C \to \mathbb C$ be an entire function belonging to the Fock space $F_\alpha^2$, that is,
$$
\int_\mathbb{C} |f(z)|^2 e^{-\alpha|z|^2} \, dA(z)
$$
with $A$ the Euclidean are measure. ...
2
votes
1
answer
86
views
Pair of positive harmonic functions with negative inner product in Drury-Arveson space
Define a reproducing kernel on the Euclidean ball in $\mathbb{C}^d$ by
$$k(z,w)=\frac{1}{1-\langle z,w\rangle}+\frac{1}{1-\langle w,z\rangle}-1.$$
Call the corresponding real reproducing kernel ...
3
votes
1
answer
538
views
Derivative of the Riemann zeta function at $z=-2$
I have a question regarding the derivatives of the Riemann zeta function. It is known that $\zeta'(-1)=\frac{1}{12}-\ln A$, where $A$ is the Glaisher-Kinkelin constant (which is an elegant ...
0
votes
0
answers
76
views
Geometry of inner products between the unit vector and several given vectors
Let $\mathcal{S}$ denote the set of all unit complex-valued $d$-dimensional vectors, i.e.,
$$
\mathcal{S} \triangleq \left\{ \mathbf{s}\in \mathbb{C}^{d} \mid \mathbf{s}^{\mathrm{H}}\mathbf{s}=1 \...
2
votes
0
answers
83
views
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{\...
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 ...
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, $$\nabla f(x_0)=0 \Rightarrow \nabla^2 f(...
5
votes
1
answer
267
views
Implicit function theorem with singularities of any order
Let $\mathcal{U}\subset \mathbb{R}\times \mathbb{C}$ a neighborhood of $(0,0)$, and $f:\mathcal{U}\to \mathbb{C}$ differentiable in the first variable and holomorphic in the second variable, with $f(0,...
8
votes
3
answers
418
views
A density claim
Suppose that $g_k\in C([1,2])$, $k\in \mathbb N$ are continuous functions such that $\|g_k\|_{C([1,2])} \leq \epsilon^k$ for some sufficiently small $\epsilon>0$. Is the following claim true:
If $f\...
1
vote
0
answers
72
views
Expectation of moduli of roots
For a complex polynomial $\sum_0^n a_i z^i$ the sum of roots is readily given by the Vieta's formula. However, after trying to find out the sum of moduli of roots in terms of the coefficients, I feel ...
0
votes
0
answers
102
views
Is this formula for 2D Fourier integral of diffraction kernel correct?
Well I have a function parametrized by $z$
$$g_z(x,y) = \frac{z}{i \lambda r^2} e^{i k r}, \quad r = \sqrt{x^2+y^2+z^2},$$
where $\lambda > 0$ is real constant and $k = \frac{2\pi}{\lambda}$. This ...
-7
votes
1
answer
228
views
Is the Klein group related to the Klein bottle? [closed]
Is the group of symmetries of the rectangle-not-square related to the Klein bottle mathematically?
The reason I am asking is because I want to put a Klein bottle coffee cup in a joke about V_4 and ...
0
votes
0
answers
233
views
how to derive this elliptic integral?
I am reading the article arXiv: 2207.09961, there are some interesting elliptic integrals, i.e. the formula (3.7) and (3.8). You can also see this image
where $p_0(z)=\sqrt{-Q_0(z)}$ and $Q_0(z)=-\...
1
vote
0
answers
90
views
Expected Number of roots in $\mathbb D (0;r)$
In the literature about roots of random polynomials there are many results about the expected number of real roots of a complex polynomial building on Kac formula especially in the the asymptotic ...
2
votes
0
answers
231
views
Where does this trig. identity hold?
Fix an integer $n\geq1$.
QUESTION. Is it possible to find ALL pair of sequences of non-negative integers $(a_k,b_k)$, for $k=1,2,\dots,n$, such that
$$\sum_{k=1}^n \sin^{2a_k}\theta\cdot \cos^{2b_k}\...
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 ...
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 ...
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 \...
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>...
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)} \...