Questions tagged [cv.complex-variables]
Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.
3,146
questions
3
votes
2
answers
509
views
Upper bound for complex integral
I am interested in obtaining a good upper bound for the absolute value of the following integral
$$
\left| \int_{0}^{\pi/3} e^{-itn} \left( 1-e^{it} \right)^{k} dt \right|,
$$
when $n>k>0$ are ...
7
votes
1
answer
456
views
On a paper by Dimitrie Pompéiu and on one (in two parts) by Edmund Landau
To celebrate the new year and the future of mathematics (or the mathematics of future), I see no better way to ask a question stemming from my researches on power series.
The two papers the title ...
1
vote
0
answers
104
views
Monomorphism/Isomorphism of $C_4$-tangent cones for complex varieties
Suppose that $(M,\mathcal{O}_M)$ is a reduced complex analytic space (or complex algebraic variety if you prefer). The tangent linear fiber space $TM$ associated to $M$ is defined as the analytic ...
2
votes
2
answers
246
views
If $\inf\{b\in\mathbb{R}\mid\sum_{n=1}^{\infty}e^{-ax_n-by_n}<+\infty\}=1-a$ for all $a\in [0,1]$, does this equality hold for all $a\in\mathbb{R}$?
Let $\left\{x_n\right\}_{n=1}^{+\infty},\left\{y_n\right\}_{n=1}^{+\infty}\subset [0,+\infty)$ be two sequences of non-negative real numbers. Suppose there exist $\lambda\ge 1, c\ge 0$ such that $\...
2
votes
0
answers
79
views
Does Kobayashi isometry map preserve complex geodesics?
Let $\gamma_1, \gamma_2$ are real geodesics in a domain $D$ and these two real geodesics are lying in the same complex geodesics, the question is, are $f\left(\gamma_1\right)$ and $f\left(\gamma_2\...
4
votes
1
answer
135
views
Derivative of riemann map from the unit disk to a Jordan domain with non rectifiable boundary
Let $\gamma$ be Jordan curve such that for each point $p \in \gamma$, there exists a neighborhood $U$ of $p$ such that $\gamma \cap U$ is non rectifiable. Let $\phi: \mathbb{D} \to \Omega$ be a ...
5
votes
0
answers
206
views
Is the Taylor map continuous?
(Skip to the bolded theorem below for my question, if you'd like)
Some context on asymptotic expansions and the Taylor map
In the setting of irregular singularities of meromorphic connections on the ...
0
votes
0
answers
63
views
Conformally embedding a finite Riemann surface of genus g
Let $R$ be a compact Riemann surface of genus $g$ and let $S \subset R$ be a Riemann subsurface. Theorem B in Maskit's paper says that we can embed $S$ into a compact Riemann surface $P$ of genus $g$ ...
1
vote
0
answers
64
views
Complex integration in a separable EDO for an eigenvalue problem
I'm trying to prove a result from Leon Cohen's book "Time-frequency Analysis", Chapter 18. Namely, I want to verify the solution of the eigenvalue problem $$\mathcal{C}s(t) = c s(t)$$ for ...
2
votes
1
answer
89
views
Mandelbrot boundary and component of $\infty$
Let $M$ be the Mandelbrot set, and $\partial M$ its boundary. So $\partial M$ is the set of those points $z\in M$ such that every neighborhood of $z$ contains a point of $\mathbb R^2\setminus M$.
Let $...
-1
votes
2
answers
81
views
Limits of integral series
Suppose we have the series of functions:
\begin{equation}
F(x)=\sum_{n=1}^{\infty} f_n(x)
\end{equation}
where convergence is uniform.
Additionally, consider the partial functions of the series:
\...
1
vote
0
answers
79
views
Finiteness of theta vanishing in the KP direction for locally planar curves
I believe the main question is Question 2 at the end, and for experts it might be completely okay to skip directly to it (assuming I'm not saying any nonsense).
My motivation comes from pure algebraic ...
1
vote
1
answer
198
views
Let $u$ be harmonic on domain $D\subset \mathbb R^d$, how far can we extend $u$ holomorphically?
If $u$ is harmonic then it is real analytic so then it can be extended locally holomorphically. I also know that if $u$ is harmonic on a ball in $\mathbb R^d$ we have that the radius of convergence is ...
0
votes
0
answers
63
views
Meromorphic functions converging in measure
Let $f_1, f_2, \ldots$, and $g$ be measurable complex-valued functions on the open unit disk. We say that the sequence $f_1, f_2, \ldots$ converges in measure to $g$ if, for all $\epsilon, \mu >0$, ...
0
votes
0
answers
95
views
Evaluating a matrix Pick function via its integral representation
In the proof of Theorem 3.1 of the paper Inequalities for M-matrices, Ando evaluates a matrix function (see equation boxed in orange below) via an integral representation of a Pick function (see ...
4
votes
0
answers
83
views
Elliptic integral as quantity associated with Riemann surface?
There are many elliptic integrals, so to show my point let me
just pick one of them (complete elliptic integral of the first
kind [1]):
$$K(k) = \int_{0}^{1} \frac {dx} {\sqrt{(1-x^{2})(1-k^{2}x^{2})}}...
3
votes
2
answers
213
views
Abel–Plana formula with fractional offset
The Abel–Plana formula compares the sum $\sum_{n=0}^\infty f(n)$ to the integral $\int_0^\infty f(x)\,dx$,
\begin{equation}
\sum_{n=0}^{\infty}f\left(n\right)-\int_{0}^{\infty}f\left(x\right)dx=\frac{...
3
votes
1
answer
111
views
Is there a generalisation of the Vivanti-Pringsheim theorem for several variables?
The Vivanti-Pringsheim theorem states that if $f(z)$ has a power series with non-negative coefficients and a radius of convergence $R > 0$, then it has a singularity at $R$. So to find the radius ...
3
votes
0
answers
96
views
What is known about the analytic continuation of Maz'ya's modified harmonic zeta function $\sum_{n=1}^{\infty} e^{-zH_n}$?
Question:
If we let $H_n = \sum_{k=1}^{n} \frac{1}{k}$ be the harmonic numbers then we can consider the modified zeta function
$$ f(z) = \sum_{n=1}^{\infty} e^{-zH_n } = \sum_{n=1}^{\infty} e^{-z(\ln(...
11
votes
3
answers
910
views
"Simple" integral equation
Let $H(z)$ be a continuous solution of the problem
$$
H(z)=\frac1{1-z}\int_z^1 \frac{2\zeta}{1+\zeta} H(\zeta^2)\,d\zeta,\ \ \ z\in[0,1);\ \ \ H(1)=1.
$$
Is it true that $H(0)=1-\ln2$? The question ...
3
votes
1
answer
147
views
$n$-th root of meromorphic functions of several complex variables
Let $f:\Omega\rightarrow\mathbb{C}$ be a "nice" meromorphic function of several complex variables on some domain $\Omega$. I wonder if the following claim is true.
Claim. $f$ admits a global ...
1
vote
1
answer
104
views
weakly separated sequences in RKHS are separated by Gleason metric
I am reading Agler and McCarthy's Pick Interpolation and Hilbert Function spaces. In Chapter 9, the authors ask to observe that weakly separated in a Reproducing kernel hilbert space implies separated ...
2
votes
1
answer
249
views
Irreducibility of an explicit complex projective variety
Let $Y\subset \mathbb P^n_\mathbb C$ be a subvariety defined by a series of homogeneous polynomials $f_1, \ldots, f_t$. Is there an effective way to determine the irreducibility of $Y$ as an algebraic ...
3
votes
0
answers
91
views
Stein manifolds admitting uniform strictly plurisubharmonic exhaustion functions
Let $(X,\omega)$ be a Kähler manifold. Call $X$ uniformly Stein with respect to $\omega$ if there exists an exhaustion map $\phi:X\to \mathbb{R}$ such that $i\partial\bar\partial \phi \ge \epsilon\...
5
votes
0
answers
312
views
Approximating $\zeta^{(r)}(s)$ by a sum
Let $\eta:[0,\infty)\to [0,\infty)$ be compactly supported, continuous and piecewise $C^1$, with its derivative $\eta'$ being of bounded variation. It is completely unsurprising that one can prove (...
0
votes
0
answers
103
views
A surprising result with the Riccati difference equation
I was looking at the Riccati difference equation with positive and negative indices
$$
R_n=\frac{aR_{n-1}+b}{cR_{n-1}+d}\quad n\in[0,N]\\
R_n=\frac{-dR_{n+1}+b}{cR_{n+1}-a}\quad n\in[-N,0]\\
$$
along ...
1
vote
0
answers
115
views
Interpolating sequences are strongly separated
I am reading Agler and McCarthy's Pick Interpolation and Hilbert Function spaces. In Chapter 9, titled "Interpolating Sequences", the authors say that every interpolating sequence is ...
0
votes
1
answer
107
views
Sufficient conditions for ensuring that a monic polynomial in $\mathbf{Z}[x]$ possesses exclusively simple roots
I am seeking sufficient conditions to ensure that a monic polynomial, denoted as $f$ in $\mathbf{Z}[x]$, possesses exclusively simple roots.
Based on an old paper (this reference), it has been ...
2
votes
0
answers
98
views
Quotient of integral representation of archimedean exterior square L-function
Let $\psi$ denote a non-trivial additive character of $\mathbb{R}$ and $r=2n$ be a positive even integer. Let $(\pi,V)$ denote an irreducible generic admissible Casselmann-Wallach representation of $...
5
votes
1
answer
276
views
Modeling the interior and exterior of a polygonal region on the Riemann sphere using Schwarz-Christoffel mappings
I am thinking about the following. I have been involved in a research project involving static magnetic fields inside and outside a polygonal magnetic material. You ended up trying to find a couple of ...
6
votes
0
answers
136
views
Fourier transform and Hodge-$*$ operator
Suppose I have a full-rank lattice $\Lambda\subset\mathbf{C}$. Then the classical Poisson summation formula says
$$\sum_{\lambda\in\Lambda}f(\lambda)=\sum_{\lambda\in\Lambda'}\widehat{f}(\lambda)$$
...
7
votes
2
answers
875
views
Polynomials having all their zeros on the unit circle
Let $P(z)=\sum_{k=0}^na_kz^k$ be a polynomial of degree $n$ having all its zeros on the unit circle. Let $M=\max_{0\leq k\leq n}\lvert a_k\rvert$. The polynomial $P(z)=z^n+1$ has $\max_{\lvert z\...
2
votes
1
answer
99
views
A modified Paley–Wiener theorem with weaker condition
Let's consider the following argument: let $f$ be a function in $L^2(\mathbb R)$ such that $\hat f$ extends to an entire function on $\mathbb C.$ Assume that for each $t>0$ and $x \in \mathbb R$
$$
...
2
votes
3
answers
256
views
Control of values of an entire function in a strip around the real line
Consider an entire function $f: \mathbb C \to \mathbb C$ such that $f|_{\mathbb R}(x)\to 0$ as $x \in \mathbb R \to \pm\infty.$ Does that imply that for each $T>0,$ we have $f(x+iy) \to 0$ as $x\to ...
7
votes
1
answer
242
views
Efficiently computing $\sum_k x^{k^2}$ modulo $p$
Let $p$ be prime. There is a whole host of "large" degree polynomials that can be computed efficiently modulo $p$. I was wondering if:
$$q(x) = \sum_{k=0}^{p-1} x^{k^2}$$
is a polynomial ...
0
votes
0
answers
46
views
pseudo inverse of a holomorphic multivariate injective map
Let $f:{\mathbb C}^n \rightarrow {\mathbb C}^N$, $N > n$, be holomorphic and injective on an open ball $B_n \subset {\mathbb C}^n$ such that the Jacobian matrices have full column rank at each ...
1
vote
1
answer
79
views
The number of roots of pseudo-exponential polynomials
Assume that $J$ is the interval $(-\pi,\pi]$. For $k=1,\ldots,2n$, suppose that $\lambda_k$s are real functions on $J$ with $|\lambda_k|=1$, meaning that $\lambda_k(t)$ is either $-1$ or $1$ where $t\...
0
votes
1
answer
73
views
Criteria for Hardy space membership
Assume that $p>2$ and let $H^p$ be the Hardy space of holomorphic functions in the unit disk $D$. It seems that $f\in H^p$ if and only if $$P(f):=\int_0^{2\pi}\left(\int_0^{1}|f'(re^{it})|^2(1-r)dr\...
0
votes
1
answer
124
views
Rational approximation for continuous function on curve $\Gamma$
Let $\Gamma \in C^{1,\lambda}$ be an oriented Jordan curve in complex plane $\mathbb{C} $, $\mathrm{R}(\Gamma)$ the set of all rational functions without poles on $\Gamma $. "$\mathrm{R}(\Gamma)$...
3
votes
0
answers
118
views
Basic obstruction to anything like holomophic symmetric functions of infinitely many variables?
The totality of all holomorphic functions on the unit disk forms some sort of infinite-dimensional complex manifold, where the coefficients of the Taylor expansion might serve as coordinates for the ...
0
votes
1
answer
93
views
Probabilistic bounds of random polynomials
This is follow-up question to my previous question about the expected number of roots .
I am considering a random polynomial given by $$p(z) = \sum_{i=0}^{n} a_i z^i$$,
where each coefficient } $a_i$ ...
2
votes
1
answer
113
views
Expected fraction of roots in the unit disc of random polynomial with Gaussian coefficients
I am trying to find the expected fraction of roots located in the unit disc for a random polynomial with Gaussian coefficients. Given a random polynomial
$$P(z) = a_0 + a_1 z + a_2 z^2 + \dots + a_n z^...
1
vote
0
answers
125
views
an eigenvalue problem for Jacobi Forms
Assume $G(q,z)$ is a Jacobi form of a certain index k. It is known that $G$ can be expanded in a Taylor series with coefficients in the ring of quasi-modular forms (generators $E_2, E_4$ and $E_6$).
$\...
3
votes
0
answers
81
views
Conformal welding and Jordan loop consequences?
In the similar context as Conformal welding of rectifiable curves
In classical conformal welding theory, we start with a homeomorphism $h$ of the unit circle and try to find a Jordan domain $D$ ...
23
votes
4
answers
2k
views
Lower bounding $|1+z+\cdots + z^{n-1}|$ when $z\approx 1$
I am trying to lower bound $|1+z+\cdots + z^{n-1}|$ when $z$ is a complex number close to $1$ (and $n$ is sufficiently large). My main concern occurs in the case $z = 1 + it$, where $t$ is small. In ...
2
votes
0
answers
103
views
Two definitions for transverse $(p,p)$ form
Let $V$ be a complex vector space of dimension $n$ and let $V^*$ be its dual. Fix any integer $1\leq p\leq {n-1}.$ A $(p,p)$-form $\alpha\in\bigwedge^{p,p}V^*$ is said to be strictly weakly positive ...
2
votes
1
answer
219
views
Zero sets for entire functions $f$ with $|f(z)| \leq C_f e^{c|z|}$
Let $c>0$ and $X$ the collection of all entire functions $f$ for which there exists $C_f > 0$ s.t.
$$
|f(z)| \leq C_f e^{c|z|}.
$$
Given a sequence $(z_n) \subset (0,\infty)$ s.t.
$$
\frac {z_n}{...
3
votes
0
answers
146
views
A Hartogs analogue?
Let $0<r<1<R$, and $A:=\{z\in \ell^2: r<\|z\|_2<R\}$.
For $n\in \mathbb{N}$, let $e_n$ denote the sequence in $\ell^2$ all of whose terms are zero, except the $n^{\text{th}}$ term, ...
2
votes
0
answers
81
views
Bounding the degree of the Weierstrass polynomial of a product of a holomorphic function and a polynomial
In brief. For a fixed holomorphic function $v$, I want to bound the degree $q$ of the Weierstrass polynomial $Q$ in the Weierstrass decomposition $v^TP = uQ$, in terms of the degree $p$ of the ...
1
vote
3
answers
160
views
Can a disk that is limit of disks in a pseudoconvex domain be partially contained in the boundary?
A holomorphic disk is the image of an injective holomorphic map $f:\mathbb D \to \mathbb C^n$ from the unit disk $\mathbb D \subset \mathbb C$ to $\mathbb C^n$.
Let $\Omega$ be a pseudoconvex domain ...