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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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 ...
user512026's user avatar
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 ...
Daniele Tampieri's user avatar
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 ...
Thomas Kurbach's user avatar
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 $\...
YC Su's user avatar
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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\...
Begginer-researcher's user avatar
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 ...
gaoqiang's user avatar
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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 ...
Brian Hepler's user avatar
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 $g$ ...
Jaikrishnan's user avatar
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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 ...
Bernardo Boechat's user avatar
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 $...
D.S. Lipham's user avatar
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-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: \...
george andrade's user avatar
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 ...
adrian's user avatar
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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 ...
user479223's user avatar
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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 $\epsilon, \mu >0$, ...
Andre Kornell's user avatar
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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 ...
Pietro Paparella's user avatar
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})}}...
Student's user avatar
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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{...
Carlo Beenakker's user avatar
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 ...
rimu's user avatar
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3 votes
0 answers
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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(...
Sidharth Ghoshal's user avatar
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 ...
AAK's user avatar
  • 283
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 ...
GTA's user avatar
  • 1,014
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 ...
ashK's user avatar
  • 83
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 ...
Pène Papin's user avatar
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\...
WilliamS's user avatar
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 (...
H A Helfgott's user avatar
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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 ...
Cye Waldman's user avatar
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 ...
ashK's user avatar
  • 83
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 ...
ABB's user avatar
  • 3,898
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 $...
Akash Yadav's user avatar
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 ...
Malkoun's user avatar
  • 4,981
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 $$\sum_{\lambda\in\Lambda}f(\lambda)=\sum_{\lambda\in\Lambda'}\widehat{f}(\lambda)$$ ...
user avatar
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\...
user159888's user avatar
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 $x \in \mathbb R$ $$ ...
Ma Joad's user avatar
  • 1,591
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 ...
Ma Joad's user avatar
  • 1,591
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 ...
mtheorylord's user avatar
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 ...
gil's user avatar
  • 265
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\...
ABB's user avatar
  • 3,898
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\...
MathArt's user avatar
  • 333
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)$...
Yidong Luo's user avatar
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 ...
David Feldman's user avatar
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 } $a_i$ ...
AgnostMystic's user avatar
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^...
AgnostMystic's user avatar
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 $E_6$). $\...
T. Amdeberhan's user avatar
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 $D$ ...
Thomas Kojar's user avatar
  • 4,414
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 ...
David Altizio's user avatar
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 ...
Invariance's user avatar
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}{...
user975628's user avatar
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, ...
Salla's user avatar
  • 31
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 ...
Sébastien Loisel's user avatar
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 ...
hife's user avatar
  • 133

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