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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The Krzyż Conjecture

What is the state of the Krzyż Conjecture? It states for that for all $f:\mathbb{D}\to \bar{\mathbb{D}}$ holomorphic and non-vanishing, the coefficients $a_n$ in the power series of $f$ are at most $2/...
mysatellite's user avatar
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Fractal covering of a plane with complex-base numeral systems - is periodicity necessary?

Taking a base $z$ positional numeral system with digits $a_k\in \{0,\ldots,n-1\}$: $$s:\left\{(a_k)\in\{0,\ldots,n-1\}^{\mathbb{Z}}: \exists_K \forall_{k>K} \ a_k=0\right \}\to \sum_{k\in\mathbb{...
Jarek Duda's user avatar
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Is a holomorphic function on a subvariety of $\mathbb C^n$ locally a restriction?

Suppose that $X\subset \mathbb C^n$ is a subvariety (locally given by holomorphic equations) and $f: X\to \mathbb C$ is a function. Suppose that $f$ is 1) continuous, 2) holomorphic on the smooth ...
Mikhail's user avatar
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Characterization of the hypergeometric function

One of the definition of the hypergeometric function $_2 F_1$ rely only on its global properties around the singularities (and not on a differential equation or a serie expansion) In modern language (...
Jeannette's user avatar
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On the embedding of manifolds into infinite-dimensional spaces

Let $X$ be a (connected, finitely dimensional) topological/smooth/complex manifold and let $i$ be a weakly continuous/continuous/smooth/holomorphic map from $X$ into the dual $F^{*}$ of a real or ...
erz's user avatar
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Is a basic hypergeometric function ${}_2\phi_1(a, b; c; q, z)$ a meromorphic function in $z$?

Here a basic hypergeometric function is the analytic continuation of the basic hypergeometric series (or called the $q$-hypergeometric series) $$ {}_2\phi_1(a, b; c; q, z) = \sum^{\infty}_{n = 0} \...
Dong Wang's user avatar
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Solving the difference equation in exotic scenarios

The difference equation, as referenced in the title, is a very specific object I'm referring to. If you have a holomorphic function $\phi$ on a domain $G$, then a solution $F$ to the difference ...
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Contour integral $\int_{\gamma}e^{-t\mu}(-\mu)^{-m}\log\sqrt{-\mu}d\mu$

Motivation: In my research I try to find the asymptotics of the heat trace $\text{tr}e^{-t\Delta}$ as $t\to0$, where $\Delta$ is Laplace operator on a manifold with singularities. First I find the ...
Asya's user avatar
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A conformal mapping onto a region bounded by convex contours (Ahlfors)

I want to solve the following exercise (from Ahlfors' text, page 261). I've tried posting it on MSE, and placed a generous bounty, but I couldn't get any answers there. *3. Using Ex. 2, show that $...
user1337's user avatar
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Lie algebra of holomorphic vector fields

It's well known that the holomorphic vector fields on a complex manifold form a Lie algebra. In simplest situations, this Lie algebra can be described explicitly. For example, take $X=\mathbb{P}^n$, ...
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Is there a connection between |roots| $\rightarrow$ 1 and Gromov's waist theorem?

Recent questions showed that roots of a random polynomial tend to lie on the unit circle ("Why do roots of polynomials tend to have absolute value close to 1?"; "Distribution of roots of complex ...
Joseph O'Rourke's user avatar
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Density of rational functions in open Stein

I repost here, after I tried here. Lately I have been wondering on this problem: if $U \subset \mathbb C^n$ is an open Stein and I denote by $\mathcal R(U)$ the set of rational functions on $\mathbb ...
Mauro Porta's user avatar
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From Selberg integral to Dyson integral

My question is about the derivation from Selberg integral to Dyson integral in this paper: Selberg integral : $$ S_n(\alpha,\beta,\gamma) := \int_0 ^1 \cdots \int_0 ^1 \prod_{j=1}^n t_j^{\alpha-1}(...
Craig Thone's user avatar
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proper mapping between Stein manifolds

My question is the following: Let $b:X \rightarrow Y$ be a proper holomorphic mapping between two Stein manifolds $X$ and $Y$. For obvious reasons, $b^{-1}(y)$ are finite subsets of $X$. Is the set $...
Jun Li's user avatar
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What is the spectrum of a ring of holomorphic rational power series?

Let $R_\infty$ be the ring of power series in a single variable with rational coefficients that converge in the whole complex plane. Let $R_\rho$ be the subring of $R_\infty$ that defines holomorphic ...
Will Sawin's user avatar
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Automorphisms of Compact Riemann Surfaces

I read a statement that for a compact Riemann surface $C$ with genus $g\geq 2$, one has for the Jacobian $J(C)$ of the curve $C$: $$ Aut (J(C))\sim Aut C$$ when $C$ is hyperelliptic and $$Aut(J(C))\...
user30246's user avatar
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Weight-2 modular forms under $\Gamma(N)$

I'm looking for explicit bases of weight 2 modular forms under $\Gamma(N)$, for small N (<16 would be enough). (Ideally in terms of Theta- or Eta-Functions) It seems to me that this should ...
phoboid's user avatar
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Studying primes via the gamma function alone: $(x+1)\prod_n \Gamma(\frac{x}{n}+1)^{\mu(n)}$

Various questions on MO concerning the "surprise" occurrence of the gamma function in the functional equation of the Riemann zeta function got me wondering whether the Gamma function alone suffice for ...
David Feldman's user avatar
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Has the Weierstass transform been used to give Hermite series representations of the Riemann zeta function?

The inverse of the Weierstrass transform expands a function as a series of Hermite polynomials $H_{n}$. There are several ways to invert the Weierstrass transform which led me to the following ...
Craig Calcaterra's user avatar
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Two meromorphic functions with overlapping sets of poles

Assume that we have two meromorphic functions $f(z,w)$ and $g(z,w)$, where $z$ and $w$ are complex (we are interested only in behavior on compact sets). Fix $z$ and assume that the sets of poles of $f(...
Semyon Dyatlov's user avatar
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Gaussian integral over a ball

How to compute the following integral? $$\int_{\|x\|^2\leq R} \exp(-x^\ast G x+2\mathcal{Re}(x^\ast a)) \,dx,$$ where $x$ is an $M \times 1$ vector ($M\gg 1$), $G$ is a positive definite matrix, and $...
F Researcher's user avatar
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1 answer
208 views

Which combinations of normality, separability, and paracompactness do complex manifolds possess?

I am interested in what kinds of non-paracompact complex manifolds may exist and which topological properties they may have. Is there a non-separable complex manifold? Can a non-separable complex ...
Joseph Van Name's user avatar
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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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Matrix product of entire functions

Suppose I have two $d \times d$ entire matrix functions $F, G$ defined on $\mathbb{C}$ with the the property that $\|FG^*\|_{L^\infty(\mathbb{C})} < \infty$. Can anything be said about $F$ and $G$, ...
Joshua Isralowitz's user avatar
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251 views

Order of growth of $\left|\frac{1}{\zeta’(\rho)}\right|$ as $\Im(\rho)\rightarrow\infty$?

Let $\zeta$ denote the Riemann zeta function, and let $\rho\in\mathbb{C}$ be a variable that takes its values among the zeros of the zeta function, so that $\zeta(\rho)=0$, and write $\rho=\sigma+it$. ...
EGME's user avatar
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Is there a dense set of Lipschitz functions in $H^\infty(U)$, each of which maps $(1,0,\ldots,0)$ to 1, where $U$ is the unit ball in $\mathbb{C}^N$?

Let $U$ be the open unit ball in $\mathbb{C}^N$, let $A(U)$ be the algebra of functions analytic on $U$ and continuous on $\bar U$, and let $u=(1,0,\ldots,0)$. Let $\mathcal{B}=\{f\in H^\infty (U): \|...
David Walmsley's user avatar
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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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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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Reverse Sobolev inequality for family of holomorphic functions

Denote by $P_m$ the space of polynomials of degree $m$ in a single complex variable $x$. This is a "Reverse Sobolev inequality": Theorem. Let $U \subset \mathbb{C}^2$ be an open domain and $...
Sébastien Loisel's user avatar
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203 views

Cartan–Remmert reduction of an algebraic variety

Let $V$ be a normal connected algebraic (say, quasi-projective) variety over complex numbers. Assume that underlying complex analytic space $V^\text{an}$ is holomorphically convex, and thus admits the ...
 V. Rogov's user avatar
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1 answer
157 views

Probability of a number being a bound for roots

Consider the polynomial $p(z)=\sum_0^na_iz^k$ where $a_n=1$ and $a_k \sim N(0,1)$, $k=0,1,2,\dotsc,n-1.$ What is the probability that 2 will be a bound of the roots of the polynomial? How can we find ...
AgnostMystic's user avatar
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66 views

harmonic envelope of holomorphy

Let $D$ be a (real) domain in $\mathbb R^n=\mathbb R^n+i\lbrace 0 \rbrace\subset \mathbb C^n$. Then, due to P. Lelong, there exists a maximal (complex) domain $\tilde D\subset\mathbb C^n$, $D=\tilde D\...
Peter Pflug's user avatar
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0 answers
444 views

Question about a paper by Franca and LeClair in analytic number theory

I am reading an article "Transcendental equations satisfied by the individual zeros of Riemann $\zeta$, Dirichlet and modular L-functions" by G. Franca and A. LeClair (2015) see here. The ...
Williams's user avatar
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Converse theorem for zeta universality

Voronin's Universality Theorem for $\zeta(s)$ is that the zeta function can uniformly approximate any non-vanishing holomorphic function to any degree of accuracy in the right-half of the critical ...
modperspec's user avatar
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255 views

Geometric interpretation of Theta functions and the Jacobi inversion problem

A great part of complex geometry and, algebraic geometry has been developed to address the theory of abelian integrals/functions. A very special problem that kept many great mathematicians busy was ...
Victor Felipe's user avatar
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221 views

Karamata's Abelian/Tauberian Theorem in the complex plane

The following result is well known (a particular case of Karamata's Tauberian Theorem for Power Series in Corollary 1.7.3 of Regular variation by Bingham, Goldie & Teugels): Fix $c, \rho>0$. If ...
Gagar's user avatar
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Trigonometric sum and residues

I am interested in the sum $$ \sum_{n=1}^k 2\biggl[\sin\biggl(\frac{n\pi}{2k+2}\biggr)\biggr]^{-2g} $$ where $k$, $g$ are integers. It is not too hard to show that this can also be expressed as $$ -1-...
Delmastro's user avatar
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Are fibers in the corona of $H^\infty$ separable?

Let $\mathcal{M}(H^\infty(\mathbb{D}))$ denote the spectrum of the Banach algebra $H^\infty$ and $\mathcal{M}_z(H^\infty(\mathbb{D}))$ the fiber over $z\in \mathbb{D}$, i.e. $\{\varphi\in \mathcal{M}:...
Stiglitz's user avatar
4 votes
0 answers
148 views

On the best constant for Carleson's embedding theorem

In "Interpolations by bounded analytic functions and the corona problem", Carleson introduced Carleson measures (for Hardy spaces) and proved the famous embedding theorem according to which $...
Stiglitz's user avatar
4 votes
0 answers
175 views

It is possible, without adding further hypotheses, to refine Rouche's theorem in order to obtain a finer localization of the zeros?

The title says it all: a now deleted question on the Mathematics Stackexchange asked more or less the same thing, and I answered by citing the work [1] of Wolfgang Tutschke, whose version of Rouche's ...
Daniele Tampieri's user avatar
4 votes
0 answers
176 views

As increasingly higher degree terms are added to a "random" polynomial, how fast do the roots approach the unit circle?

As increasingly higher degree terms are added to a "random" polynomial, the roots of a polynomial can be proven to approach the unit circle. For example, see the MathOverflow question Why ...
Likes Algorithms's user avatar
4 votes
0 answers
610 views

What is a holomorphic foliation?

For a smooth foliation $F$, there are three equivalent definitions: the leaves of $F$ are tangent to a smooth vector field; the foliation chart $\phi:U\to \mathbb R^k\times \mathbb R^{n-k}$ is ...
Mjr's user avatar
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Walsh-Lebesgue type theorem in $\Bbb R^{2m}$ for $m>1$

Is someone aware of any analogue of the Walsh-Lebesgue theorem in $\mathbb{R}^{2m}$ for $m>1$ and dealing with polyharmonic polynomials? In this post, $\phi$ is said to be polyharmonic in $\mathbb{...
Pierre's user avatar
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106 views

Quasi-crystaline generalization of elliptic functions

I came across some meromorphic function, call it $f(z)$, which is "quasicrystalline" in the following sense: one can write $f$ as: $$ f(z)=\frac{\sum_i a_i e^{i(q_{i,x}x+q_{i,y}y)}}{\sum_i ...
Yarden Sheffer's user avatar
4 votes
0 answers
145 views

How to show the set of stable polynomials equals to the set of Lorentzian polynomials in degree 2

Give a homogenous polynomial $f\in \mathbb{R}[x_1,\dots,x_n]$ of degree $2$ in $n$ variables, we can consider $f$ as a quadratic form. We call $L_n^2:=$ the set of quadratic forms with nonnegative ...
ypl's user avatar
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0 answers
254 views

Four infinite series involving Riemann zeta function

Can you provide a proof for at least one of the claims given below? It is known that $\pi=\displaystyle\sum_{n=1}^{\infty}\frac{3^n-1}{4^n} \cdot \zeta(n+1)$ where $\zeta$ denotes Riemann zeta ...
Pedja's user avatar
  • 2,673
4 votes
0 answers
189 views

Approximation of a holomorphic function vanishing at a submanifold by polynomials

Suppose $M$ is a complex affine algebraic manifold in ${\mathbb C}^n$ (I mean, a set of common zeroes of a system of polynomials on ${\mathbb C}^n$, which is at the same time a smooth manifold). ...
Sergei Akbarov's user avatar
4 votes
0 answers
181 views

A more general version of the Fejér-Riesz theorem

A classical result, known as the Fejér-Riesz theorem, states that any Laurent polynomial $p(z)=\sum_{|k|\leq N} c_kz^k$ (the coefficients $c_k$ are complex numbers) which is nonnegative on the torus $\...
Stefano Rossi's user avatar
4 votes
0 answers
454 views

Understanding vector-valued analytic functions vs holomorphic functional calculus

Let $A$ be a unital Banach algebra over complex numbers and call elements of $A$ "vectors". Let $\Omega$ be an open set in $\mathbb{C}$ and $H(\Omega)$ the space of analytic functions on $\...
Stanley Chan's user avatar
4 votes
0 answers
164 views

Finding roots of equation with gamma functions

Encountered this function in one of my research problems $$\frac{\Gamma \left(1-\dfrac{i c}{a}-\gamma \right) \Gamma \left(1+\dfrac{i c}{a}+\dfrac N 2-\gamma \right)}{\Gamma \left(1+\dfrac{i c}{a}-\...
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