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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3
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0answers
38 views

Two questions on asymptotic expansion of confluent hypergeometric functions for real variable $x, |x| \to \infty$

I'm looking into the asymptotic expansion for confluent hypergeometric function $_1F_1(a;b;z) \equiv M(a;b;z)$ and I've two quick questions regarding its asymptotic behavior for real values $x,$ i.e. ...
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58 views

Real integrals with complex analysis [closed]

I don't have a clear formal viewpoint on this problem. Resolving the Euler-Lagrange equations for the string with a point mass perturbation: $$ \frac{\partial^2 \phi }{\partial x^2} = \delta (x-a)$$ I ...
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18 views

What can be concluded from the analyticity in a half-plane of a sum of functions?

If I have two functions $F$ and $G$ for which $F(s)+G(s)$ is analytic in some half-plane ${\rm Re}(s)>a$, what can be concluded about these functions individually with respect to their analyticity? ...
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113 views

Does the maximum principle hold in this pluriharmonic setting?

Let $U \subseteq \mathbb{C}^m$ be open, and let $F: U \to \mathbb{C}$ be a holomorphic function, with real part $u$. We are given a subset $S \subseteq U$ given by finitely many real equalities and ...
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38 views

Regarding supremum over a set

Let $\Delta$ be a Compact Hausdorff space in $\mathbb{C^n}$. Let $A$ be a closed sub algebra of $C(\Delta)$(space of all complex valued continuous functions on $\Delta$) which contains the constant ...
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1answer
118 views

Holomorphic function bounded in a sector with angle $>\pi$ [closed]

I know that according to Liouville’s theorem, if a holomorphic function is bounded on all of C, it is constant. This got me thinking if I could find holomorphic non-constant functions that are bounded ...
11
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1answer
312 views

A density question for the Hilbert transform

Let $\mathscr Hf$ denote the Hilbert transform of a function $f$ defined on the real-line $\mathbb R$. Are the set of functions $$ \{(f+\mathscr Hf)_{|_{(0,1)}}\,:\, f \in C^{\infty}(\mathbb R)\quad \...
6
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2answers
115 views

holomorphy in infinite dimensions (holomorphic families of operators)

Let $X$ be a Banach space (over $\mathbb C$), and let $\mathcal L(X)$ be its algebra of bounded linear operators. Let $U\subset \mathbb C^N$ be an open subset, and $f:U\to \mathcal L(X)$ a function ...
3
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0answers
50 views

Infinitely many deformation equivalent Hodge diamonds II

Let $S$ be a connected smooth complex-analytic space. Let $\phi:X\to S$ be a proper holomorphic submersion with connected fibers. Can the fibers of $\phi$ have infinitely many distinct Hodge diamonds? ...
3
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2answers
230 views

On a variation of Hartogs' separate analyticity theorem

Let $f(z_1,z_2,\ldots,z_n)$ be a function on $\mathbf{C}^n$ such that for all $i$, the restriction $$ [z_i\mapsto f(z_1,z_2,\ldots,z_n)] $$ is a "rational function". (added: to be precise ...
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1answer
116 views

Theta series analogues for higher degree forms

It is simple to see that the following series converges absolutely and uniformly on $\mathcal{H}$ for all k positive: $F_{2k}(z) = \sum_{n \in \mathbb{Z}} q^{n^{2k}}$ And this series being a ...
2
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1answer
590 views

Complex manifold defined over $\mathbb{R}$

Let $M$ be a connected closed complex manifold with an antiholomorphic involution. Must there be an atlas and a choice of a reference point in each chart such that the transition functions are ratios ...
2
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1answer
119 views

Infinitely many deformation equivalent Hodge diamonds

Let $S$ be a connected open complex manifold. Let $\phi:X\to S$ be a proper holomorphic submersion with connected fibers. Can the fibers of $\phi$ have infinitely many distinct Hodge diamonds? An ...
5
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1answer
213 views

Hodge diamonds of complex threefolds

There is no closed complex curve or surface with $h^{1, 0}-h^{0, 1}=1$. Now consider threefolds. Can this condition be satisfied? Is Serre duality in fact the only restriction on the Hodge diamond?
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95 views

Does the minimal surface system in the plane have the weak unique continuation property?

Let $\Omega \subset \mathbf{R}^2$ be a domain in the plane and suppose that $u : \Omega \to \mathbf{R}^k$ is a smooth function for which the graph of $u$ is a smooth minimal surface in $\Omega \times \...
6
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183 views

All complex surfaces embed into a common complex manifold

Is there a closed complex manifold into which every closed complex surface embeds?
4
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104 views

Regarding the boundary of the range of a holomorphic map on the unit disc

Let $\Omega$ be a convex domain in $\mathbb{C}^n$. Let $f:\mathbb{D}\longrightarrow \bar\Omega$ be a holomorphic function. Let $z\in f(\mathbb{D})\cap \partial\Omega$. Let $\phi$ be a linear ...
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213 views

How singular can a holomorphic submersion over the punctured disk be?

Let $f : X \to \mathbb{D}^{\ast}$ be a holomorphic submersion from a compact Kähler manifold of dimension $n>1$. We say that $f$ admits a meromorphic extension $\widetilde{f} : \mathcal{X} \to \...
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41 views

Characterizing a set of functions

Let $\mathscr C$ be the space of measurable functions $f$ on the interval $[0,1]$ that can be written in the form $$ f(t)=\sum_{k=0}^\infty c_k e^{-kt},$$ for some square summable series $\{c_k\}_{k=1}...
11
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1answer
264 views

Why should I look at the resolvent formalism and think it is a useful tool for spectral theory?

Wikipedia calls resolvent formalism a useful tool for relating complex analysis to studying the spectra of a linear operator on a Banach space. Sure, I believe you because I've seen results that use ...
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2answers
1k views

Algebraic independence of shifts of the Riemann zeta function

Let $\zeta(s)$ denote the Riemann zeta function. Is the set $\{ \zeta(s-j)\, \colon\, j\in\mathbb{Z}\}$, or even $\{\zeta(s-z)\, \colon\, z\in\mathbb{C}\}$, algebraically independent over $\mathbb{C}$?...
5
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2answers
157 views

Necessary and sufficient conditions for a holomorphic function defined in the unit disk to be univalent?

de Branges has proved de Branges's theorem (the famous Bieberbach conjecture) that if a holomorphic function $f(z) = z+\sum_{n=2}^{\infty} a_nz^n$ in the unit disk $D = \{z\in \mathbb{C},|z| \leq 1\}$ ...
5
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1answer
154 views

First order PDE in complex variables?

Consider the equation $$f'(x)+ g(x)f(x)=0$$ This equation is an ODE and has a solution $$ f(x)=C e^{ \int_1^x g(x) \ dx}.$$ Similarly, we can look at complex variables and consider the equation and ...
3
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0answers
53 views

Real part of a holomorphic section of a vector bundle

Let $F\to M$ be a holomorphic vector bundle over a complex manifold $M$ and let $s:M\to F$ be a no-zero section. Let $E$ be the complexification of $F$, and suppose that $E$ admits a holomorphic ...
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69 views

About the definition of lineal convexity

I have been trying to understand the definition of lineal convexity. I am reading the article Duality of functions defined in lineally convex sets by Christer O. Kiselman. For a set $A\subset \mathbb{...
3
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107 views

A problem on polynomial operators

Let $p(z)=\sum_{k=1}^na_kz^k$ be a polynomial of degree $n.$ Then we have these two results; If $p(z)\neq 0 $ in $\{|z|<1\}$ then $$ np(z)+(\alpha-z)p'(z)\neq 0\label{1}\tag{1} $$ for all $z,\...
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76 views

A Riemann Hilbert problem on the unit square

Let $p_0=0$, $p_1=1$, $p_2=1+i$ and $p_3=i$ be the four vertices of a square $Q$ on the complex plane $\mathbb C$. Let $f \in C^{\infty}_c((0,1))$ and consider the following Riemann-Hilbert problem on ...
2
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1answer
146 views

Reconstructing the metric on $CP^2$ with special one forms

I know that $(z_1,z_2)$ are the affine\inhomogeneous coordinates on the complex projective space $CP^2$. Now I have four one forms $(Y_1,Y_2, Y_3, Y_4)$. I want to rewrite the Fubini Study metric on $...
23
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3answers
3k views

Cardioid-looking curve, does it have a name?

The curve, given in polar coordinates as $r(\theta)=\sin(\theta)/\theta$ is plotted below. This is similar to the classical cardioid, but it is not the same curve (the curve above is not even ...
5
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1answer
146 views

A domination property for the Hardy space $H^1$

In the theory of Hardy spaces of the unit disc, a fact that is implicitely used quite often is that if $f\in H^p, 1<p<\infty$, then there exists a function $F\in H^p$ such that $|f(z)| \leq |F(z)...
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0answers
95 views

Engineering mathematics course and the order of the teaching of its topics

I am going to teach a course names "Engineering Mathematics" and the topics in it, are: Fourier series and integrals; including the motivation and computational aspects and their ...
6
votes
2answers
132 views

Growth of (integral of) Laplace transform of a function of compact support as $Re \to -\infty$

Let $f:[0,\infty)\to \mathbb{R}$ be supported on $[0,1]$, with $\int_0^1 f(x) dx = 1$. Let $\mathcal{L} f$ be its Laplace transform. How slowly may $$\int_{-\infty}^\infty |\mathcal{L} f(\sigma+i t)| ...
2
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0answers
69 views

On the upper bound for the real part of the logarithmic derivative of a polynomial

Let $P(z)=a_0+a_1z+\cdots+a_nz^n, a_n\neq 0$ with $\max_{|z|=1}|P(z)|=1.$ May it be true that for all $z$ on $|z|=1,$ for which $P(z)\neq 0,$ $$\Re{\frac{zP'(z)}{P(z)}}\leq n-\frac{1+|a_0|-|a_n|}{1+|...
0
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1answer
72 views

Cross-ratios of $4$ boundary points on a continuous family of disks in $\mathbb C^1$

Let $S^1=\mathbb R^1/\mathbb Z$. Consider a family $\varphi_t$ of pieceswise smooth injective maps $\varphi_t:S^1\to \mathbb C^1$ depending continuously on $t$. Then each curve $\varphi_t(S^1)$ is a ...
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0answers
46 views

Bergman Kernel for $D_R (R>0)$

Let $R>0,$ $D_R=\{z\in \mathbb{C} / |z|<R\}$; I am trying to show that if $f$ holomorphic on $D_R$ and continue on $\bar D_R$, and $w$ is an arbitrary point in $D_R$, then $$f(w)=\frac{R^2}{\pi}\...
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1answer
421 views

On Soundararajan's explicit formula

I'm reading Soundararajan's https://arxiv.org/pdf/0705.0723.pdf, and on page 5, one has $$\sum_{n\leq x} \frac{\Lambda(n)}{n^z} \log (x/n) = -\frac{\zeta'}{\zeta}(z)\log x - \Big(\frac{\zeta'}{\zeta}(...
3
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1answer
64 views

Are injective analytic maps between non-archimedean spaces open?

Let $\Omega$ be a non-archimedean complete field, $n\in\mathbb N$ and $f:\Omega^n\to\Omega^n$ be an injective analytic map. Is the application $f$ open? In the complex case, this is a consequence of a ...
6
votes
0answers
162 views

Proof of Tian's constant

Basically Tian's invariant relies fundamentally on a lemma of Homander which says that $e^{- \phi}$ is integrable for $\phi$ plurisubharmonic on a ball of radius $1$ under some extra assumption. I am ...
3
votes
1answer
128 views

Mittag-Leffler for non-compact Riemann surfaces

Quote from Grauert & Remmert's Theory of Stein spaces: 'Behnke and Stein showed in 1948 that the Mittag-Leffier Partial Fraction Theorem and the Weierstrass Product Theorem (i.e. the Cousin ...
5
votes
1answer
256 views

Factorization in formal power series versus in convergent power series over the complexes

Let $R=\mathbb C\{x_1,...,x_n\}\subset S=\mathbb C [[x_1,...,x_n]]$ denote the ring of convergent, respectively formal, power series over $\mathbb C$. Suppose $f\in R$ is irreducible in $R$. Does it ...
2
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0answers
59 views

Particular Ehrenpreis factorization for covariance function

Let $f:\mathbb{R}^d\to\mathbb{R}$ be a smooth compactly supported covariance function of a stationary random fields (hence positive definite). Is there a compactly supported function $g:\mathbb{R}^d\...
17
votes
1answer
642 views

About the function $\prod_{k \in \mathbb{N}}(1-\frac{x^3}{k^3})$

I'm wondering if the function $$f(x)=\prod_{k \in \mathbb{N}}\left(1-\frac{x^3}{k^3}\right)$$ has a name, or if there are any properties (especially about derivatives of $f$) have studied so far. I ...
18
votes
1answer
427 views

Hadamard factorization of L-functions

I have already asked this question here in a different form, but really need an answer. Let $L(s)$ be a "standard" $L$-function, say with Euler product, functional equation, etc... (Selberg ...
1
vote
1answer
82 views

Subharmonic function in unbounded regions

The harmonic majorization for a subharmonic function $h$ is well-known for bounded regions $\Omega \subset \mathbb{C}$: $$h \le 0 \text{ in }\partial \Omega \Longrightarrow h \le 0 \text{ in }\Omega.$$...
12
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2answers
877 views

Short research articles

I am a masters student. I am interested in short articles which have counter examples and very few references. I want to write a short and interesting article. For example; One of the best known ...
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0answers
48 views

Extension to all dimensions of complex line integral

Let $\Gamma$ be a smooth curve in $\mathbb{C}^d$. Since $\mathbb{C}^d$ can be seen as $\mathbb{R}^{2d}$, one can define the line integral of functions $f:\Gamma\to \mathbb{C}$ using for instance ...
3
votes
0answers
103 views

When are two complex Tori biholomorphic

Let $g \ge 1$ be a natural number and $\mathbb{C}^g$ complex vector space which is isomorphic to $\mathbb{R}^{2g}$ is real vector space. An additive subgroup $\Gamma \subset \mathbb{C}^g$ is called a ...
6
votes
0answers
166 views

Tsuchiya-Ueno-Yamada's proof that sheaves of conformal blocks are locally free

I'm referring to Tsuchiya-Ueno-Yamada's (TUY hereafter) celebrated paper Conformal Field Theory on Universal Family of Stable Curves with Gauge Symmetries. One of the main goals of their paper is to ...
5
votes
1answer
186 views

A functional equation in two complex variables

Let $X$ be a compact metric space, or just $X=\mathbb T$, the unit circle, if it helps. We consider only continuous, complex-valued functions on $X$. Let $\varepsilon >0$. Is there $\delta > ...
5
votes
0answers
130 views

On the asymptotics of some sum involving the Mertens function

Let $a_n$ be a sequence of nonnegative real numbers such that $\sum_{n\leq x} a_n \gg \frac{\sqrt x}{\log x}$ for large enough $x$. Denote by $\mu$ the Mobius function, and let $M(N)=\sum_{n\leq N} \...

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