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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A converse of the maximum modulus Theorem

W. Rudin in Real and Complex Analysis (262) mentioned that Theorem Suppose $M$ is a vector space of continuous complex functions on the closed unit disc $\bar U$, with the following properties: (a) $...
azalea's user avatar
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3 answers
871 views

Boundary behavior of a holomorphic function on $D$ ?

Hi, I have two related questions. $D$ = open init disk in the complex plane $C$. A. Let $f: D \to C $ be a holomorphic function. Then is it possible that $\forall q \in S^1$,there exists a ...
Analysis Now's user avatar
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Factorization in the Wiener algebra on the unit disc.

Consider the Banach algebra $W^+=\ell^1(\mathbb{Z}^+)$, viewed upon as the analytic functions $f$ on the unit disc $\mathbb{D}$ such that $$\|f\|=\sum_{k\ge0}|a_k|<\infty$$ where $$f(z)=\sum a_kz^k$...
AD - Stop Putin -'s user avatar
4 votes
2 answers
606 views

The link of a singular quintic hypersurface in CP^4

Given a family of quintic hypersurfaces in $\mathbb{CP}^4$ by $x_1^5+x_2^5+x_3^5+x_4^5+x_5^5+(5+\epsilon)x_1x_2x_3x_4x_5$ we get a singular variety for $\epsilon=0$ with 125 singular points. I know ...
Peter Miller's user avatar
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2 answers
439 views

Elementary functions with zeros only at the positive integers

Does there exist a (meromorphic) elementary function $f(z)$ that is zero at all the positive integers $z = 1, 2, 3, \ldots$ and only at those points? Edit: an elementary function can be written as a ...
Fredrik Johansson's user avatar
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1 answer
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Does $P_xP_y+Q_xQ_y=0 \implies$ "non-existence of limit cycle" for $P\partial_x+Q\partial_y$"? (Complex dilatation and limit cycle theory)

Let $X=P\partial_x+Q\partial_y$ be a vector field on the plane $\mathbb{R}^2$. Assume that we have :$$P_xP_y+Q_xQ_y=0$$ Does this imply that the vector field $X$ is a divergence-free vector field ...
Ali Taghavi's user avatar
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A kind of holomorphicity of maps on Hilbert space

Let $H$ be an infinite dimensional seperable Hilbert space. Is there an Irreducible involutive sub algebra $D$ of $B(H)$ with the following properties?: 1)For every open set $U\subset H$ and every ...
Ali Taghavi's user avatar
4 votes
1 answer
193 views

Distribution boundary value of analytic function and wave front sets

Assume $f(z)$ is analytic in the tube domain $\mathbb R^n\oplus iC$, where $C\subset \mathbb R^n$ is a convex cone. Under the assumption $|f(x+iy)|\leq 1/|y|^k$, we know by a Theorem of Martineau (see ...
Dima's user avatar
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Conformal mappings for domains whose complement is totally disconnected

Consider a conformal homeomorphism $f \colon \Omega \to \Omega'$ between domains $\Omega, \Omega'$ in the plane. Let $E = \mathbb{R}^2 \setminus \Omega$ and $E' = \mathbb{R}^2 \setminus \Omega'$. My ...
mdr's user avatar
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A question about openness theorem

The openness theorem says that: If $\varphi$ be a negative plurisubharmonic function in the unit ball $B(0,1)$ in $\mathbb{C}^{n}$ satisfying $$ \intop_{B(0,1)}e^{-\varphi}<\infty, $$ then there ...
Haji's user avatar
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Regarding outer functions

Please see the definition of Hardy spaces on the unit disc here. Let $0<p\leq\infty$. Let $f\in H^p$ with $\|f-1_e\|_p<1$ (Where $1_e$ Is the constant function one). Then is $f$ an outer ...
user510271's user avatar
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Are there algorithmic tools for computing poincare residues?

In Schnell's note on Computing Picard-Fuchs Equations he gives a recursive method for computing residues on hypersurfaces. In short, if you have a meromorphic differential form $$ \frac{dw}{w^k}\wedge ...
54321user's user avatar
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Riesz inequality

Let $p>2$ and $$\|b\|^p_p:=\int_0^{2\pi} |b(e^{it})|^p dt.$$ Assume that $a$ is a bounded holomorphic on the unit disk and assume that $a(0)=0$. What is the best constant $C_p$ in the inequality $$\...
djole's user avatar
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Integral Expression in Complex Dynamics

Let $\phi\in \mathbb{C}(z)$ be a degree $d\geq 2$ rational map, which we can write as $\phi = \frac{f}{g}$ for $f,g\in \mathbb{C}[z]$. Let $\omega_{FS}$ denote the Fubini-Study form on $\mathbb{P}^1(\...
Ken Jacobs's user avatar
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upper bound on derivatives of a function defined on an arc

This is a simple question I asked in math.SE last month but unfortunately no one gives any comment. So I decided to try some luck here. You can skip examples below and read from "General setting" at ...
booksee's user avatar
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351 views

Quantifying the monotonicity property of the hyperbolic metric

Suppose that $X$ and $Y$ are connected hyperbolic Riemann surfaces, with $X\subsetneq Y$. Let $\rho_{X,Y}$ be the density of the hyperbolic metric of $X$ with respect to that of $Y$; then $\rho_{X,Y} &...
Lasse Rempe's user avatar
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2 answers
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existence of a special conformal mapping

Sorry I don't know how to give an appropriate title. In the complex plane, suppose there is a graph $x+if(x)$ separating the plane into two unbounded components, where $f(x)$ is smooth and bounded, ...
qingtang's user avatar
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1 answer
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Laurent expansion of a principal value integral

Let $f(t)$ be a nice Hölder continuous function. Also, suppose that $f$ is even. I'm interested in evaluating integrals of the form: $$\oint (1-z)^{k+1}\int_0^1 \frac{f(t)}{(1-zt)^{n+1}}dtdz,$$ ...
Alex R.'s user avatar
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1 answer
479 views

Weighted projective space with rational or real weights

The most common formulation of the weighted projective space is perhaps the global quotient $$ (\mathbb{C}^{n+1} \setminus \{(0,\ldots,0\}) / \mathbb{C}^\ast $$ with the $\mathbb{C}^\ast$ group action ...
ssquidd's user avatar
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2 answers
310 views

Algebraic function with extra condition, what can it be?

I have an unknown function $\psi(\xi_1,\dots,\xi_n)$, such that $\psi$ satisfy an (unknown) polynomial equation with coefficients polynomials in the $\xi_i$. The function is homogeneous, that is, $\...
Per Alexandersson'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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152 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{...
Tomas's user avatar
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0 answers
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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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130 views

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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0 answers
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
4 votes
0 answers
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
4 votes
0 answers
124 views

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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0 answers
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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0 answers
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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0 answers
143 views

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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0 answers
119 views

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
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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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4 votes
0 answers
103 views

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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0 answers
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
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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}-\...
user824530's user avatar
4 votes
1 answer
158 views

Restricted Perron-Bremermann envelopes

Consider an upper semicontinuous function $\phi: \Omega \to (-\infty, \infty]$, in the sense that $\phi = \phi^*$, where $\phi^*$ denotes the upper semicontinuous regularization $$ \phi^*(z) = \...
mrf's user avatar
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4 votes
0 answers
90 views

Tauberian theorem converse (Wiener-Ikehara)

Jacob Korevaar provides a nice converse to to Wiener-Ikehara tauberian theorem on p. 125 of his Tauberian Theory book: For the non-decreasing, locally of bounded variation function $s$, if we have $\...
Hamid Enki's user avatar

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