# Questions tagged [cv.complex-variables]

Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.

2,266
questions

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65 views

### Stokes's Theorem with singularities on projective line

Let $X$ be a complex manifold and $\omega\in \Omega(X\times \mathbb{P}^1)$ a form. I met the following identity:
$$\int_{\mathbb{P}^1}(\partial_z\bar{\partial}_z\omega) \log|z|^2=\int_{\mathbb{P}^1}\...

**4**

votes

**1**answer

325 views

### An open problem on polynomials

Let $P(z)$ be a polynomial of degree $n$ with $|P(z)|\leq 1$ on $|z|=1$ and $P_m(z)$ be a partial sum of $P(z).$ How large $P_m(z)$ can be on $|z|=1?$ It is an open problem and I did not find any ...

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votes

**0**answers

41 views

### Holomorphic semigroups vs analytic semigroups

Is there any difference between the two notions in the theory of semigroups?
In the literature, we find some monographs use the farmer while others use the latter. I expect that they are always the ...

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votes

**0**answers

57 views

### Derivative of a polynomial $P(z)$ and the derivative of the conjugate reciprocal of $P(z)$

Let $P(z)$ be a polynomial of degree $n$ having no zeros in $|z|<1.$ Let $Q(z)=z^n\overline{P(1/\overline{z})}.$ Then it is an easy exercise to show that $\Re\left(zP'(z)/P(z)\right)= \sum_{k=1}^n\...

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72 views

### Theta-function in the lower half-plane

Standard theta function
$$\vartheta(q)=\sum_{n=-\infty}^\infty q^{n^2} \qquad\qquad(1)$$
has a natural boundary of analyticity at $|q|=1$. This means that it can not be used to regularize expressions ...

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votes

**0**answers

27 views

### Fixing coefficients using analytic structure

I am trying to understand an exercise in a set of lecture notes on random matrices - Eynard - Random matrices - given in the paragraph following (4.6.60) (pp. 70–71).
Specifically, I am trying to fix ...

**0**

votes

**1**answer

60 views

### Logarithms of $L$-functions of irreducible characters of Galois group

We know that the $L$ functions of Dirichlet characters $\chi$ of $(\mathbb Z / m\mathbb Z)^\times$ satisfy the property that $\log L(s, \chi)$ is holomorphic for $\Re(s) \geq 1$ if $\chi$ is a ...

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votes

**1**answer

76 views

### Existence of analytic continuation of Dirichlet series corresponding to the indicator sequence of a complement of a special multiplicative set

Let $K/ \mathbb Q $ be a finite Galois extension and let $X$ be a proper non-empty subset of the Galois group $G=Gal(K/ \mathbb Q)$ that is closed under conjugation. Consider a set of integer primes $...

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votes

**1**answer

75 views

### A question on preimage of a locally injective meromorphic function

Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be a meromorphic function such that $f'(z) \ne 0$ for all $z \in \mathbb{C}$. Let $\Gamma \subset \mathbb{C}$ be a curve which has no self intersections. If ...

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votes

**1**answer

185 views

### To show holomorphicity of a certain infinite series of functions

I have the following sequence of holomorphic functions on $(f_n(s))_{n \geq 1}$ on the closed region $R:= \{ s \in \mathbb C : \Re(s) \geq 1\}$ by
$$f_n(s) :=
\begin{cases}
\log(1+p^{-s})-p^{-s}\...

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votes

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205 views

### On the logarithmic derivative of an analytic function

I have reached upto this situation while exploring some properties of analytic functions which are bounded on the unit circle.
Suppose $f(z)$ be such an analytic function in the closed unit disc ...

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votes

**1**answer

199 views

### Fun examples relating to Hopf surfaces

A Hopf surface is a compact complex surface whose universal cover is complex analytically isomorphic to $\mathbb{C}^2 \setminus \{ 0 \}$. I would like to know whether anyone has any of the following ...

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votes

**1**answer

110 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 ...

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votes

**1**answer

303 views

### Sendov's conjecture

It has been more than fifty years for famous Sendov's conjecture which states that if $p(z)$ is a polynomial of degree $n$ having all its zeros in the unit disc $|z|\leq 1$ then each of the n ...

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51 views

### zero extension of positive currents are always positive

In Demailly's Complex Analytic and Differential Geometry page 139:
He said the trivial (zero) extension of the positive current $T$ (on $X\setminus E$), which denoted by $\tilde T$ is always positive ...

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votes

**1**answer

105 views

### The state of art of the singular Levi problem — and hyperkähler quotients

One of the versions of the classical Levi problem asks the following:
Let $X$ be a complex manifold. Is it true that $X$ is Stein iff
$X$ admits a smooth exhaustion strictly plurisubharmonic ...

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votes

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49 views

### Resources for divergent / asymptotic series

This series is divergent; therefore, we may be able to do something with it. -- Oliver Heaviside
Other than the usual references given in Wikipedia and Mathworld, which resources have you found ...

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votes

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39 views

### maximum modulus of polynomials

Suppose $p(z)$ is a polynomial of degree $n$ having no zeros in $|z|<1.$ Then for any $R\geq1$
we have a result $
\max_{|z|=R}|p(z)|\leq \frac{1+R^n}{2}max_{|z|=1}|p(z)|,\;\;R\geq 1.$
I could not ...

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48 views

### Separating a Riemann-Hilbert problem

Consider a RHP on the real line a jump is piece-wise H\"older continuous(or $L^2$), say for example the jump is
$$g(x)=g_1(x)\chi_1+g_2(x)\chi_2,$$
where $g_j(x)$ are Holder continuous functions and $...

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votes

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84 views

### Exponential iterates of a complex number

Let $f:\mathbb C\to \mathbb C$ be defined by $f(z)=e^z-1$. Let $f^n$ denote the $n$-fold composition of $f$.
In my new paper Erdős space in Julia sets I show that $$Z:=\{z\in \mathbb C:\lvert\...

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votes

**1**answer

81 views

### Space of holomorphic functions multiplied by smooth functions taking real values

Suppose we have a fixed function $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ (sufficiently regular, say $C^\infty$ ). The question is: for which $f$ there exists a scalar function $g: \mathbb{R}^2 \...

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**1**answer

50 views

### weak convergence of positive currents vs. $L^1$ convergence of normalized potentials

I have run into the following statement in the literature (e.g. here, p.5, after Theorem 1.1): that weak convergence of positive $(1,1)$-currents on a complex manifold is equivalent to $L^1$ (I ...

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**0**answers

43 views

### Locally connectedness and accessibility in $\mathbb{C}$

Suppose $\Omega$ to be a bounded area in the complex plane $\mathbb{C}$ with a locally connected boundary $\partial\Omega$, then every point of $\partial\Omega$ is accessible from Ω.Here accessibility ...

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votes

**1**answer

333 views

### Complex plane minus Cantor set admits non-constant bounded harmonic function

Let $K\subset [0,1]$ denote the usual 1/3 Cantor set. I know that $\mathbb{C}\backslash K$ has no non-constant bounded analytic function, since the singularity $K$ can be removed. However, a statement ...

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63 views

### Construction of weight function to satisfy condition on given functional

(Sorry for similar and trivial looking question ; But could have potential application in prime number theory )
Consider the following function :
$$F(z) = \omega(z)\frac{\sin^2\left(\frac{c\Gamma^2(...

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votes

**2**answers

352 views

### if $f\circ f=g$ has no solution does this imply $f\circ f=g+g^{-1}$ also has no solution with $g^{-1}$ being a compositional inverse of $g$?

This question is related to solving $f(f(x))=g(x)$.
Assume that $g$ is a bijective function $g:\mathbb{R}\to \mathbb{R}$. If there is no continuous function $f : \mathbb R \to \mathbb R\,$ for which $...

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vote

**1**answer

184 views

### Entire even functions of order 1 have infinitely many zeros?

Let $f$ be an entire even function of order 1 such that $f(0)\neq 0$. Does $f$ have infinitely many zeros?

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147 views

### cohomology classes of complex submanifolds

I was wondering if there were restrictions in what the cohomology classes corresponding to complex submanifolds of a complex manifold could be.
For example, say $T^4$ is regarded as a complex ...

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150 views

### What is decoupling theory means on Tao Blog ? And what is its purpose in mathematics? [closed]

I accrossed on Tao Blog a new theory for me which it is called "Decoupling theory", But I didn't find in the web its definition and its purpose , I find only this article in wiki but this very far ...

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59 views

### How to solve a problem from Frank Olver's book

I'm learning Frank Olver's book, called Asymptotics and Special Functions. There is an difficult exercise.
Problem. Suppose that $f,\frac{1}{f}$ possess the following asymptotic expansions :
$$f(...

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votes

**3**answers

438 views

### Acting with all rational rotations on a subset of the circle having positive measure do you fill almost the whole circle?

Set $\Gamma$ for the group of the roots of the identity: $\Gamma=\{z\in \Bbb C | z^n=1$, for some $n\geq 0\}$ and for $E\subset S^1$ set
$\Gamma E=\{z\zeta, z\in \Gamma, \zeta\in E \}$
A trivial but ...

**3**

votes

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76 views

### Existence of a maximal rank CR Lie subalgebra

Let $\mathfrak{g}$ be a real Lie algebra of dimension $2n+1$ and let $\mathfrak h \subset \mathfrak g \otimes \mathbb C$ be a subalgebra of complex dimension $n+1$ satisfying $\mathfrak h + \overline{\...

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votes

**1**answer

67 views

### Rate of convergence of Padé approximants

Let $f$ be an entire function of order $1$. Two questions:
1) Can one assert that the diagonal Padé approximants converge to $f$ (pointwise or uniformly over compacts of $\mathbb C$)?
2) if yes, can ...

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vote

**1**answer

41 views

### Stieltjes transform of a compactly supported measure : behaviour at the boundary

I ask here the same question I asked on Mathematics to maybe reach other poeple :
I am studying the Stieltjes transform $$ G_\mu(z) = \int_a^b \frac{1}{z-s} d \mu(s) $$ of some positive finite ...

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**0**answers

117 views

### What is the closed form of this integral?

Consider the Chebyshev first function $\psi(y):=\sum_{p^j \leq y} \log p$, where $p$ is a prime. Define $$F(s, k) = s\int_{1}^{\infty} \psi(x + x^k)x^{-s-1} \mathrm{d}x$$ for $ \Re(s) >$ max $(1, ...

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**1**answer

77 views

### A density question

Suppose $\Omega= (0,1)\times(0,1)\subset \mathbb R^2$. Assume that $f, g \in C^{\infty}(\Omega)$ and that
$$ \int_\Omega \left(f(x_1,x_2)- \frac{m}{(n+1)}g(x_1,x_2)\right) x_1^n \,x_2^m \,dx_1\,dx_2 = ...

**2**

votes

**2**answers

93 views

### Reference request for the integral representation of the Hadamard product of two infinite series

Define $F(x) = \sum_{n\geq 1} f_{n}x^n$ and $G(x) = \sum_{n\geq 1} g_{n}x^n$. Then the Hadamard product of $F$ and $G$ is
$$H(x):=(F*G)(x) = \sum_{n\geq 1} f_{n}g_{n}x^n.$$
The author of Riesz ...

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votes

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36 views

### Asymptotics of a certain integral in singularity theory

Let $f:\mathbb{C}^2\to \mathbb{C}$ be an isolated plane curve singularity. Consider the versal deformation space $\mathbb{C}^\mu$ parameterizing deformations $f_\lambda$ for $\lambda \in \mathbb C^\mu$...

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72 views

### 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 $D$ together with two conformal isomorphisms $f_1 \colon \mathbb D \to D$ ...

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votes

**1**answer

64 views

### Is a domain biholomorphic to the unit ball a Runge domain?

Let $\Omega \subset \mathbb C^n$ be a bounded domain which is biholomorphic to the unit ball $B^n=\{|z|<1 \mid z\in \mathbb C^n\}$. Can we show $\Omega$ must be a Runge domain? By definition, $\...

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votes

**1**answer

112 views

### Subharmonic in any holomorphic coordinates = Plurisubharmonic?

An upper semi-continuous function $u : \Omega \to \mathbb{R}$, $\Omega \subseteq \mathbb{C}^n$ is said to be subharmonic if it satisfies the submean inequality $u(a) \leq \mu_S(u;a,r)$, where $\mu_S(...

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votes

**0**answers

50 views

### Limit of Hankel function for large complex order, fixed real argument

Consider the Hankel function $H_\nu(z)$ where $\nu=re^{i\theta}$ (real $z>0$, $r>0$, $0\leq\theta<\pi$) as $r\rightarrow\infty$. I am aware that the Bessel function $J_\nu(z)$ has the ...

**2**

votes

**1**answer

75 views

### Conformal isomorphism uniquely determined by boundary identification?

Let $\Gamma$ be a smooth Jordan arc, and let $\Phi \colon \hat{\mathbb C} \backslash \overline{\mathbb D} \to \hat{\mathbb C} \backslash \Gamma$ be a conformal isomorphism that fixes the point at $\...

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votes

**1**answer

102 views

### Reference on boundary behavior of conformal maps

I am looking for some results on the boundary behavior of conformal maps between simply connected domains. In particular I am interested in conformal maps between $\mathbb{C}-\Delta$, where $\Delta$ ...

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votes

**1**answer

89 views

### How to check if you have the asymptotic solution of some equation? [closed]

Suppose I have an analytic function $f : \mathbb{R} \to \mathbb{R}$ and I have the asymptotic expansion of some $x_0$ up to a few terms in terms of $\epsilon$ for some $\epsilon \to 0$ which I believe ...

**0**

votes

**1**answer

73 views

### Transformation which “opens up” an arc

I am reading Harold Widom's paper "Extremal Polynomials Associated with a System of Curves in the Complex Plane". At the beginning of section 11 he states that:
[There is] a simple transformation ...

**3**

votes

**1**answer

131 views

### Isoperimetric inequality for analytic functions on an annulus

Let $f$ be an anylytic function on the unid disk $|z|<1$. It is well known that
$$\left (\int_0^{2\pi}f(e^{i\theta})d \theta \right)^2 \geq 4\pi \iint_{|z|<1} |f(r e^{i\theta})|^2r dr d \theta.$...

**8**

votes

**1**answer

518 views

### A question on the Riemann zeta function

Yesterday, a certain very talented and passionate young student from Southern Africa asked me the following question about the Riemann zeta function $\zeta(s)$. He says he "thinks" he knows the answer,...

**1**

vote

**1**answer

119 views

### Commuting matrices of complex functions

If $A(z) :=[A_{ij}(z)] $ and $B(z) :=[B_{ij}(z)] $ are two invertible $n\times n$ matrices of entire complex valued functions entries $A_{ij}(z)$, and $B_{ij}(z) $ with
(1). $AA^{\#}=A^{\#}A$ ...

**10**

votes

**1**answer

225 views

### Convex Julia sets

Consider the classical Julia set $J_f$ associated with $f(z)=z^2+c$.
Since $J_c$ is completely invariant,
we know that $f^{-1}(J_f) \subseteq J_f$.
Now, let $H_f$ be the convex hull of $J_f$.
Is it ...