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

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34 views

### Is a domain of a holomorphic flow pseudoconvex?

Let $Z$ be a holomorphic vector field on $\mathbb{C}^n$. I would like to know whether (it seems that it is) the domain $D_\phi \subset \mathbb{C} \times \mathbb{C}^n$ of a maximal flow $\phi: D_\phi \...

**2**

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**1**answer

90 views

### A question about distribution of fractional part of $2^k\alpha$

Let $\{x\}$ be the fractional part of $x$, i.e. $\{x\}=x-[x]$, where $[x]$ is the biggest integer $\leq x$.
The question might be well known but I don't know where to look for: Assume $\alpha$ is an ...

**2**

votes

**1**answer

210 views

### Does $\sum_{n=1}^\infty \frac{\mu(n)}{n^s}$ converge for $\sigma > \frac{1}{2}$?

Looking at @Lucia's answer to this question it appears $\sum_{n=1}^\infty \frac{\mu(n)}{n^s}$ converges for $\sigma > \frac{1}{2}$. Can someone point me to a proof or provide proof for this? If I ...

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**1**answer

231 views

### Cohomology of neighborhood of $\mathbb{C}\mathbb{P}^1$ in $\mathbb{C}\mathbb{P}^n$

Let $\mathbb{C}\mathbb{P}^1$ be embedded linearly to $\mathbb{C}\mathbb{P}^n$ with $n>1$. (Such an embedding is given in coordinates by $[x:y]\mapsto [x:y:0:\dots: 0]$.)
Is it true that for any ...

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**0**answers

70 views

### Borel-Ecalle re-summation and resurgence: Criteria and Results

This is about the theory of Borel-$\acute{\textrm{E}}$calle re-summation and resurgence, see Refs below.
This states that the perturbative series (say of the vacuum expectation value of an operator $\...

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**0**answers

113 views

### Cohomology of a tubular neiborhood of submanifold vs cohomology of the formal neighborhood

Let $X$ be a smooth complex analytic manifold. Let $Z\subset X$ be a smooth compact analytic submanifold. Let $\hat Z$ be the formal neighborhood of $Z$. Let $U$ be an open neighborhood of $Z$.
Is ...

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**1**answer

404 views

### Are there enough meromorphic functions on a compact analytic manifold?

Let $X$ be a compact complex analytic manifold, $D\subset X$ an irreducible smooth divisor, given as zeroes of a global meromorphic function $f\in {\mathfrak M} (X)$. Are there enough other ...

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**0**answers

80 views

### Is there a probabilistic proof/interpretation of Mergelyan Theorem

I came across Mergelyan's Theorem:- Let K be a compact subset of the complex plane C such that C∖K is connected. Then, every continuous function $f : K \to C$, such that the restriction f to int(K) ...

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222 views

### Cohomology of complex manifold vs cohomology of its complex submanifold

Let $X$ be a smooth complex analytic manifold. Let $Z\subset X$ be a smooth compact analytic submanifold. Let $A$ be a holomorphic vector bundle over $X$. Assume that
$$H^i(Z, A|_Z)=0 \mbox{ for any } ...

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**1**answer

266 views

### A variant of Cauchy-type functional equation conjecture

Let $f:\mathbb{C}\to \mathbb{C}$ be a complex function such that
$$|f(x-y)|=|f(x)-f(y)|,\qquad x,y\in\mathbb{C}.$$
Is it true that $$f(x+y)=f(x)+f(y),\qquad x,y\in\mathbb{C}?$$
The answer is ...

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votes

**2**answers

108 views

### Lelong numbers and integrability of psh functions

Let $\varphi$ be a plurisubharmonic function in the unit ball $B_1\subset \mathbb{C}^n$ with $\varphi\le 0$. Suppose that the Lelong number $\nu(\varphi,0)<k$ for some $k>0$. Does it follow that ...

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votes

**1**answer

906 views

### A conjecture of Littlewood

The following is a conjecture due to Littlewood.
For any set of distinct non-zero integers $n_1,\ldots,n_k$ the inequality
$$\int_0^{2\pi}|1+e^{in_1x}+\cdots+e^{in_kx}| \, dx\geq C\log k$$ holds....

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**1**answer

169 views

### An inequality of T. Carleman

I'm looking for the name and some references for the proof of the inequality below. I founded that is due to T. Carleman but no reference was given.
Let $f(z)$ be an analytic function on a subdomain $...

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**1**answer

89 views

### Algebraic independence of certain values implies algebraic independence of functions?

It is quite general and elementary question.
Is it possible that some holomorphic functions $f_1,\cdots,f_m $ on a region $\Omega $ of $\mathbb C$ satisfies:
Whenever $(f_1(z), \cdots, f_m (z)) $ is ...

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**2**answers

104 views

### Simple Closed Hyperbolic Geodesics on Punctured Spheres

Thinking of $\mathbb {CP^1}$ as the sphere $S^2\subset\mathbb R^3$, we can define the notion of a circle on it to be a subset that is got by a hyperplane section of $S^2$ inside $\mathbb R^3$. This ...

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**0**answers

402 views

### Is it possible that the following integral is $0$?

Given any integer $n\geqslant1$, let $E,F$ be two subsets of $\{\{i,j\}:1\leqslant i<j\leqslant n\}$ such that every two sets in $F$ are disjoint.
It is not difficult to see that
$$\int_{1<|z|&...

**3**

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**0**answers

126 views

### Analytic Aspects of Rational Maps

I would like some help finding references for a analytic treatment of rational maps between compact complex manifolds (that is holomorphic maps defined away from a codimension at least 2 subvariety). ...

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**0**answers

72 views

### Analytic continuation of an NLS soliton

The attractive nonlinear Schroedinger equation $i \partial_t \psi = - \frac {1} {2} \Delta \psi - |\psi|^{p-1} \psi$ in the $H^1$-subcritical case $1 < p$, $\frac {d} {2} + \frac {2} {p-1} < 1$ ...

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**3**answers

193 views

### about the Hausdorff dimension of Removable singularities of PDE

There are some interesting phenomenons about removable singularities (or extension problems).
In the theory of functions of several complex variables, we know the classical Hartogs theorem:
Let f ...

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**2**answers

354 views

### Conformal mappings that preserve angles and areas but not perimeters?

Conformal mappings from $U$ to $V$, both subsets of $\mathbb{C}$, locally preserve angles.
But, in general, such mappings neither preserve areas nor preserve perimeters.
Q. Are there examples of ...

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**0**answers

42 views

### What is the ring of functions on the open unit disc with polynomially bounded Maclaurin coefficients called?

Let $R$ be the the set of complex-valued (analytic) functions $f$ on the open unit disc $\mathrm D:=\{z\in\Bbb C:|z|<1\}$ for which there exist constants $a_0$, $a_1$, ... in $\Bbb C$ and $n$ in $\...

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**4**answers

168 views

### PDE with Laplacian and squared of the gradient

Let $u$ be a real function in $\mathbb{R}^2$. Does anybody know that the following PDE
$$\Delta u+|\nabla u|^2=0$$
has any non-constant general solution or not? It would be appreciated if any one ...

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vote

**1**answer

177 views

### Roots of modular functions

Let $\mathfrak f(\tau)=e^{-\pi i/24}\frac{\eta\left(\frac{\tau+1}{2}\right)}{\eta(\tau)}=q^{-1/48}\prod_{n=1}^{\infty}\left(1+q^{n+1/2}\right)$ be the Weber modular function. The function $\mathfrak f$...

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**7**answers

3k views

### Pathology in Complex Analysis

Complex analysis is the good twin and real analysis the evil one:
beautiful formulas and elegant theorems seem to blossom spontaneously
in the complex domain, while toil and pathology rule the ...

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vote

**1**answer

257 views

### Residues of Zeta-like Function

I'm looking for the residues of the following function $$s \mapsto\sum^\infty_{m,n =1} (m+n) \left[ amn + (m-n)^2 \right]^{-s}$$
at $s=\frac{1}{2}$ and $s=\frac{3}{2}$, where $a$ is some real positive ...

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33 views

### Smooth curves and arc length parametrization

Assume that $z(t)=x(t)+i y(t), t\in[-1,1]$ is smooth injective curve except in $0$ so that $\frac{\dot z(t)}{|\dot z(t)|}= e^{i\varphi(s(t))}$. Here $s(t)$ is the arc-length parameter. My question is ...

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64 views

### Integral Representation of Analytic Functions

Lang's Complex Analysis 3ed, Lemma XV 1.1 (pg. 392):
Let $I$ be an interval of real numbers, and $U$ an open set of complex numbers. Let $f(t,z)$ be continuous on $I \times U$. Suppose further that
...

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**1**answer

353 views

### Estimating the derivative of a polynomial on the unit circle

Let $P(z)=\sum_{k=0}^na_kz^k $ be a polynomial of degree $n$ and $z_k (1\leq k\leq n)$'s be $n$th roots of $-1$. Then when $\theta=0$ the inequality
$$|P'(e^{i\theta})|\leq \frac{4}{n}\left|\sum_{k=...

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126 views

### Extremal polynomial majorants of $\log{|f|}$: a multivariate extension of a theorem of Carneiro and Vaaler

Carneiro and Vaaler have proved, as an application of their work on Beurling-Selberg extremal majorants, that for any non-zero complex polynomial $f(z) \in \mathbb{C}[z]$, the infimum value of the ...

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**2**answers

164 views

### Unramified map of Riemann surfaces

Let $f:S \to T$ be a surjective, unramified, holomorphic map between connected Riemann surfaces. If $S$ is not compact is it always true that $f$ is a covering?
This is of course true if $S$ is ...

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**1**answer

162 views

### Analytic Continuation of Zeta-like function

Reading a paper about eta invariants I came across a zeta-like function.
I'm looking for the analytic continuation of $$\sum_{k=1}^\infty k(k+a)^{-s}$$ at $s=0$, where $a$ is positive.
In the paper ...

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**1**answer

81 views

### Estimate for radius of convergence of solutions given by Cauchy-Kovalevskaya Theorem

I'm sure you can extract it from the proof, but does anyone know of a reference where the radius of convergence (in terms of radius of convergence of the initial data and PDE) of the solution given by ...

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**0**answers

100 views

### Does Hartogs's Theorem for complex-analytic functions hold for real-analytic functions? [duplicate]

Recall a very famous theorem due to Hartogs for complex analytic functions of several variables.
Hartogs's Theorem Let $f$ be a holomorphic function on a set $G \setminus K$, where $G$ is an open ...

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**1**answer

262 views

### Chow theorem in $\mathbb{C}^2$

I have the following question the answer to which I cannot find in the literature (but it must have been studied):
Suppose that $M\subset\mathbb{C}^2$ is a real surface which may locally be written ...

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81 views

### A circle separating the critical points

A circle $C$ is said to be a separating circle for a set $S=\{z_1,z_2,\cdots,z_n\}$ if either there are points from $S$ in the interior and the exterior of the circle $C$ or all points of $S$ are on $...

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**0**answers

164 views

### Sobolev space under Mellin transform

The Mellin transform is known to be an isomorphism see wikipedia
between $M:L^2(0, \infty) \rightarrow L^2(-\infty, \infty)$
where $$M(f):= \frac{1}{\sqrt{2\pi}}\int_0^{\infty} x^{-\frac{1}{2} + is} ...

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99 views

### A property of a polynomial of degree 2 [duplicate]

If $P(z)=(z-z_1)(z-z_2)$ where $z_1,z_2$ are any complex numbers with $|z_1|\geq 1, |z_2|\geq 1.$ Then let us try the proof for a naive inequality
$$\max_{|z|=1}|P'(z)|\leq \max_{|z|=1}|P(z)|.$$ I am ...

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144 views

### An inequality with rotation

Let $P(z)=\sum_{m=0}^na_mz^m$ be a polynomial of degree $n\geq 1$ having no zeros in $|z|<1,$ then for any complex number $\alpha$ with $|\alpha|=1,$ is it true on $|z|=1$ that
$$\left|\alpha zP'(...

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79 views

### Universal cover of ladderly puntured complex plane

The twice punctured complex plane $\mathbb{C}-\{0,1\}$ has as its universal cover the upper half plane via elliptic modular function.
I am looking for the constructions of the covering map from the ...

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**1**answer

94 views

### Zeros of Multivariate Complex Functions [need reference]

I am looking for a good accessible reference that would summarize properties of zeros of complex analytic functions.
For my purpose, it would be interesting to see a discussion on the following ...

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42 views

### Finding positive powers of a Laurent series

Consider the functions $G(z)$ defined as a Laurent series
$$
G(z)=\sum_{n=-\infty}^\infty g_n z^n, \; z > 0,
$$
and $G(1)=0$. Also consider the function $f(z)$ defined through $G(z)$ as
$$
f(z) =...

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33 views

### Holomorphic function has radial limit at z=0 [duplicate]

Let f(z) be a holomorphic function defined on C-{0} satisfying f(r*e^iθ)→0 as r→0 for each θ∈R.
Must z=0 be a removable singularity?

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37 views

### Is singular Cauchy operator bounded in Morrey spaces?

The singular Cauchy operator is defined by
$$S_\Gamma :f \to \int_\Gamma \frac{f(\xi)}{\xi-z} d\xi , z\in \Gamma.$$
Is this operator bounded in Morrey spaces and weighted Morrey spaces? i.e.
is there ...

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**0**answers

100 views

### On a real part of a series with complex numbers

Let $P(z)=\sum_{m=0}^na_mz^m $ be a polynomial of degree $n$ having all its zeros in $|z|\leq 1.$ Then what is the best value for 'L' in
$$\Re\left(\sum_{k=1}^nP(zw_k)\frac{w_k}{(w_k-1)^2}\right)\geq ...

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**4**answers

201 views

### Complex differential equations

I'm looking for a gentle an concise introduction to complex-variable differential equations. Eventually, I need to look at complex PDEs, but I assume one starts with complex ODEs.
Mostly, I'm just ...

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69 views

### Derivative of complex matrix pseudo inverse with respect to real and imaginary components

I have a complex non-square matrix $\mathbf{Y}\in\mathbb{C}^{n \times m}$ whose inverse I compute using the Moore-Penrose pseudo inverse, $\mathbf{Z}=\mathbf{Y^+}$.
I am interested in evaluating the ...

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51 views

### Non-compact analogue of Hartog's extension theorem?

Suppose a function $f(z,w)$ is analytic in the open polydisk $\Delta^2$ with $\Delta = \{z \in \mathbb C | |z| < 1 \}$. I am interested in the limit $f(z,w)$ as $w \to 1$. This limit may be ...

**3**

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**1**answer

289 views

### Induction principle on proving an inequality

If $P(z)$ having no zeros in $|z|<1,$ then
$$\frac{\max_{|z|=1}|P'(z)|}{\max_{|z|=1}|P(z)|}\leq \frac{n}{2}.$$
Can we prove this by induction on $n$? or is there any alternative way?
Attempt at ...

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**0**answers

149 views

### The generalization of Hartogs' Theorem

I know a version of Hartogs' Theorem in the book An introduction to complex analysis(p 30) by Hormander, namely
Hartogs' Theorem when K compact with complement being simply connected
I also have ...

**3**

votes

**1**answer

93 views

### compare N(f,a,r) with T(f,r)

I'm reading William Cherry and Zhuan Ye's book 'Nevanlinna's theory of value distribution, the second main theorem and its error terms'. In Section 1.12, they explains why $N$ and $T$ is used in ...