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

2
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0answers
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
votes
1answer
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
1answer
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 ...
6
votes
1answer
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 ...
8
votes
0answers
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 $\...
3
votes
0answers
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 ...
5
votes
1answer
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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0answers
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) ...
8
votes
0answers
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 } ...
6
votes
1answer
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 ...
5
votes
2answers
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 ...
13
votes
1answer
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....
4
votes
1answer
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 $...
0
votes
1answer
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 ...
3
votes
2answers
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 ...
12
votes
0answers
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
votes
0answers
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). ...
2
votes
0answers
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 $1 < p$, $\frac {d} {2} + \frac {2} {p-1} < 1$ ...
5
votes
3answers
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 ...
2
votes
2answers
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 ...
2
votes
0answers
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 $\...
1
vote
4answers
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 ...
1
vote
1answer
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 $\mathfrak f$...
29
votes
7answers
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 ...
1
vote
1answer
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 ...
0
votes
0answers
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 ...
0
votes
0answers
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 ...
1
vote
1answer
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=...
6
votes
0answers
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 ...
4
votes
2answers
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 ...
0
votes
1answer
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 ...
7
votes
1answer
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 ...
0
votes
0answers
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 ...
3
votes
1answer
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 ...
1
vote
0answers
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 $...
6
votes
0answers
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} ...
0
votes
0answers
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 ...
3
votes
0answers
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'(...
2
votes
0answers
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 ...
1
vote
1answer
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 ...
0
votes
0answers
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) =...
0
votes
0answers
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?
3
votes
0answers
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 ...
2
votes
0answers
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 ...
2
votes
4answers
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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vote
0answers
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 ...
1
vote
0answers
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
votes
1answer
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 ...
2
votes
0answers
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
1answer
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 ...