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Questions tagged [cv.complex-variables]

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

9
votes
2answers
398 views

Why only $\bar\partial$ but not $\partial$ in Dolbeault cohomology

While I learn about $\partial$ and $\bar{\partial}$ operators, I had some questions about the reason why people prefer $\bar\partial$ over $\partial$. Specifically, When defining Dolbeault ...
1
vote
1answer
96 views

How to prove that weighted Bergman space is separable.

Let $D$ be a bounded domain in $\mathbb{C}^n$ and $\varphi$ be a non-positive plurisubharmonic function on $D$. The weighted Bergman space $A^2(D,e^{-\varphi})$ is the space of holomorphic functions ...
0
votes
2answers
122 views

An upper bound for minimum

For any polynomials of degree $n$ having all its zeros in $|z|\leq K,K\geq 1,$ is it true $\max_{|z|=1}|nP(z)+(a-z)P'(z)|\geq n\min_{|z|=K}|P(z)| $ where $a$ is any complex number with $|a|\geq K?$
0
votes
1answer
198 views

Bounding the derivative of a holomorphic function on a disk by its absolute value

Let $f(z)$ be a holomorphic function defined on the disk $|z|\le 2$. Suppose $|f(z)|<1$ for $|z|\le 2$. It looks like there is a constant $c>0$ such that $|f(z)'|<c$ on the disk $|z|\le 1$ (...
1
vote
1answer
102 views

Plancharel-Pólya inequality for functions of exponential type

If $f(z)$ is an entire function of exponential type $\tau$ and $p$ a positive number such that that $$\int_{-\infty}^{+\infty}|f(x)|^pdx<\infty$$ then it can be proven that $$\int_{-\infty}^{+\...
1
vote
1answer
143 views

Is there always a polynomial with real zeroes between two polynomials with real zeroes?

Suppose that we have two complex polynomials $p(z)=\sum_{k=0}^n p_kz^k$ and $q(z)=\sum_{k=0}^n q_kz^k$ and also that we have $|p_k|<|q_k|$ for $k=0,1,...,n$. We say that a polynomial $r$ is ...
1
vote
0answers
39 views

Which complex maps with branch cuts have a representation by Dirichlet series?

Which complex maps with branch cuts have a representation by Dirichlet series? I am aware of the work of A.F. Leont'ev on general Dirichlet series, and the theorems of representation of analytic ...
10
votes
4answers
1k views

Why the unreasonable applicability of complex numbers in physics/engineering? [duplicate]

After years of using complex numbers in every kind of analysis of physical and electrical engineering problems I am starting to wonder: why is this particular algebra so effective in modelling the ...
38
votes
2answers
2k views

Abel and Galois (and Arnold)

Question Is there a connection between Abel and Galois theories of polynomial equations? Recall that for every polynomial $p(x)\in \mathbb{Q}[x]$ (say, without the free coefficient), Abel considered ...
5
votes
2answers
275 views

The largest disk contained by a 'product' of two simply connected plane regions with unit conformal radii

Consider a pair of holomorphic functions $f,g \in \mathcal{O}(\Delta)$ on the complex unit disk $\Delta = \{|z| < 1\}$ that both satisfy $f(0) = g(0) = 0$ and $f'(0) = g'(0) = 1$. Does the domain $$...
1
vote
1answer
74 views

Upper bound of the dimension of automorphism group of compact Kähler manifolds

It is well-known that the dimension of the isometry group of an $n$-dimensional compact Riemannian manifold is no larger than $\frac{1}{2}n(n+1)$, which is attained precisely by $S^n$ and $\mathbb{R}P^...
0
votes
1answer
136 views

The cohomology of meromorphic functions

Let $A$ be a sheaf such that $$A(U) = \{ f \in \mathbb M(U): f \in \mathbb{O}(U \backslash\{p_1,\ldots, p_n\}) \ \mbox{with at worst a simple pole at}\ p_i \} $$ where $\mathbb M(U)$ means the set of ...
7
votes
1answer
222 views

Integral inequality involving an analytic function

I have been trying to prove for any $\delta>0,$ $$ \int_0^{2\pi}\left|1+ e^{i\theta}f(e^{i\theta})\right|^{\delta}d\theta\leq \int_0^{2\pi}\left|1+e^{i\theta}\right|^{\delta}d\theta $$ for any ...
5
votes
0answers
714 views

Fractal covering of a plane with complex-base numeral systems - is periodicity necessary?

Taking a base $z$ positional numeral system with digits $a_k\in \{0,\ldots,n-1\}$: $$s:\left\{(a_k)\in\{0,\ldots,n-1\}^{\mathbb{Z}}: \exists_K \forall_{k>K} \ a_k=0\right \}\to \sum_{k\in\mathbb{...
1
vote
1answer
76 views

Does the boundary of immediate basin contain a fixed point?

Let $f$ be a rational map of degree $d\geq 2$, and $B$ is a simply connected immediate basin of an supper-attracting fixed point of $f$. I want to know whether there exists a fixed point of $f$ ...
5
votes
3answers
463 views

Closed, sum-free form for the $n$-th derivative of $\operatorname{arcsinh}(\frac1x)$ in $x=1$

During research involving the Born–Jordan quantization I came across the expression $$ \frac{d^k}{dx^k}\operatorname{arcsinh}\Big(\frac1x\Big)\Big|_{x=1}\tag1 $$ for $k\in\mathbb N_0$. It is not too ...
7
votes
1answer
299 views

Reference for flatness in complex-analytic geometry

What is a good reference for flat morphisms of complex-analytic spaces? (The book by Grauert and Remmert doesn't treat them). Topics I'm interested in: openness of flat maps, descent for coherent ...
2
votes
0answers
117 views

example of torsion of higher direct image sheaf

I'm reading kollar's paper about higher direct image of dualizing sheaf. Suppose f: X-Y is morphism, X smooth,Y normal. He mentioned usually the higher direct image of structure sheaf is "bad," and ...
1
vote
0answers
97 views

Error term in França-LeClair approximation of zeta zeros

The imaginary part of the $n$th critical zero of the Riemann zeta function with positive imaginary part is asymptotically $$ t_n \sim 2\pi\frac{n}{\log n} $$ and has been approximated [1] as $$ t_n \...
3
votes
0answers
86 views

Geometric or topological flavored proof of Nevanlinna five valued theorem?

In a very early state of the development of the Nevanlinna theory, Nevanlinna proved what is now called Nevanlinna five valued theorem, Let $f$ and $g$ be two transcendental meromorphic function. ...
5
votes
2answers
257 views

An equality relation for complex numbers off the nonnegative real axis [closed]

For every complex number $z$ off the nonnegative real axis there exist positive numbers $p_0,... ,p_n$ such that $\sum_{i=0}^n p_iz^i = 0$. Finding difficulty in proceeding with the problem. Need ...
1
vote
0answers
45 views

Supremum norm of certain quantity II

Can anyone solve the maximization problem...$\max_{|z_i|=1}\Big|\sum_{i,j=1}^nz_iz_j+\sum_{i,j=1}^n|z_i-z_j|\Big|$?
2
votes
0answers
58 views

Non singularity of a generalised Vandermonde matrix through Hadamard product

I'm currently trying to prove the following. Consider $k_1,\dots,k_N$ complex numbers not lying on the real and imaginary axes. Then consider \begin{equation} W_N(x)= \text{Wronskian}\big(\cosh(k_1x),...
0
votes
0answers
77 views

Meromorphic function with prescribed growth and poles

I was wondering if you can construct a meromorphic function $f$ with no poles with small imaginary part, let's say no pole in $\{z\in\mathbb{C},\ |Im(z)|<a \}$ with some $a>0$, while having an ...
0
votes
1answer
186 views

extension of bounded holomorphic function on the disk

Let $f$ be a bounded holomorphic function defined on the open unit disk in the complex plane. Is it true that $f$ could always be extended to a Hölder continuous function on the closed disk?
4
votes
1answer
100 views

Neumann DBAR problem with tempered distributions

It is well-known that the operator $$\frac{\partial}{\partial \overline{z}} : C^{\infty}(\mathbb{C}) \to C^{\infty}(\mathbb{C})$$ is surjective. (And it also works if we replace functions by Schwartz ...
6
votes
1answer
216 views

Bounding Taylor coefficients of $f(z)$ with $f(0)=1$, $f(z)\ne 0$ for $|z|\le 1$

Let $f(z)=1+a_1z+\ldots+a_nz^n+\ldots$ be a complex analytic function defined on the unit disk $|z|\le 1$. Suppose $f(z)\ne 0$ for $|z|< 1$. I would like to know what kind statements one can make ...
2
votes
0answers
172 views

Understanding the branch cut and discontinuity of the hypergeometric function

DISCLAIMER: This question comes from math.stackexchange (where it has an active bounty). The link is here. UPDATE: the question has been answered on math.stackexchange at the previous link, and the ...
7
votes
1answer
218 views

Model theory of the restricted complex analytic functions

Let $\mathbb{C}_{an}$ be the expansion of the structure $(\mathbb{C}; +,-,×,0,1)$ by adding the restricted complex analytic functions. This is the complex analog of the familiar $\mathbb{R}_{an}$ in O-...
1
vote
1answer
105 views

Boundedness of a finite subharmonic function

Let $$u\colon B^n(0,1)\to \mathbb{R}$$ be a subharmonic function in the open unit ball in $\mathbb{R}^n$. The crucial assumption is that $u$ never equals $-\infty$. Is it true that $|u|$ is ...
8
votes
0answers
197 views

Analytic space not embeddable in any complex manifold

I am looking for an example of a compact complex analytic space, reduced and irreducible, which does not admit any holomorphic embedding into any (smooth) complex manifold (possibly non-compact). I ...
10
votes
1answer
273 views

Is every endomorphism of the sheaf of holomorphic functions on a disk a differential operator?

Let $D= \{z\in \mathbb{C}:|z| < 1\}$ be the unit disk. And consider the sheaf of holomorphic functions $\mathcal{O}_{D}$. Question (?) : Is there a sheaf endomorphisms $\phi : \mathcal{O}_D \to \...
4
votes
1answer
152 views

Does exist a Kahler-Einstein metric on the blow-up of $\mathbb{P}^3$ along a smooth plane cubic?

This might be well known for the experts but I am not able to find a reference. I was wondering if there exists a Kahler-Einstein metric on the Fano threefold given by blow-up of $\mathbb{P}^3$ along ...
1
vote
1answer
109 views

Sampling set: relatively dense and uniformly discrete

The Paley-Wiener space of a domain $\Omega\subset\mathbb{R}^d$ is the set $$PW_\Omega:=\{f\in L^2(\mathbb{R}^d):\text{supp}\widehat{f}\subset\Omega\}.$$ We say that a discrete set $\Lambda\subset\...
5
votes
1answer
148 views

Supremum norm of certain quantity

Is there any easy way of finding supremum of the quantity $$\sum_{i,j=1}^n|z_i-z_j|,$$ where $|z_i|=1$ for $1\leq i\leq n$ ? We are considering complex variables of course.
0
votes
0answers
45 views

Questions on the behaviour of functions of exponential type 1

I am interested in understanding the properties of entire functions of exponential type 1. I have few questions about their growth. How many sectors can a function of exponential type have, in which ...
2
votes
2answers
242 views

Can the “Bisector” be represented by a holomorphic function?

Note: In this question, a complex number is counted as a vector initiated from the origin. ______________________________________________________________- Is there a holomorphic function $B:\...
0
votes
0answers
54 views

Description of the set of analytic functions satisfying certain (uniform) inequality in a disc

Given some positive constant $R>0$ how can we characterize the set of all functions $f(z)=\sum_{k=2}^\infty \frac{a_k}{k!}z^k$ analytic in $|z|<R$ for which there exists $C>0$ (same for all $...
3
votes
0answers
154 views

Prove a certain function maps to upper half plane

Suppose $M$ is a bounded self-adjoint operator on space of complex valued functions on the real line $S_1=L^2(\mathbb{R},a(x)dx)$, where $a(x)$ is a nice real positive analytical function ( I have in ...
1
vote
1answer
107 views

Seeking the derivation of the Fourier Sine Transform of $x^{2\nu}(x^2+a^2)^{-\mu-1}$

In this answer on math.stackexchange.com the Fourier Sine Transform of $x^{2\nu}(x^2+a^2)^{-\mu-1}$ is given in terms of the generalized hypergeometric function: $$\frac{1}{2}a^{2\nu-2\mu}\frac{\Gamma(...
1
vote
1answer
201 views

Plurisubharmonic function having log pole along divisor

Let $R$ be a compact Riemann surface. For a given point $p\in R$ identified to the origin $z=0$ in a coordinate chart, then the function $z$ defines a local holomorphic section vanishing along the ...
7
votes
1answer
596 views

A natural residue formula

A residue formula I have strong evindence to believe that the following identity holds: $$ \frac{n!}{2\pi i}\oint_{|z-1|=\epsilon} \frac{z^{a-1} \mathrm{d}z}{(z^d-1)^{n+1}} = d^{-n-1}\prod_{j=1}^{n}\...
2
votes
2answers
137 views

Boundary behavior of power series vs. boundedness of partial sums

Let $f\left(z\right)=\sum_{n=0}^{\infty}a_{n}z^{n}$ be a power series with $0$s and $1$s as its coefficients ($a_{n}\in\left\{0,1\right\}$ for all $n$) with a radius of convergence of $1$. I call such ...
1
vote
1answer
112 views

Locus of roots of all convex combinations of two monic polynomials, II

This post contains a revised conjecture to a conjecture I posed previously which was shown to be false. Let $p, q \in \mathbb{C}[t]$ be two monic polynomials of degree $n \ge 1$. For $\alpha \in [0,1]...
3
votes
0answers
34 views

What are good ways to 'relax' a uniform approximation into independent saddle-point expressions once the uniform approach is no longer needed?

I am doing long-running project that involves asymptotic saddle-point estimation of integrals (for flavour, it's this sort of stuff) and I would like to ask if there are established ways in the ...
11
votes
1answer
488 views

How do analysts think about functions with poles at all roots of unity?

In branches of algebra impinging on the enumeration of partitions, one often encounters formulas like $$\prod_i \left( \frac{1}{1-q^i} \right)^{n_i}$$ for some integers $n_i$. E.g., with $n_i = 1$, ...
2
votes
1answer
173 views

Defining “addition” on the Riemann surface of log(z)

The title of this question is a bit awkward, as adding two points together on a manifold is usually not considered possible, but in this case there appears to be a nice little hack. Consider the ...
6
votes
3answers
709 views

Roots of a polynomial inside the unit circle

Let $k$ be a even positive integer. Now, consider the polynomial $$ p(x)=x^k-px^{k-1}-qx^{k-2}-x^{k-3}-\cdots -x-1, $$ with $p$ and $q$ integers satisfying $q-1>p\geq 1$. How to prove that this ...
18
votes
2answers
665 views

Zeros of MacLaurin polynomials for the exponential function

Asked but never answered at MSE. Let $\exp_n(z)$ denote the nth degree Taylor polynomial of $e^z$ : $\exp_n(z) = 1 + z + z^2/2! + ... + z^n/n! \;$ . The zeros of $\exp_n(z)$ were studied by ...
0
votes
0answers
57 views

Countable dense subset of functions of exponential type 1 that decay along the positive real axis

I am interested in the space of all holomorphic function of exponential type one, that decay exponentially along the positive real axis. I tried to define it as follows. Let $$\|f\|_n = \sup_{z\in\...