# Questions tagged [cv.complex-variables]

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

**9**

votes

**2**answers

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

**1**answer

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

**2**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**4**answers

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

**2**answers

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

**2**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**3**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**2**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**2**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**3**answers

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

**2**answers

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

**0**answers

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\...