# Questions tagged [cv.complex-variables]

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

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### Modulus of image of a curve family in a rectangle

I don't expect to get a positive answer to this question but I may as well try.
Let $R$ be the rectangle in $\mathbb{C}$ given by $\{z=x+iy: 0\leq x \leq l, 0 \leq y \leq h\}$ for some $l,h>0$. ...

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**1**answer

58 views

### meromorphic extension of dirichlet series

Suppose $\{a(n)\}_{n\ge 1}$ is a bounded complex sequence. Let $\phi(s)=\sum_{n\ge 1} \frac{a(n)}{n^s}$. Obviously, the Dirichlet series $\phi(s)$ is absolutely convergent for $\mathcal{R}(s)>1$. I ...

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**1**answer

112 views

### Must $q$ be analytic?

I have a continuous function $q:\mathbb{R}^+ \to \mathbb{R}^+$. An interesting property of this function is that
$$F(s) = \frac{e^{-q(s)}q(s+1)}{1-e^{s-q(s)}}$$
which also takes $\mathbb{R}^+ \to \...

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47 views

### Differences of $\omega$-plurisubharmonic functions

Let $X$ be a complex manifold, and $\omega$ a Kähler form on $X$.
A smooth function $\phi$ on $X$ is $\omega$-plurisubharmonic ($\omega$-psh for short) if the form $\omega+\sqrt{-1}\partial\bar{\...

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**1**answer

156 views

### Zeros of derivatives of Dirichlet Eta function

Let
$$
\eta^{(d)}(z) =
\sum_{n=1}^\infty
\dfrac
{(-1)^d(-1)^{n-1}\ln(n)^d}
{n^z}
$$
be the derivative of Dirichlet Eta function of order $d$.
Does it exist any known or not known zero of $\eta^{(d)}...

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150 views

### Regular functions vs holomorphic functions

Let $X$ be an affine smooth variety over the complex numbers, $X^{an}$ its associated smooth complex analytic space, and $\mathcal{O}$, resp. $\mathcal{O}^{an}$ the respective structure sheaves.
Is ...

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**1**answer

64 views

### Existence of Laurent series with zeroes at $𝑒^2𝑛$ (𝑛∈ℕ0 ) and even faster coefficient decay

This is an extension of an earlier question of mine which corresponds to the case $A = 1$. Precisely, my question is as follows:
Given $A > 0$ fixed but arbitrary, is there a non-trivial ...

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**0**answers

248 views

### Riemann–Hilbert-type problem

Let $P$ be a fixed pentagon in the hyperbolic plane $\mathbb H^2$ with all the angles equal to $\pi / 3$. Let $w_1, w_2, \dots, w_5$ be the sides
of $P$ going in the counterclockwise order. We are ...

**0**

votes

**1**answer

104 views

### What are the holomorphic automorphism groups of unit ball, polydisc, and Hartogs domain in C^n ( n>1)? [closed]

What are the holomorphic automorphism groups of unit ball, polydisc, and Hartogs domain in $\mathbb{C}^n$ ($n>1$) ? I would be pleased if you tell me.

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43 views

### Continuity of Complex function [closed]

enter image description here
For the second part, it is just a formula of Cauchy Integral formula, so F(x) = f(x) for all |z| <= 1, but the only thing I know is that it is continous on the circle. ...

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2k views

### Meaning of Cauchy integral theorem - the (co)homology viewpoint

I'm not sure what follows is not just a complicated way to deduce a blatant triviality, or if is even correct. Let's try.
In the elementary theory of analytic functions of $1$ complex variable, one ...

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**0**answers

68 views

### Is there any possibilities that the following partial sum of the Dirichlet eta function can be zero?

If $s_o$ is one of the non-trivial zeros of the Riemann zeta function with $0 <Re(s_o)<1$ , we know:
$$\eta(s_o ) = \left(1-2^{1-s_o}\right) \zeta(s_o)= \sum_{n=1}^\infty \frac {(-1)^{n-1}} ...

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**1**answer

141 views

### Existence of Laurent series with zeroes at $e^{2n}$ ($n \in \Bbb{N}_0$) and extremely fast coefficient decay

I am working on a problem in harmonic analysis, which I converted into the following existence problem concerning Laurent series. I am a bit at a loss concerning this problem, since my knowledge of ...

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**2**answers

1k views

### A sum involving roots of unity

Let $n$ be a positive integer and $\zeta$ be a primitive $n$th root of unity. It is not hard to show that
\begin{align*}
\sum_{k=1}^{n-1}\frac{\zeta^k}{1-\zeta^k}=\frac{1-n}{2}.
\end{align*}
Since $\...

**3**

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**1**answer

93 views

### Decay of the binomial expansion of $f^{\circ k}$

Suppose $f$ is a holomorphic function in a neighborhood of zero fixing zero. Suppose $f'(0) = \lambda$ and $0<\lambda < 1$. It's not so hard to prove that $f^{\circ k}(z) = f(f(\ldots\text{$k$ ...

**3**

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32 views

### Extremal metric for image of a curve family

Let $U\subset \mathbb{C}$ be a domain and $\Gamma$ some family of curves in $U$ with $\textrm{mod}(\Gamma)<\infty$ and such that $\rho$ is an extremal metric for the modulus. Suppose we are given a ...

**3**

votes

**1**answer

433 views

### Help with a difficult integral [closed]

Referring to a previous question, I am trying to do the following integral :
$$\phi(s)=i\int_{0}^{\infty}\frac{\log \left[1+\frac{\left(s\log\sqrt{1+ix} \right )^{2}}{\pi ^{2}} \right ]-\log \left[1+\...

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77 views

### Residue theorem with winding numbers [migrated]

In my studies, I've been uniquely using the Residue Theorem with no more than a single winding around each singularity. Actually, my professor has never mentioned winding-numbers in the Residue ...

**0**

votes

**0**answers

83 views

### A certain ratio condition for polynomials with real coefficients

Let $p:\mathbb{C} \longrightarrow \mathbb{C}$ be a polynomial with real coefficients and suppose that $p$ satisfies
\begin{equation}
\frac{p(y)}{y} \le \frac{p(x)}{x} \tag{*} \label{ratcond}
\end{...

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82 views

### Fourier inversion formula for compactly supported distributions

I know that the Fourier transform of a compactly support distribution $u\in \mathscr{E}'(\mathbb{R}^{n})$ is smooth and also satisfies
$$
|\hat{u}(\xi)|\leqslant C_{N}(1+|\xi|)^N,\label{1}\tag{1}
$$
...

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**3**answers

192 views

### Origin of term Ahlfors-David regular

Much of the literature on analysis in metric spaces makes use of an assumption called Ahlfors regularity or Ahlfors-David regularity. Let $q>0$. A metric space $(X,d)$ is Ahlfors(-David) $q$-...

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**1**answer

77 views

### Generalized Lambert W Function

I am looking for inverse functions for the following family of functions:
$
\begin{aligned}
f_0(z) &= z+e^z \\
f_1(z) &= ze^z \\
f_2(z) &= z^z \\
&\cdots \\
f_{n+1}(z) &=...

**5**

votes

**1**answer

106 views

### Failure of Borel-Schmid quasi-unipotency theorem in non-algebraic case

Let $\mathcal{X} \to \Delta^{\times}$ be a smooth family of compact Kähler manifolds over a punctured disc.
When this family is algebraic, the celebrated quasi-unipotency theorem (I think, due to ...

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**0**answers

61 views

### How to compute expansion factors for hyperbolic rational maps?

It is a commonly-referenced result about certain rational maps acting on $\mathbb{\hat{C}}$ that they are expanding on a neighborhood of their Julia sets. A sufficient condition to be expanding is ...

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**2**answers

150 views

### Dominant root of a family of polynomials

Let $f(x)=x^5-x^4-x^3-x^2-x-c$, where $c>2$ is a real number. It is easy to prove that there exists a positive real root $\alpha>2$ of $f(x)$ and all the other roots are non real.
Also, by ...

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**1**answer

169 views

### On the roots of Bernoulli polynomials

Consider the Bernoulli polynomials denoted by $B_n(z)$. Now, start plotting the set of all (combined) complex roots $\mathcal{A}_N$ of $B_n(z)$, say for $n=1,2,\dots,N$ for some enough large $N$. It ...

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**0**answers

73 views

### Proper projections

Let $D \subset \mathbb{C}^k$ be your favorite complex domain.
Suppose we are given a proper holomorphic mapping $f \colon D \to \mathbb{C}^{k+2}$.
Let us take $k+1$ generic linear functions $l_i \...

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**1**answer

151 views

### Unconditionally convergent series in some functional spaces

Linked with [this question and discussion](
Bilinear product of two summable families), I am very
interested in counterexamples/results about the following questions (cf the end).
First, I recall ...

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**0**answers

153 views

### When is entire function bounded on a ray?

Let $f(z)=\sum_{n=1}^\infty c_n z^n$ be entire function on the complex plane. May we express the property $\int_0^\infty |f(x)|^2 /x dx<\infty$ or some other property controlling the behavior for ...

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**1**answer

93 views

### Modulus bounded by Nevanlinna characteristic in several variables

Let $f:\mathbb{C}^n\to\mathbb{C}$ be an entire holomorphic function of $n$ complex variables. Then its Nevannlinna characteristic equals
$$
m_f(r)=\int_{\partial B(r)}\log^+|f(z)|d\eta(z),\quad\forall ...

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**1**answer

91 views

### Compatible solution of PDE

Let $c=c(z, \bar z)$ be a complex function satisfying $\partial_{z} \bar c=\partial_{\bar z} c$. It follows that there exists a real function $f$ such that $\partial_{\bar z} f=-c$. Would it be ...

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**0**answers

52 views

### Notion of contractive map for certain set-valued functions

A map $f:\mathbb{C} \to \mathbb{C}$ is contractive (and Lipschitz) if
for every $z_1,z_2$, we have $|f(z_1)-f(z_2)|\le K|z_1-z_2|$ for some positive $K<1$. This implies the existence of a unique, ...

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**1**answer

653 views

### Zeros of an infinite series

Let $\sum_{j=1}^{\infty}a_{j}$ be a convergent series of positive numbers and $\{z_{j}\}_{j=1}^\infty$ a closed discrete subset of the open unit disc $\mathbb{D}$. Then $h(z):=\sum_{j=1}^{\infty}\frac{...

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**2**answers

568 views

### Bounds on the derivative of a Riemann map

Let U be a simply connected domain with smooth boundary in the complex plane, and let $\mathbb D$ be the unit disc. Is there a nice sufficient condition for the existence of a biholomorphic map $f:\...

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**0**answers

129 views

### Chern number of projection-Topological magic in physics

I enclosed a computation from a well-known paper in the field of mathematical physics where the Chern number of the first Landau level is computed (the result claimed is $-1$) and the full paper can ...

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**0**answers

41 views

### Entire analytic functions with entire analytic Fourier transform, and corresponding distributions

I'm interested in the Fourier transform on a space of distributions that includes more than the usual tempered distributions, and in particular allows for $\delta$-distributions supported at complex ...

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**2**answers

141 views

### Density of Lacunary Functions

I'm curious whether lacunary functions (functions in the complex plane that are holomorphic in some open ball about the origin, but cannot be analytically extended past that ball) are typical or the ...

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**1**answer

64 views

### Bringing a Heun equation into canonical form

It is a well known fact that any second order Fuchsian differential equation on the complex plane $$u''(x) + p(x)u'(x) + q(x)u(x)=0$$ with exactly $4$ regular singular points may be suitably ...

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**1**answer

126 views

### Poles of equivariant meromorphic functions on Riemann surfaces

Let $p:\Sigma\to \mathbb{P}^1$ be the cyclic cover of $\mathbb{P}^1$ with Galois group $\Gamma$. Let $\Gamma\cdot p$ be a free $\Gamma$-orbit on $\Sigma$. Given any character $\chi$ of $\Gamma$, does ...

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**1**answer

140 views

### positive real matrix-valued function as linear combination of positive-real functions

In my question I am considering $z$ as the complex variable in the $z$-transform $X(z)$ of a discrete-time sequence $x[n]$:
I have $M$ square complex matrices $\mathbf{R}_m$ of size $N\times N$.
I ...

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**2**answers

257 views

### On finite extensions of the field of meromorphic functions

Let $\mathcal{M}$ be the field of meromorphic functions of one (complex) variable and $w = w(z)$ an analytic function satisfying a polynomial equation
$P(w; z) := w^n + a_{n-1}(z) w^{n-1} + \cdots + ...

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**1**answer

506 views

### Dual of the space of all bounded holomorphic functions

Let $\mathbb{B}$ be the open unit ball in $\mathbb{C}^n, n\geq 1$ and let $H^\infty (\mathbb{B})$ be the space of all bounded holomorphic functions on $\mathbb{B}$. It is well known that $H^\infty (\...

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**1**answer

131 views

### Compilation of representations of holomorphic functions

Holomorphic functions are my muse. As my muse, I love drawing them different ways. Allow me to frame this as though an artist talking about his muse.
A holomorphic function $f$ on the unit disk $\...

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464 views

### Lacunar series with an interesting (in-formula) symmetry.

So, I wrote out a table of functions like so:
$\sum_{n=1}^{\infty} (-1)^{n+1}q^{n}=$ $+q^{1}$ $-q^{2}$ $+q^{3}$ $-q^{4}$ $+q^{5}$ + $\ldots$
$\sum_{n=1}^{\infty} (-1)^{n}q^{n^{2}}=$ $-q^{1}$ $+q^{4}...

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votes

**1**answer

103 views

### Extended Abel-Jacobi theorem

Let $X$ be a compact Riemann surface, $u$ a meromorphic function on $X$ with divisor supported on a set of points $\{P_1, ..., P_n\}$, and $f$ a meromorphic function on $X$ such that $f$ has no pole ...

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**1**answer

104 views

### Variation of steepest descent/Laplace methods for non-exponential integrands

I was wondering if versions of the Laplace/steepest descent methods exists for integrals of the type
$$\int_C f(z) M(\lambda g(z)) dz$$
for $\lambda >>0$ functions $f(z), g(z): \mathbb C \...

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**1**answer

287 views

### Topological properties of complex valued Riemann sum limit curve and a particular integral inequality

I am studying under what conditions the following integral inequality would hold ($a$ real, $a>0$):
$$ \int_{-\infty} ^{\infty} \frac{f(ix)}{a\pm ix}dx\ = 0 \ \ \ \ \Rightarrow \ \ \ \int_{-\...

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**1**answer

553 views

### An integral involving the argument of the Gamma function and the Riemann Hypothesis

Evaluate $$I=\int_{0}^{\infty} \frac{t\arg
\Gamma(\frac{1}{4}+\frac{it}{2})}{(\frac{1}{4}+t^2)^2}\mathrm{d}t$$
where $\Gamma(s)=\int_{0}^{\infty}e^{-x}x^{s-1}\mathrm{d}x.$
Note that $I$ converges ...

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**3**answers

650 views

### Analog of Newlander–Nirenberg theorem for real analytic manifolds

It is well known that one can specify a complex structure on a real $C^\infty$ manifold in two equivalent ways: an atlas with holomorphic transition functions between charts and an integrable almost ...

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**1**answer

226 views

### Bounded holomorphic functions on a Riemann surface separating points

Let $R$ be a Riemann surface that admits a non-constant bounded holomorphic function. Then is it true that any two points of $R$ can be separated by a bounded holomorphic function? This is easy to see ...