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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.

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0answers
21 views

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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1answer
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 ...
2
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1answer
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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0answers
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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1answer
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)}...
4
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0answers
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 ...
3
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1answer
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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0answers
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 ...
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1answer
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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0answers
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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0answers
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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0answers
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}} ...
9
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1answer
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 ...
27
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2answers
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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1answer
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$ ...
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0answers
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
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1answer
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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0answers
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
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0answers
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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0answers
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 $$ |\hat{u}(\xi)|\leqslant C_{N}(1+|\xi|)^N,\label{1}\tag{1} $$ ...
5
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3answers
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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1answer
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
1answer
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 ...
2
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0answers
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 ...
0
votes
2answers
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 ...
4
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1answer
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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0answers
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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1answer
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 ...
4
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0answers
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 ...
2
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1answer
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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1answer
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 ...
2
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0answers
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, ...
12
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1answer
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{...
6
votes
2answers
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:\...
3
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0answers
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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0answers
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 ...
4
votes
2answers
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 ...
0
votes
1answer
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 ...
0
votes
1answer
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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vote
1answer
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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2answers
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 + ...
6
votes
1answer
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 (\...
4
votes
1answer
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 $\...
6
votes
0answers
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}...
2
votes
1answer
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 ...
3
votes
1answer
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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1answer
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_{-\...
1
vote
1answer
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 ...
14
votes
3answers
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 ...
8
votes
1answer
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 ...