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

learn more… | top users | synonyms (2)

1
vote
0answers
83 views

Is there a general connection between value distribution and zero distribution for functions representable by Dirichlet series?

Some time ago I read part of a book in which the author made some conjectures outlining what kind of zero distribution is expected for functions representable by Dirichlet series with completely ...
0
votes
1answer
171 views

A condition on isolated singularity

Suppose $F: {\mathbb C}^N \to {\mathbb C}$ defines a singularity at the origin (for simplicity one can assume that $F$ is a quasi-homogeneous polynomial). Suppose it is nondegenerate, i.e., $dF(z) = ...
1
vote
0answers
43 views

Determine the position of the contour with the value of corresponding contour integral

Let $C$ be the contour of the unit square with lower left corner at origin. We define a function $g(z)=\int_{z+C} f(w)dw$ for a given (not necessarily holomorphic) function ...
4
votes
2answers
635 views

What does analyticity imply in complex analysis? [closed]

In complex analysis, we're constantly faced with problems about the analyticity of a function, on which many theorems are developed. I of course know a bunch of formulas and theorems, but could not ...
7
votes
0answers
75 views

What does this number tell me about a convex lattice polygon?

EDIT: I realized I'd tricked myself by working with a too special case of $f$, the question is now updated (boundary lattice points replaced vertices). Suppose I have a convex lattice polygon $P$, ...
6
votes
1answer
322 views

Applications of the Small and Great Theorems of Picard

I just presented the two famous theorems of Picard (Small and Great) in a graduate course, but I have not managed to discover a good number of interesting applications. List of applications (rather ...
0
votes
0answers
86 views

Non-trivial global solution for Dirichlet eigenvalue problem

Suppose $f:\mathbb{R}^2\to\mathbb{R}$ is smooth everywhere except a set of measure zero.(i.e. A set of area zero) and satisfies the equation $\Delta f=\lambda f$ for some constant $\lambda$ off this ...
6
votes
0answers
189 views

What function space does holomorphic functional calculus give us?

Let $A$ be a unital Banach algebra, $U$ be an open subset of $\mathbb{C}$, and $A_U:=\{x\in A:\sigma(x)\subset U\}$. Holomorphic functional calculus says that any holomorphic function ...
12
votes
1answer
215 views

$\pm1$-polynomials with a maximal non-real root

For given $n$, consider a polynomial $\sum_{k=0}^na_kz^k$ with all coefficients $a_k\in\{\pm1\}$. I am interested in the following: How big can the modulus of a non-real root of such a ...
1
vote
1answer
174 views

meromorphic extension of a function

Let $\Lambda\in \mathbf{C}$ be a discrete subset. We assume that $\mathrm{Re}(\lambda)<0$ for all the $\lambda\in \Lambda$. For $i\in \mathbf{N}$, $\lambda\in \Lambda$, let $m_{i,\lambda}\in ...
2
votes
1answer
161 views

Meromorphic functions with finitely prescribed zeros and poles on annuli

I hope that this (probably) naive question will not bother those experts. Anyway, please allow me to ask this question here: We set $$\mathbb{A}(1/2, 2) = \Big\{z \in \mathbb{C}: 1/2 < |z| < ...
2
votes
3answers
128 views

closed integral formula for a non-zero solution of a homogeneous linear ODE of order 2

Let $$ (\star) \;\;\;\;\;\;\;\;\;\;\;\; y''+p(x)y'+q(x)y=0, $$ be a homogeneous linear ODE of order $2$ with $p(x)$ and $q(x)$ complex valued analytic functions in a small neighbourhood of $0$. Q: ...
0
votes
2answers
173 views

“Convolution” for Multiplying Random Variables

The following situation arises frequently in probability. Suppose we have two independent continuous random variables $X$ and $Y$ and we consider their sum, $Z=X+Y$. Then the pdf of $Z$ is the ...
7
votes
5answers
481 views

Bound on sum of complex summands involving binomial coefficients

I am trying to find the asymptotic behavior of the sum: $$ \sum^n_{i=0} \begin{pmatrix} 2n \\ i \end{pmatrix} x^i y^{2n-i}$$ as $n\rightarrow\infty$. Here $x$, $y$ are complex numbers and I have ...
1
vote
0answers
143 views

Marten's proof of torelli theorem

I am trying to read the proof of torelli theorem by Henrik H.Martens "A new proof of torelli's theorem" Annals of mathematics vol78 no. 1 .The proof seems to me like using mysterious combination of 3 ...
1
vote
2answers
186 views

Two questions related to $TS^{2}$ as a holomorphic manifold

We consider $TS^{2}$ as a 2 dimensional holomorphic manifold and fix an explicit holomorphic structure on $TS^{2}$ as it is indicated in the answer of Mike Usher to the following question. ...
6
votes
1answer
270 views

Does there exist harmonic function with that property?

Can one construct a harmonic function $f$ defined in the unit disk with the condition $f(0)≥1$ such that area of $\{z∈D:f(z)>0\}$ is small enough, i. e. for every $\epsilon>0$ does there exist ...
9
votes
3answers
412 views

How many isolated roots can a polynomial in $z$ and $\overline{z}$ have?

This is probably well-known by I can't find it now online. My guess is that if the degree is $n$ then it's $2n$ but it's just a hunch. EDIT: This is an edited version. Before I asked about roots ...
0
votes
1answer
217 views

Is there any algorithm to decide whether a series with integral coefficiens is a algebraic function? [closed]

Given a series with integral coefficiens as following: $$F(x)=\sum_0^i a_i x^i,\text{where }a_i\in \mathbb{N}\bigcup 0 $$$$\text{and there is a computable function $\psi$ such that } \forall i ...
3
votes
2answers
462 views

Intuition behind $\zeta(2) = \frac{\pi^2}{6}$ [closed]

In literature, there are many proofs of the well-known result $$\zeta(2) = \frac{\pi^2}{6}.$$ However, as far as I know, they do not offer an intuitive explanation of why this result should be true. ...
3
votes
1answer
72 views

Conditions conformal mapping to be expansive

Let $\Omega' \subset \Omega$ be simply connected domains in the complex plane. The Riemann mapping theorem tells us that there a biholomorphism $f: \Omega' \to \Omega$. What conditions guarantee that ...
5
votes
1answer
101 views

Special Kähler normal coordinates around a point

Let $(M,\omega)$ be a compact Kähler manifold and suppose there are holomorpic vector fields vanishing at a point $p$. As a consequence we have a group $G_{p}$ of biholomorpisms fixing $p$. Let ...
5
votes
1answer
368 views

Why are the angular differences of these random complex polynomial coefficients almost constant?

This is based on მამუკა ჯიბლაძე's (not-)answer here. I guess it is better to make up a new thread for it. Let me repeat the setup here: We consider polynomials whose complex roots are randomly ...
6
votes
1answer
196 views

How to prove an elementary functional equation for polylogarithms?

Let $Li_s(z)$ denote the usual polylogarithm. The elementary functional equation $$Li_{-n}(z)=(-1)^{n-1}Li_{-n}(1/z)$$ holds for $n\geq 1$. I remember only that the proof used some reproducing ...
5
votes
0answers
178 views

Is there a connection between |roots| $\rightarrow$ 1 and Gromov's waist theorem?

Recent questions showed that roots of a random polynomial tend to lie on the unit circle ("Why do roots of polynomials tend to have absolute value close to 1?"; "Distribution of roots of complex ...
0
votes
1answer
84 views

Location of the zeros set of holomorphic function [closed]

Recently I proved the following result. "If a holomorphic function $f$ maps the unit disc $\Delta$ into the unit disk $\Delta $ with $0<|f(0)|$ then $f$ doesn't vanish in the disk $D(0,|f(0)|)$. " ...
7
votes
3answers
329 views

A learning roadmap to the Schramm-Loewner evolution (SLE) for the complex analyst

I would like some good references to learn about the Schramm-Loewner evolution (SLE), for a complex analyst with no background in probability. A quick google search gave a lot of references on SLE ...
4
votes
2answers
544 views

Power series with funny behavior at the boundary

Consider a power series $$ \sum_{n=0}^{\infty}a_nz^n $$ where $a_n$ and $z$ are complex numbers. There is radius $R$ of convergence. Let us assume that is a positive real number. It is well known that ...
232
votes
15answers
32k views

Why do roots of polynomials tend to have absolute value close to 1?

While playing around with Mathematica I noticed that most polynomials with real coefficients seem to have most complex zeroes very near the unit circle. For instance, if we plot all the roots of a ...
1
vote
0answers
121 views

Existence of zero-free strip of a Laplace transform (edited ..)

Problem Let $\beta$ be a probability measure on $\mathbb{R}$, and define $$ K = \left \{z \in \mathbb{C}: g\left(z\right)=\int_{-\infty}^{\infty}\exp\left(z x\right)\beta ( dx ) \text{ is ...
6
votes
1answer
267 views

Can we normalize a complex analytic space in a covering of an open subset?

Let $X$ be a normal connected complex analytic space, $x\in X$ a point, $f$ a nonzero holomorphic function vanishing at $x$. Denote by $U\subseteq X$ the locus where $f$ is nonzero. Suppose that ...
0
votes
2answers
115 views

Metric properties of a quadratic differential at an essential singularity

Let $f(z)dz^2$ be a holomorphic quadratic differential on the punctured disk $\{0<|z|<1\}$, which gives rise to a Riemannian metric $g=|f(z)|\,|dz|^2$ and hence a volume form $\nu=|f(z)| ...
1
vote
0answers
42 views

singularity of the solution to an integral equation

I consider a function $x\mapsto f(x)$ which is the positive solution to the integral equation $$\int_{f(x)}^\infty \frac{du}{u^\gamma-x}=1, \quad x\ge 0,$$ where $\gamma\in (1, 2]$ is some ...
7
votes
0answers
201 views

Cesaro summation of a particular Dirichlet series associated with $\zeta(s)$

If you've investigated the error in Perron's formula in general, you've probably noticed that Cesaro summation $$\lim_{x\rightarrow\infty}\sum_{n\leq x} \left(1-\frac{\log n}{\log ...
12
votes
2answers
598 views

Special values of $\zeta$ outside the real line and the critical strip

The values of Riemann's function at the integers have been extensively studied. I was wondering, is there anything interesting known (or conjectured) to happen arithmetically outside the real line ...
0
votes
1answer
127 views

Estimating the height required to find a given small value of $|\zeta(s)|$ near the line $\sigma=1$

There are some qualitative theorems of Bohr, Jessen and Titchmarsh (e.g. The Theory of the Riemann zeta function, E.C. Titchmarsh, pages 306-308) proving that there is a $K=K(a,\alpha,\beta)$ such ...
2
votes
0answers
125 views

Questions about transformation or integral transformation

I have asked several mathematicians about the following questions,but all of them think they are good questions,but can not give a complete answer.Now I have to come here to ask mathematicians all ...
1
vote
1answer
66 views

Minimum of two plurisubharmonic functions

I know that in general for $u,v\in PSH$ (plurisubharmonic) $\min\{u,v\}$ is not a $PSH$ function. Are there any known results under which conditions on $u$,$v$ a function $\min\{u,v\}$ is $PSH$? I ...
6
votes
0answers
182 views

Non-trivial bounds for polynomials at a fixed point

Let $f$ be a polynomial of degree $d$. Of course $|f(z)|\sim C|z|^d$ as $|z|\rightarrow\infty$ but also, since any polynomial is completely determined by its values at any $d+1$ points, we may ask how ...
0
votes
0answers
91 views

Holder continuous analytic function

Assume that $0<\alpha<1/2$ and $f$ is analytic in the unit disk $D$ with $|f(z)-f(w)|\le M|z-w|^\alpha$. Can we state that in general $$\int_D |f'(z)|^2 dx dy <\infty?$$
3
votes
0answers
140 views

Continuous extension of Riemann maps and the Caratheodory-Torhorst Theorem

If $G\subsetneq\mathbb{C}$ is a simply-connected plane domain, then by the Riemann mapping theorem there is a conformal isomorphism $\newcommand{\D}{\mathbb{D}}\varphi:\D\to G$, where $\D$ is the unit ...
3
votes
3answers
139 views

Integrability of second derivative of conformal mappings

a) How to construct a conformal mapping $f$ of the unit disk $D$ onto a Jordan domain with $C^1$ boundary such that $$\int_D|f''(z)|^2 dxdy =\infty.$$ (This is done in two different ways in the ...
9
votes
1answer
192 views

On the conformal removability of Jordan curves

We say that a compact subset $E$ of the Riemann sphere $\mathbb{C}_\infty$ is (conformally) removable if every homeomorphism of $\mathbb{C}_\infty$ conformal outside $E$ is actually conformal ...
8
votes
1answer
224 views

Different notions of convergence of complex subvarieties

Let $X$ be a smooth complex algebraic variety (or, better, complex analytic manifold). Let $\{C_i\}$ be a sequence of compact algebraic subvarieties (resp. analytic reduced subspaces) which converges ...
3
votes
1answer
225 views

Non-bijective conformal maps between annuli

I need to answer the following question, hopefully in the negative. Question: Does there exist a conformal map $f$ of degree $1$ from the annulus $\{1<|z|<R\}$ to the punctured disk ...
1
vote
0answers
165 views

Proving identity involving delta-functions

Lately I came across the following identity: ...
0
votes
0answers
95 views

Periodicities of a Complex Dynamical System

Consider A function $f:\mathbf{C}^2\rightarrow \mathbf{C}$ defined as $$f_{\alpha, \beta}(z,w)=\frac{\alpha}{z}+\frac{\beta}{w}$$ where $\alpha$ and $\beta$ both are complex number. It is easy to ...
1
vote
1answer
166 views

The locus of rational/elliptic curves on a special surface in $\mathbb{P}^3$

Let $P$ and $Q$ be two general polynomials of the same degree $d>5$. Consider the surface $S: z^2=P(x)Q(y)$ in $\mathbb{P}^3$ (after homogenization by the variable $w$). One can show that these ...
0
votes
0answers
88 views

Boundedness and Convergence of a Complex sequence

Consider a dynamical systems over complex numbers $$ z_{n+1}=\frac{\alpha}{z_{n}}+ \frac{\beta}{z_{n-1}},\qquad n=0,1,\ldots $$ where the parameters $\alpha, ~\beta$ are complex numbers, and the ...
4
votes
2answers
215 views

A question on certain elliptic PDE

Consider the elliptic PDE "CR" $$\begin{cases} U_{xx}=V_{yy}\\U_{yy}=-V_{xx} \end{cases}$$ And its consequence "LAP" $$U_{xxxx}+U_{yyyy}=0$$. Somehow, these equations are similar to the Cauchi ...