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

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1answer
106 views

A question on $J(f)$ and $J(f')$

I was confused by the following question for a long time: Does there exists a transcendental entire function $f$ such that $J(f)\cap J(f')=\emptyset$ ? where $J(f)$, ($J(f')$) is the Julia set of ...
5
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1answer
152 views

Which combinations of normality, separability, and paracompactness do complex manifolds possess?

I am interested in what kinds of non-paracompact complex manifolds may exist and which topological properties they may have. Is there a non-separable complex manifold? Can a non-separable complex ...
5
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3answers
128 views

Do the bounded isophase lines of a complex polynomial $f$ through the zeroes of $f’$ define a spanning tree?

Let $f: \mathbb{C} \to \mathbb{C}$ be a polynomial and let $\arg(f(z))$ be the phase of $f(z) = | f(z)| \exp(\mathrm{i} \arg(f(z)))$. The zeroes of $f'(z)$ are saddle points of $\arg(f(z))$, i.e. ...
4
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1answer
307 views

Roots of characteristic function of “reciprocal gamma measure”

Let us call a measure $\mu$ on the Borel $\sigma$-algebra $\mathfrak{B}_{(0,\infty)}$ of subsets of $(0,\infty)$ a reciprocal gamma measure if it is absolutely continuous with respect to the Lebesgue ...
6
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1answer
392 views

Analytic Chern classes

I have two questions on Chern classes, following Huybrechts' Complex Geometry. Are the analytic Chern forms just the elementary symmetric polynomials of the eigenvalues of the curvature? I googled ...
0
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1answer
49 views

criterion for a differential of the third kind to be a logarithmic derivative of a function

Let $X$ be a compact Riemann surface of genus $g\geq 1$. If $f$ is a meromorphic function on $X$ then, the meromorphic differential $\omega=\frac{df}{f}$ is a differential of the third kind with ...
11
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2answers
696 views

Is Every Holomorphic Near an Entire?

Let $K\subset \mathbb C$ be a closed subset of the complex plane, not necessarily bounded. Let $U$ be the interior of $K$. Let $f:K\to \mathbb C$ be a continuous bounded function, whose restriction ...
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1answer
76 views

Reproducing Kernel of a RKHS of continuous functions may not be continuous in two variables together

Let $\mathcal{K}$ be a Hilbert Space of continuous functions on some topological space, where point evaluations are continuous linear functional on $\mathcal{K}$. That is $\mathcal{K}$ is RKHS, ...
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0answers
59 views

solutions of elliptic linear pde depending analytically on a parameter

Fix $ \Omega$ a bounded smooth domain in $ R^N$ and suppose $0<w(x)$ is a smooth solution of $ -\Delta w(x)=w(x)^2$ in $ \Omega$ with $ w=0$ on $ \partial \Omega$ (were are assuming $2< ...
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0answers
103 views

Steepest descent path and Picard-Lefschetz theory

Assume that an ordinary integral of the form $$I=\int_{-\infty}^{\infty}dx e^{-f(x)} $$ for some real function $f(x)$ is given where $f(x)$ is well defined over all $\mathbb{R}$ and the integral is ...
6
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2answers
312 views

Does the proof of Picard's theorem become simpler by increasing the number of points that are not attained?

Let $f$ be an entire analytic function which attains all but $k$ complex numbers $z_1,\ldots,z_k$. Is there any elementary proof, for some $k$, that $f$ is constant?
2
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0answers
65 views

Tensor product of bounded analytic functions

I asked this question on math.SE, but couldn't get an answer. Let $H^\infty(\mathbb{D})$ denote the set of functions holomorphic and bounded on $\mathbb{D} = \{z \in \mathbb{C}: |z| < 1\}$. ...
1
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0answers
116 views

Ricci flow in complex analysis [closed]

Occasionally, I find a paper http://arxiv.org/abs/math/0505163 written by Chen, Lu and Tian. In this paper, the uniformalization theorem was proved by Ricci flow. I think it is a very interesting ...
2
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1answer
95 views

What is the image of the Ramanujan Delta function?

Consider the Ramanujan $\Delta$ function as a map from the upper half plane to the complex plane. We know that the image of $\Delta$ is unbounded and that it does not contain the point $0$. What else ...
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0answers
58 views

Finding singularities from power series

I am sorry beforehand for the length of my post, but I thought I should give some details. I try to figure out where are the singularities of a rather complicated power series. This series comes from ...
1
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0answers
64 views

When does analytic in the operator norm imply analytic in the trace class norm?

This is a crosspost from MSE. It's been up there for a few weeks now. A 200 rep bounty yielded no results (or even comments). I'm hoping someone here has some helpful ideas. See this post for the ...
1
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1answer
178 views

Class of functions between $C^{\infty}$ and $C^{\omega}$

I am always curious about that whether there exists a class of function which seems that more smooth than the $C^{\infty}$ class, while it is far from $C^{\omega}$ analytic function . From my point ...
5
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2answers
77 views

Equivalence of Definitions of Quasiconformal Surfaces?

I have been reading John H. Hubbard's book Teichmüller Theory vol. 1 and I am a little bit concerned with his definition of Quasiconformal Surface. Definition: A Quasiconformal surface $S$ is a ...
1
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1answer
324 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 ...
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0answers
65 views

Poisson Kernel and Triangles

The Poisson Kernel is an approximation to the identity, meaning $P_r(\theta) \approx \delta(\theta)$; here is the formula on $\mathbb{D}$: $$ P_r(\theta) = \sum_{n \in \mathbb{Z}} r^{|n|} ...
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0answers
43 views

Non interacting complex unit

How to work with two non interacting complex units say i and j. These two imaginary complex unit represent different quantities. For example i is for periodicity in theta and j is for frequency or ...
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1answer
230 views

Quintic Equation [closed]

Can we solve the following polynomial quintic equation by radicals x^5 + x^4 = 1 I found one real root which is algebraic solution (no approximation method ...
0
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0answers
46 views

On sequences of rational functions [duplicate]

Let $\{f_n\}_{n=0}^\infty$ be a sequence of rational functions of the following form: $$ f_n(z) = \sum_{m=1}^\infty \frac{C_{m,n}}{z-m}$$ with $C_{m,n} \in \mathbb{Z}$, $C_{1,n} = 1$, and for each $n ...
3
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1answer
121 views

Dominated convergence to characteristic function

Let $\phi_m(x):=\chi_{[0,1]} * \chi_{[0,1]} *...* \chi_{[0,1]}$ be the m -times convolution (so $m+1$ characteristic functions are involved). Then the Fourier transform of this function is given by ...
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6answers
2k views

Is this a rational function?

Is $$\sum_{n=1}^{\infty} \frac{z^n}{2^n-1} \in \mathbb{C}(z)\ ?$$ In a slightly different vein, given a sequence of real numbers $\{a_n\}_{n=0}^\infty$, what are some necessary and sufficient ...
0
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1answer
132 views

When can two Cauchy transforms intersect?

Given two polynomials $p$ and $q$ over reals and being guaranteed that both have all roots real I want to know if there is any characterization of the solutions of the equation $\frac{p'}{p} = ...
4
votes
1answer
95 views

Numerical equality testing

I am working on developing an online homework system. One thing I would like to have is something which compares a student's answer (like $2\sin(x)\cos(x)$) with the intended answer (maybe ...
2
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0answers
64 views

What are “minimal lines” in complex geometry?

Schwarz reflection across a real analytic arc $C$ in $\mathbb{R}^2$ is usually defined analytically. Thinking of the arc as the image of an interval in $\mathbb{R}$ under an invertible holomorphic map ...
1
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0answers
57 views

Related to derivative of Modified Bessel I function wrt the order

I recently met some problems related to the modified Bessel I funtions. Let $I(\nu,x):=I_\nu(x)$, and $I'_\nu(\nu,x):=\dfrac{\partial}{\partial \nu}I(\nu,x)$. Using maple, it seems that ...
3
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2answers
213 views

Conformal map and Jordan curve

Here is my question : Suppose you have a simple (analytic) closed curve $\gamma$ in an open simply connected domain $\Omega \neq \mathbb{C}$. Does there exist a conformal bijection $f : \Omega ...
0
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1answer
64 views

A Generalization of growth exponents

The growth exponent of a function $f(\sigma + it)$ is defined as the least nonnegative real number $\psi(\sigma)$ satisfying $$ f(\sigma + it) \ll |t|^{\psi(\sigma) + \epsilon} $$ as $t \to \infty$, ...
1
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1answer
51 views

Oscillation of subharmonic functions of slow growth

Given a sequence of real numbers $c_k\to-\infty$, is there always a $C^\infty$ subharmonic function $f$ on $\mathbb R^2$ and a sequence $z_k\to\infty$ with $|z_k|<k$ such that ...
3
votes
2answers
292 views

Smooth paths on affine varieties

I have the following question which is in some way related to an application of Randell Isotopy Theorem to complex hyperplane arrangements. Let $h,k\geq1$ be integer numbers and let ...
0
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0answers
56 views

Hurwitz's theorem for a system of functions

First, let me define a notation of $H(G_1\times G_2 \times \ldots \times G_m)$. We say that $f\in H(G_1\times G_2 \times \ldots \times G_m)$ if $$f:G_1\times G_2 \times \ldots \times G_m \rightarrow ...
7
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0answers
63 views

Extremal length of graphs in surfaces

Given a surface $\Sigma$ with conformal structure $\omega$, the extremal length of a homotopy class $\gamma$ of curves in $\Sigma$ is defined to be $$ \sup_{g \in \omega} ...
0
votes
0answers
58 views

Derivative of a conjugation of matrices

Let $\mathcal{M}_n$ be the space of complex $n\times n$ matrices. Let $\Phi\colon \mathbb{D}\to \mathcal{M}_n$ and $\psi \colon \mathbb{D}\to \mathcal{M}_n$ be holomorphic functions. Consider the ...
1
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1answer
172 views

Singular homology of the zero loci of polynomials

I am very sorry but apparently I am really weak in cohomology flavored questions. I try to reformulate my problem in a very simple and hopefully clear way. This question is related with a problem in ...
1
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1answer
116 views

Interesting property of analytic functions

Let $f:(t_0-\varepsilon, t_0+\varepsilon)\to\mathbb{C}$, be an analytic application, such that: $f(t)=0\Longleftrightarrow\ t=t_0$. Is it true that there is an analytic function $g:(t_0-\varepsilon, ...
0
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0answers
63 views

prove that a function is approximatively three dimensional

Let $D_n(x)$ be a diagonal matrix of size $N\times N$ where the $k$th element is $\exp(2\pi\jmath x(n+(k-1)/N)$. Let $P_n$ be a random diagonal $N\times N$ matrix where each diagonal element is a ...
1
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0answers
53 views

Analytic continuation of an integral

Let $$f(y)=\frac{y_1^{1/3}y_2^{1/3}}{y_1+y_2+1}$$. Consider the following integral: $$F(s_1,s_2)=\int_{\mathbb{R}_+^2}f(y)^{s_1}f(y^{-1})^{s_2}\frac{dy_1dy_2}{y_1y_2}$$ where ...
9
votes
1answer
291 views

Higher Fano varieties and Tsen's theorem

The rational connectivity of (complex) Fano manifolds ($c_1(T_X) > 0$) is one of the major, and surely most memorable achievements of Mori's bend-and-break method. To this day, despite intensive ...
3
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1answer
594 views

Is the integral always nonzero?

Let $$I_{n,p,a}:=\int_0^{\infty-} \frac{g_{n,a}(t)}{t^{p+1}} \, dt,$$ where $$(*)\qquad\qquad\qquad n\in\mathbb N,\quad -\infty<a<\infty,\quad p_{n,a} < p < ...
3
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2answers
281 views

Residue for the generating function of the Euler totient function

Let $\varphi$ be the Euler totient function, and let us define the function $f(z)$ by the series $$ f(z) := \sum_{n=1}^{\infty} \varphi(n) z^n $$ Since $0\le \varphi(n)\le n$, I believe this gives a ...
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0answers
38 views

Euler transformation of pochhammer symbol

From the Euler transformation of Pochhammer symbol $$\sum_{n=0}^{\infty}\frac{(b)_n}{n!}a_nz^n=(1-z)^{-b}\sum_{n=0}^{\infty}\frac{(b)_n}{n!}\Delta^na_0(\frac{z}{1-z})^n$$ the following ...
1
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0answers
56 views

Geodesic equation and radial metric

Assume that $g(z)=f(|z|)$ is a radial metric on the unit disk in complex plane, where $f$ is a smooth real function. Is there any simple equation of geodesic lines w.r.t. metric $g$, e.g. ...
0
votes
3answers
164 views

Extension of conformal map and annulus

My question is the following : suppose you have a doubly-connected open set $\Omega \subset \mathbb{C}$, that is a domain bounded by 2 non-intersecting circles $C_1$ (the interior) and $C_2$ (the ...
2
votes
1answer
304 views

Real and imaginary part of an holomorphic function

I guess this could be a very elementary question. Anyway I can not find an answer in literature. Let $f:U\rightarrow\mathbb{C}$ be an holomorphic function on an upen subset $U\subseteq\mathbb{C}$. ...
16
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2answers
447 views

Classification of complex structures on $\mathbb{R}^{2n}$

Is there anything known about classification of complex structures on $\mathbb{R}^{2n}$ up to isomorphism for $n>1$? Say, are there finitely or infinitely many isomorphism classes? If there is a ...
1
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1answer
101 views

If $f$ is separately holomorphic on $\Omega$ then $f\in\mathcal{C}^0(\bar\Omega)\Leftrightarrow f\in L^1(\Omega)$

Let $\Omega\subseteq\Bbb C^2$ be open bounded (and connected), $f:\Omega\to\Bbb C$ separately holomorphic (i.e. $f$ is holomorphic in each variable when the other is fixed). Hartogs theorem is not ...
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2answers
88 views

Integrability at $z$ of the 2-form $ d\omega=\frac{\partial_{\bar{\zeta}}g(\zeta)}{\zeta-z}d\zeta\wedge d\bar{\zeta} $

Given $g\in\mathcal{C}^1(\bar\Delta)$, and $z\in\Delta$, how can i prove that the 2-form $$ d\omega=\frac{\partial_{\bar{\zeta}}g(\zeta)}{\zeta-z}d\zeta\wedge d\bar{\zeta} $$ is integrable in $z$? At ...