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

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61 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
265 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 ...
1
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1answer
146 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 $t \to \infty$, ...
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1answer
66 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
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2answers
325 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 ...
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0answers
74 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 ...
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74 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
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0answers
71 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 ...
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1answer
176 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 ...
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1answer
132 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, ...
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0answers
83 views

Proof sought that a certain function is approximatively 3-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 ...
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0answers
67 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
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1answer
372 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 ...
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1answer
857 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
339 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
79 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. ...
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3answers
318 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 ...
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1answer
508 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 $U\subseteq\mathbb{C}$. ...
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2answers
511 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
114 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
96 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 ...
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1answer
324 views

Acyclicity of the sheaf of real analytic differential forms

Let $M$ be a real analytic manifold. In the book "Sheaves on Manifolds" by Kashiwara and Schapira it is claimed on p. 127 (without reference or proof) that the Poincare lemma holds for the de Rham ...
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1answer
149 views

Zeros of Polynomial with decreasing coefficients [closed]

If $n_1<n_2<n_3\cdots<n_m$ are positive integers. Does the polynomial $a_0+a_1z^{n_1}+a_2z^{n_2}+\cdots+a_mz^{n_m}$ satisfying $$ 0<a_0\leq a_1\leq \cdots\leq a_m $$ has all its zeros ...
3
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1answer
154 views

intersection of holomorphic curve with hyperplane

Let $f : \mathbb{C} \rightarrow \mathbb{C}^n$, $n>1$ be an entire function. Assume for simplicity that $f(0)=0$. Let $B$ be the closed ball of centre $O$ and radius $R$. Is there an upper bound ...
3
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1answer
88 views

About extending plurisubharmonic function

I came across a problem like this. Suppose that $\Omega$ is an open subset of $\mathbb{C}^{n}$ and $V$ is a complex submanifold of $\Omega$ of codimension 1. Now given a plurisubharmonic function ...
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2answers
151 views

convergence radius of Pochhammer symbol series

What is the convergence radius of the series $$\sum_{n=0}^{\infty}\frac{(-1)^n z^{(n)}}{n!},$$ where $z^{(n)}$ is Pochhammer symbol?
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1answer
243 views

generalization of fundamental theorem of algebra for several complex algebra [closed]

I am looking for a generalization to fundamental theorem of algebra for several complex variables functions or systems. If such theorem exists, it should concisely relates the number of zeros of ...
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0answers
44 views

Meromorphic extensions of $\zeta-$functions

Suppose we have a series $\zeta_x(s)=\sum_{n\geq1}x(n)n^{-s},$ where $x=\{x(n)\}_{n\geq1}$ is a bounded sequence. Clearly, $\zeta_x$ is analytic function for $\Re(s)>1.$ Question: is there a ...
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2answers
207 views

Connected complement manifold

I'm working on some problem in algebraic geometry. I need a reference to the following result: Let $h\in\mathbb{N}$ with $h\geq1$ and let $F\in\mathbb{C}\left[x_{1},\ldots,x_{h}\right]$ be a non ...
0
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1answer
102 views

Dirichlet series without order term

is there a name in use for Dirichlet series without the order term, analogously to Laurent or Puiseux polynomials? Is there work known about such expressions? $D(s) = \sum_{0<n<N}a_n/n^s$ The ...
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2answers
190 views

Generalized Schwarz Lemma for near-zeros

In approximation theory, it is classical to use a result that can be considered a generalization of the Schwarz Lemma: Let $f:[-1,1]\rightarrow\mathbb{C}$ be a function that is analytic in a domain ...
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0answers
97 views

Complex Hessian Signature

It' all, simply, about the signature of a matrix. Let $\Omega\subseteq\Bbb C^n$ open, $r:\Omega\to\Bbb R$ twice differentiable (real differentiable, not necessarely complex differentiable, i.e. not ...
9
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1answer
195 views

Continuous Weierstrass map

Let $\mathbb C$ be the complex plane, $H(\mathbb C)$ the set of all entire functions, and $D(\mathbb C)$ the set of all non-negative divisors in $\mathbb C$. Consider the map $Z:H(\mathbb C)\to ...
2
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0answers
255 views

What is the status on questions related to Bhargava's factorial function?

In Manjul Bhargava's The Factorial Function and Generalizations he motivates a new type of factorial $n!_S$ using by generalizing a few theorems like: For $k, l \in \mathbb{Z}$, we have $k! \times ...
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0answers
90 views

holomorphic curves invariant by lattices

Suppose I have an entire function $f : \mathbb{C} \longrightarrow \mathbb{C}^n$ for $n \geq 1$. Let $C$ be the curve $f(\mathbb{C})$ in $\mathbb{C}^n$. Let $\Lambda$ be a lattice in $\mathbb{C}^n$ ...
2
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1answer
130 views

Prove or disprove an inequality concerning zeros of a polynomial

If a polynomial $p(z)$ of degree $n$ with zeros $z_1,z_2,\cdots,z_n$ assumes maximum at $w$ on $|z|=1.$ Prove or disprove that the Harmonic mean of $|z_k-w|,$ $k=1,2,\cdots,n$ is greater or equal to ...
2
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1answer
295 views

If a polynomial $p(z)$ omits a value, then $p(z)-\dfrac{(1-e^{i\psi})}{n}zp^{\prime}(z)$ also omits that value

Suppose that a polynomial $p(z)$ of degree $n$ does not assume the value $w$ for $|z|<1$, that is $p(z)\neq w$ for $|z|<1.$ Show that $p(z)-\dfrac{(1-e^{i\psi})}{n}zp^{\prime}(z)\neq w$ for ...
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2answers
535 views

Certain inverse problem related to moments

Suppose $D\subset \mathbb C$ is a smoothly bounded domain and it contains the origin. Let $ds$ denote the arc length measure on $\partial D.$ I am interested in the following two inverse problems ...
3
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1answer
154 views

Determining the Fourier transform

Let $d>2$. Let $M$ be a 2-dimensional submanifold of $\mathbb{R}^d$. For instance (and this is the type of example I primarily care about) we could have $M$ being the set of scalar multiples of a ...
3
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0answers
145 views

Quadrature domains for arc length

Is ellipse a quadrature domain for arc-length? More precisely does there exist points $z_1,\cdots,z_n$ inside an ellipse $E$ and non zero constants $c_1,\cdots,c_n$ such that $$\int ...
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1answer
102 views

About the upper bound on the roots of the matching polynomial

Heilman and Lieb had proven that if a graph had $d$ as its maximum vertex degree then the roots of the matching polynomial are bounded from above by $2\sqrt{d-1}$. Is there a modern exposition of ...
2
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1answer
126 views

hyperbolic metrics

Let $D_1\subset D_2$ be simply connected domains in the complex plane. Let $\lambda_1$ and $\lambda_2$ be the corresponding hyperbolic (Poincare) metrics. It seems intuitive to me that $\lambda_2$ is ...
6
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1answer
183 views

Is the Poincaré metric continuous with respect to the domain?

Suppose $K \subset \mathbb{C}$ is a Cantor set and let $u:\mathbb{C} \setminus K \to \mathbb{R}$ be the maximal smooth function such that the conformal metric $e^{2u}(\mathrm{d}x^2 + \mathrm{d}y^2)$ ...
2
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1answer
204 views

A conjecture regarding the integral of the square of an entire function

Can some help me prove or disprove the following assertion which I encountered in research? Thanks! Let $f:\mathbb R\to\mathbb R$ be an analytic function. If for $\forall c > 0$, we can find some ...
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1answer
342 views

Holomorphic Function on Disk

Let $f$ be holomorphic function on unit disk and it is continuous on boundary of the disk. It is known that $f$ is constant and equal to zero if $f$ is vanishing on sub-arc of boundary (Maximum ...
3
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0answers
86 views

Multivariate ML inequality and holomorphic functions on the closed unit ball

There exists a dimensional constant $C_n$ such that, for each holomorphic function $f:\overline{B(1)}\to \mathbb{C}$ on the closed unit ball centered at the origin of $\mathbb{C}^n$ and each ...
2
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1answer
216 views

Exact reference for Liouville theorem

It seems hard for me to find that the solution of the following equation $$ \Delta u+e^u=0 $$ defined on a simply-connected domain $D\subset R^2$ must be of form $$ ...
1
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1answer
194 views

Cohomology of lattice with coefficients in field of rational functions

In my research, I came across a 1-cocycle in the following group cohomology complex: Let $\Lambda_\mathbb{Z}$ be a lattice (i.e. isomorphic to $\mathbb{Z}^n)$; let $\Lambda_\mathbb{C} = ...
14
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2answers
355 views

Vanishing of Dolbeault cohomologies and Steinness

That Stein manifolds have all $(p,q), p \geq 0, q \geq 1$ vanishing Dolbeault cohomology groups is more or less standard. I am a little bit confused about the reverse implication: whether the ...
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1answer
240 views

Degree of a rational function [closed]

I would like to have a simple proof for the following result: Let $f=\frac{p}{q}:\mathbb{C}\longrightarrow\mathbb{C}$ be a quotient of polynomials (of course, at some points it may be undefined). ...