**1**

vote

**1**answer

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**

votes

**0**answers

65 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**

vote

**0**answers

54 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

**1**answer

294 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**

votes

**1**answer

670 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**

votes

**2**answers

287 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 ...

**0**

votes

**0**answers

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**

vote

**0**answers

59 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

**3**answers

176 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

**1**answer

335 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**

votes

**2**answers

452 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**

vote

**1**answer

102 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 ...

**1**

vote

**2**answers

90 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 ...

**6**

votes

**1**answer

281 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 ...

**1**

vote

**1**answer

128 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**

votes

**1**answer

141 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**

votes

**1**answer

81 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 ...

**1**

vote

**2**answers

139 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?

**1**

vote

**1**answer

158 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 ...

**1**

vote

**0**answers

39 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 ...

**0**

votes

**0**answers

28 views

### Is the Mellin transform of a measure nongrowing at imaginary infinity everywhere, or just on the fundamental strip?

Let $\mu$ be a measure on the positive real numbers. Its Mellin transform is a complex function defined by
$$
M_\mu (s) =\int x^{s-1} d \mu(x)
$$
on the set $S_\mu$ of $s \in \mathbb{C}$ where
$$
...

**2**

votes

**2**answers

177 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**

votes

**1**answer

96 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 ...

**1**

vote

**2**answers

167 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 ...

**1**

vote

**0**answers

72 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**

votes

**1**answer

187 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 ...

**1**

vote

**0**answers

202 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 ...

**1**

vote

**0**answers

87 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**

votes

**1**answer

118 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**

votes

**1**answer

210 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 ...

**1**

vote

**2**answers

509 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**

votes

**1**answer

143 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**

votes

**0**answers

130 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 ...

**1**

vote

**1**answer

94 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**

votes

**1**answer

121 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**

votes

**1**answer

160 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**

votes

**1**answer

192 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 ...

**-1**

votes

**1**answer

240 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**

votes

**0**answers

69 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**

votes

**1**answer

188 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**

vote

**1**answer

162 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**

votes

**2**answers

309 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 ...

**-2**

votes

**1**answer

182 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). ...

**1**

vote

**1**answer

55 views

### About convex combinations of real-stable multivariable complex polynomials

Say $f: \mathbb{C}^{n+1} \rightarrow \mathbb{C}$ is a real stable multivariable polynomial on the variables $(z,w_1,w_2,...,w_n)$. (a "real-stable" polynomial is one which has no zeroes in the open ...

**8**

votes

**2**answers

293 views

### Implicit Function Theorem on Singular Varieties

Let $X$ and $Y$ be two complex reduced affine algebraic or analytic varieties, possibly singular. Take a regular proper function
$$f\colon X \to Y $$
and assume that it is bijective at the level of ...

**6**

votes

**1**answer

474 views

### Bound on the sum of arguments

Problem: Show that for all real $s,t,u$ and all complex $z$ with $|z|<1$ one has
$$(*)\qquad \arg\frac{1-zf(s-u)}{1-zf(s+u)}
+\arg\frac{1-zf(t+u)}{1-zf(t-u)}<\pi,
$$
where $f$ is the ...

**5**

votes

**1**answer

384 views

### Structure of the automorphism group of a Riemann surface

I was wondering if anything is known about the possible structure of $\mathrm{Aut}(S)$ for a Riemann surface $S$. More precisely, are there known obstructions for a finite group $G$ to be such an ...

**18**

votes

**1**answer

668 views

### Why would the roots of the generating functions of the number of k-almost primes less than x have negative real parts?

Specifically, I find it appealing to count only squarefree numbers having $k$ prime factors, so I define
$$\pi_k(x)=\#\{n\leq x: \omega(n)=k;\mu(n)\neq0 \}$$
and consider the generating functions
...

**0**

votes

**0**answers

106 views

### Explicit formula for Bergman kernel on the unit ball

On page 173 in Krantz's book "Explorations in Harmonic analysis" in the proof of Lemma 7.1.21 there is a part that I really don't understand. What I don't understand is why is ...

**2**

votes

**0**answers

107 views

### Real-rooted polynomials and higher rank matrices

For $A$ and $B$ being matrices of the same dimension and $B$ being rank $1$, one knows that $det(A+tB)$ is a linear polynomial in $t \in \mathbb{R}$. Hence by Taylor series it follows that $det(A + ...