**18**

votes

**1**answer

669 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 + ...

**2**

votes

**1**answer

297 views

### Triviality of holomorphic vector bundles over contractible Stein manifolds

If I have correctly undrestood,it is a result of the so called Grauert-Oka principle that all holomorphic vector bundles over contractible Stein manifolds are holomorhically trivial.Does any one knows ...

**0**

votes

**0**answers

186 views

### Does the Euler product converge at $s=1$ for the Dirichlet $L$ function?

For the Riemann Zeta function, the Euler product converges on $\{Re(s)=1\}$ except at $s=1$.The zeta series diverges everywhere on $\{Re(s)=1\}$. But the $L$ series converges on $\{Re(s)>0\}$. What ...

**1**

vote

**1**answer

93 views

### Lifting quadratic forms on the cotangent bundle to higher level forms

Backround
In several complex variables, an essential tool is Hormander's machinery for solving the $\overline{\partial}$ problem with $L^2$ estimates.
If $\alpha$ is a $(p,q+1)$ form on a domain ...

**0**

votes

**0**answers

53 views

### Maximum modulus principle and restriction

Suppose I have a complex multivariate polynomial $f\in \mathbb{C}[z_1,\ldots,z_n]$ and a connected, bounded open set $U$. The maximum modulus principle says $f$ achieves its maximum value on the ...

**5**

votes

**1**answer

176 views

### Are polynomials dense in holomorphic $L^p(\mathrm{Gauss})$ for $p < 1$?

Let $\mu$ be standard Gaussian measure on $\mathbb{C}^n$, i.e. $d\mu = \frac{1}{(2 \pi)^n} e^{-|z|^2/2}\,dz$, and fix $0 < p < 1$ (note carefully).
Suppose $g$ is holomorphic on ...

**1**

vote

**1**answer

139 views

### How to find isothermal coordinates equivalent to circles in far limit?

I am trying to find the most general rotational coordinate systems for Euclidean 3-space, with the following two defining characteristics: 1) being equivalent to spherical coordinates in the limit of ...

**3**

votes

**0**answers

124 views

### Implicit function theorem for singularities

I am looking for an implicit function theorem which holds also on singular spaces, at least if the singularities are "mild".
For example, let $0 = z^2 - x y + z w + w^2 + \epsilon w$ define a ...

**5**

votes

**1**answer

291 views

### Does pointwise convergence of holomorphic functions on the boundary imply pointwise convergence in the interior?

Let $\Omega$ be a simply connected open set in the complex plane and $\gamma$ be a simple path inside $\Omega$. Suppose $f_n$ is a sequence of holomorphic functions converging pointwise to 0 on ...

**3**

votes

**0**answers

170 views

### A difficult integral which the Risch algorithm shows is not elementary

For reasons which aren't conceptually related to the problem a few of my colleagues and I are in need of finding an expression for the following integral in terms of $a$ and $\delta$:
...

**2**

votes

**0**answers

116 views

### Does this symmetrization operator have a name? Any theory?

Consider a function $f(x_1,\ldots,x_n)$ of $n$ complex variables. Define
$$f_{\mathrm{symm}}(x_1,\ldots,x_n) =
2^{-n}\sum_{\varepsilon_1,\ldots,\varepsilon_n=\pm 1}
...

**0**

votes

**1**answer

115 views

### holomorphic continuation

consider the function given by $f(t):=\sum\limits_{n=0}^{\infty}e^{-\left(n+\frac{1}{2}\right)^2t}$ for $t\in (0,\infty)$.
This function can be continued holomorphically for all complex numbers with ...

**1**

vote

**1**answer

134 views

### Sequence of smooth maps converging to the identity [closed]

Let $\{g_{n}:B_{1}(0) \rightarrow \mathbb{R}^{2}\}_{n\in \mathbb{N}}$ be a sequence of smooth maps where $B_{1}(0)=\{x\in \mathbb{R}^{2} \mid |x|<1\}$ is the unit ball in $\mathbb{R}^{2}$. Assume ...

**1**

vote

**1**answer

162 views

### Finding Laurent Series of a function [closed]

I've been assigned to write a computer program which then calculates the Laurent series of a function. Of course I'm familiar with the concept, but I've always calculated the Laurent series in an ad ...

**1**

vote

**0**answers

111 views

### Coordinate charts on converging Riemann surfaces

Let $S$ be a $2-$dim manifold and $q \in S$. Furthermore, let $j_{n}$ be a sequence of complex structures on $S$ converging in $C^{\infty}_{\text{loc}}$ to a complex structure $j$ on $S$ as ...

**10**

votes

**1**answer

230 views

### Is there a solvable point on any variety over the field of complex rational functions?

Let $K = \mathbb{C}(T)$ be the field of complex rational functions in one variable, and let $V$ be a variety defined over $K$.
Must $V$ have a solvable point?
The variety $V$ is assumed ...

**2**

votes

**1**answer

135 views

### Flatness of a morphism of complex analytic spaces

Let $f\colon X\to D$ be a morphism of a complex analytic space $X$ into the 1-dimensional disk $D$. Assume for simplicity that $X$ has a single irreducible component which may not be reduced.
...

**3**

votes

**1**answer

160 views

### A weak analytic version of the valuative criterion of properness

EDIT: Let $f\colon X\to Y$ be a morphism of complex analytic spaces (not necessarily smooth or reduced). Assume that
(a) $f$ is injective on points;
(b) $f$ is local imbedding near each point $x\in ...

**1**

vote

**0**answers

107 views

### Possible generalizations of Hadamard's three line lemma

Let $f$ be an analytic function on a sector
$$
S=\left\{re^{i\theta}:0<r<\infty,\; 0<\theta<\gamma<\frac{\pi}{2}\right\}
$$
with opening angle $\gamma$ at the origin. Suppose $f$ is ...

**2**

votes

**1**answer

161 views

### polynomial inequality in complex variable (generalized)

Let $f(w)=\frac13+\frac12 w+\frac16 w^3$. If $\vert f(w)\vert\leq1$ or simply $\vert f(w)\vert=1$, show that $\vert \frac{w}2 f(\frac{w}2)\vert\leq1$. Here, $w$ is a complex number.
What happens if ...

**0**

votes

**1**answer

167 views

### existence of positive curved line bundles on a compact Riemann surface

Can anyone suggest a proof of the existence of positive line bundles on a compact Riemann surface, avoiding Hodge decomposition. (I am aware of the method in Dror Varolin's book, but I consider that ...

**0**

votes

**0**answers

58 views

### Analytic functions space on Riemann surface

I have some questions about the analytic function space on Riemann surface and distinguished varieties:
Let S be a compact Riemann surface and $\Omega\subset S$ be a domain with piecewise smooth ...

**7**

votes

**1**answer

214 views

### Analytic diffeomorphisms of the circle from complex domains

Let $\gamma \subset \mathbb C$ be a simple closed analytic curve and let $\Delta$ be the closure of the disk it bounds. The Riemann mapping theorem gives two biholomorphisms:
$$\phi : (D^2,S^1) \to ...

**1**

vote

**0**answers

201 views

### Estimating the kernel of Poisson semigroup $e^{-z\sqrt{-\Delta}}$ for complex $z$

Let $f(z,a)$ be an analytic function on $C^+=\{\Re z>0\}$ for each fixed $a>0$, and we have the following (weaker) estimates
$$
|f(re^{i\theta},a)|\leq C (r\cos\theta)^{-n}, ...

**2**

votes

**0**answers

77 views

### Is logarithmic convexity of the heat kernel with complex time a general fact?

Suppose $H$ is a non-negative self-adjoint operator acting on $L^2(\mathbb{R}^n)$, and generates an analytic semigroup with kernel satisfying a Gaussian upper bound, i.e., if we denote $K(t,x,y)$ the ...

**7**

votes

**0**answers

165 views

### Complex structures on $\Bbb R^4$

Calabi & Eckmann proved that $S^{2p+1} \times S^{2q+1}$ admits an integrable complex structure fibred by holomorphic tori, and this implies that $\Bbb R^{2p+2q+2}$, obtained by removing a point in ...

**3**

votes

**1**answer

161 views

### Complex function for mapping a circle to a superellipse

I was wondering if anyone knows an analytic complex function that would map a circle to a superellipse, or vice versa. Any ideas, comments, or functions are much appreciated!
Thanks,
Kayvan

**3**

votes

**0**answers

103 views

### Dimension of certain polynomial spaces

Let $(\omega_1, \eta_1） \dots (\omega_n, \eta_n)$ be $n$ pairs of complex numbers where $\omega_i \ne \omega_j$ for all $1 \leq i \ne j \leq n$. We define the following polynomial space
$$
Z_n^d(\eta, ...

**4**

votes

**2**answers

308 views

### Is there any elementary proof of No wandering domain for polynomials

It seems that it is almost impossible to give a elementary proof of Sullivan's no wandering domain for rational map or even more general class of maps.
I think it is interesting to ask whether we ...

**0**

votes

**0**answers

44 views

### on the existence of holomorphic coordinates under bounded curvature

Let $(X,g)$ be a complete Kahler manifold with the Riemann curvature and its derivatives bounded. Suppose also the injectivity radius of this manifold is bounded below by a constant $i_0>0$. My ...

**1**

vote

**1**answer

83 views

### An optimization problem in complex space

Consider the following optimization problem
$$
\min \| \textbf{Ax-B}\|
$$
$$
s.t.|x_i|=1,i=1,...,n
$$
where $\textbf{x}\in \mathbb{C}^{n}$ is the optimization varaible, $x_i$ is the $i$-th ...

**8**

votes

**1**answer

226 views

### Conformal map of polygon with circle segments

I am looking for a conformal map from a "polygon" to eg the upper half plane, which consists of circle segments instead of lines. So for example, it could be a quadrilateral ABCD, but where AB is a ...

**13**

votes

**3**answers

477 views

### Entire function bounded at every line

I would like to ask about, does there exists an entire function which is bounded on every line parallel to $x$ - axis , but unbounded on the $x$ - axis.

**0**

votes

**0**answers

73 views

### kostant partition function vs Haar measure

I am trying to understand the relationship between the Kostant partition function and the Haar measure. Both seem to involve the Vandermonde determinant:
$$ \Delta(\theta) = \prod_{i< j} ...

**4**

votes

**2**answers

130 views

### Bound areas of disks with respect to a quadratic differential

Let $q = q(z)\,dz^2$ be a quadratic differential on the unit disk $D(1)$, normalized so that $\int_{D(1)}|q| = 1$. I have a fairly convoluted argument that for any smaller disk $D(r)$ with $r < 1$, ...

**7**

votes

**2**answers

194 views

### Gauss--Lucas type theorem for tracts and higher derivatives of a polynomial

The Gauss--Lucas Theorem states that all zeros of a degree $n$ complex polynomial $p(z)$ are contained in the convex hull of the zeros of $p$. By iteration, this implies that the zeros of ...

**9**

votes

**0**answers

125 views

### How useful is knowing every torsionfree $\mathcal O(D)$ module is flat?

One of the corollaries of Weiertrass' factorization theorem plus the theorem of Mittag Leffler is that $\mathcal O(\Bbb C)$, more generally $\mathcal O(D)$ for some region $D$ is such that every ...

**1**

vote

**1**answer

225 views

### Residues and values of Riemann Zeta function at some points

I need the following computational results for proving something.
Let $1/2 + i\gamma_0$, be the first nontrivial zero of Riemann zeta function, $\zeta(s)$,
i.e. $\gamma_0\sim 14.134...$.
1) what is ...

**4**

votes

**0**answers

119 views

### residue and regulator

Let $C$ be a curve defined over $\mathbb{Q}$. The regulator is a map
$$
reg: K_2(C)_{\mathbb{Q}} \longrightarrow H^1(C(\mathbb{C}), \mathbb{R}).
$$ Here $K_2(C)_{\mathbb{Q}}$ is the K-group tensor ...

**2**

votes

**0**answers

144 views

### weak form of Sendov conjecture

Suppose $p$ is a polynomial of degree $n$ and all roots $z_1,\cdots,z_n $ of $p$ are inside the unit disk. Then how to show that every disk of radius $\sqrt{2}$ and centered at $z_k$ for ...

**5**

votes

**1**answer

174 views

### Starshapeness of polynomial tracts with respect to the (entire collection of) critical points contained in the tract

I recently found out (Piranian, "The Shape of Level Curves") that a polynomial tract (ie a connected component of a set of the form $G=\{z:|p(z)|<\epsilon\}$ for some $\epsilon>0$) need not be ...

**0**

votes

**0**answers

122 views

### Solution to system of polynomial equations

Suppose $A$ is a set of polynomials:$$P_1(x,y_1,\dots,y_n)=0,$$
$$P_2(x,y_1,\dots,y_n)=0,$$
$$\vdots$$
$$P_k(x,y_1,\dots,y_n)=0$$
is a system of equations with coefficients over $\mathbb{Z}$, and ...

**32**

votes

**1**answer

2k views

### $\zeta(0)$ and the cotangent function

In preparing some practice problems for my complex analysis students, I stumbled across the following. It is not hard to show, using Liouville's theorem, that
$$\pi\cot(\pi ...

**0**

votes

**0**answers

48 views

### what is the first non-constant term in the Kronecker Limit formula?

The Kronecker Limit formula gives the constant term in the Laurent expansion about s=1 of the Eisenstein series E(s,\tau). What is the next term? I.e., the coefficient of the first power of (s-1)? I ...

**5**

votes

**0**answers

61 views

### Obstruction to the existence of global resolution of coherent sheaf

It is well known that any coherent sheaf on a complex manifold (or more generally a complex space) admits locally a resolution with locally free sheaves. It is also well known that for non-algebraic ...

**0**

votes

**0**answers

102 views

### Asymptotic analysis of a sum of complex summands using integral

I'm trying to find the exact asymptotics of a sum:
$$A = \sum^n_{i=0} \begin{pmatrix} 2n \\ i \end{pmatrix} x^{i} y^{2n-i} $$
as $n\rightarrow\infty$. Here $x,y$ are complex numbers, $|x|\leq1, ...

**1**

vote

**0**answers

118 views

### Proper monomorphisms in complex analytic spaces

In the algebraic geometry of schemes, we know that monomorphisms which are universally closed (= every pullback is a closed map) and of finite type are closed immersions. See Gortz, Wedhorn "Algebraic ...

**1**

vote

**2**answers

92 views

### A question on deficient values of entire functions

Recently I come cross a question about deficient values of entire functions.
I find that many examples in the book about functions $f$ whose deficient values are singularities of the inverse ...