**2**

votes

**0**answers

89 views

### A parametrix for the $\bar\partial$ operator adapted to a holomorphic foliation

Let $X$ be a compact complex manifold with (regular) holomorphic foliation given by a holomorphic subbundle $\mathcal F$ of the tangent bundle. The foliation induces a filtration on differential ...

**2**

votes

**2**answers

391 views

### What is the closure of space of polynomials in a dense subspace along with a marked point equal to?

EDIT
Let $\mathbb{C}^{m*}$ be the space of non zero polynomials of degree at most
$d$ in two variables. So an element of this space is essentially
$$ f:=f_{00} + f_{10} x + f_{01} y + \ldots ...

**4**

votes

**0**answers

137 views

### semiclassical proof of Wigner semicircle

In Terence Tao's discussion of the Gaussian Unitary Ensemble, he derives the Dyson and Airy kernels. The GUE is the probability distribution of the eigenvalues of a random Hermitian matrix.
\[ \int ...

**12**

votes

**3**answers

695 views

### Do all the roots of the polynomial lie in the unit disk?

How to prove (or to disprove) that all the roots of the polynomial of degree $n$ $$\sum_{k=0}^{k=n}(2k+1)x^k$$ belong to the disk $\{z:|z|<1\}?$ Numerical calculations confirm that, but I don't see ...

**28**

votes

**1**answer

1k views

### Zeros of polynomials with real positive coefficients

The following problem arose in collaborative work with Subhro Ghosh:
Question: To any polynomial $P_n(z)=\sum_{i=0}^n a_i z^i =a_n \prod_{i=1}^n(z-z_i)$, attach the empirical measure of zeros
...

**2**

votes

**1**answer

153 views

### Interpolating delta like functions by trigonometric polynomials of bounded modulus and fast decay

Consider a grid of points $T=\{t_0,t_1,\ldots,t_m\}$ with $0\le t_i\le 1$. I would like to find a function $f(t):[0,1]\rightarrow \mathbb{C}$ of the form
\begin{equation*}
f(t)=\sum_{k=-n}^n c_k ...

**5**

votes

**1**answer

208 views

### Zariski's main theorem in the complex analytic category

Hello,
I am looking for a reference to something like that: if $f\colon X\to Y$ is a finite (i.e., proper with finite fibers) morphism of reduced and irreducible normal (or at least smooth) complex ...

**3**

votes

**1**answer

203 views

### Which real analytic functions of two variables locally are magnitudes of complex-analytic functions [closed]

Assume we have a real-analytic function $f(x, y)>0$ in some neighborhood of 0. When does there exist a complex-analytic function $w(z)$ such that $|w(z)|=f(x,y)$ for $z=x+iy$.
One necessary ...

**9**

votes

**2**answers

377 views

### Constructing Riemann maps using Brownian motion?

There's a relation between two-dimensional Brownian motion and conformal maps, see e.g. Thurston's answer to this question. Given two non-empty simply-connected domains $U$ and $V$ in the complex ...

**2**

votes

**2**answers

378 views

### Mellin Transform

What is the inverse Mellin transform of (s-1/2)^k on the vertical line Re(s)=a where
0 < a <1 and k is a natural number?

**2**

votes

**2**answers

156 views

### What is the moduli space of germs of one-sided complex structures near the circle?

Consider a one-sided ( say, internal) neighborhood $U$ of the unit circle $S$ ( $U$ contains $S$) on the plane with a choice of smooth complex stricture $\tau$ on $U$.
By smoothness of $\tau$ on $U$ ...

**2**

votes

**2**answers

298 views

### What is known about this product?

I bet the product
$$
\prod_{n=2}^\infty\frac 1 {1-n^{-s}},
$$
which is convergent for ${\rm Re}(s)>1$, has been studied before. Can it be analytically extended across the line ${\rm Re}(s)=1$? If ...

**5**

votes

**4**answers

542 views

### Visualizing functions with a number of independant variables

I need to graph real valued functions ( for exposition and analysis)
the issue is the independent variables are more so that the conventional graphing method cant be used and further i don't want to ...

**7**

votes

**1**answer

320 views

### Defining holomorphic functions in terms of Banach algebras, and similarly for C*-algebras

Let $C$ be the category of commutative Banach algebras and let $U : C \to \text{Set}$ be the usual forgetful functor. The holomorphic functional calculus guarantees that every holomorphic function $f ...

**1**

vote

**2**answers

400 views

### On the existence of a holomorphic logarithm

Hi,
The following is probably well-known, but I couldn't find anything in the literature. Any reference would be nice.
Let $\Omega$ be a domain in the complex plane, and let $f$ be holomorphic and ...

**5**

votes

**0**answers

162 views

### proper mapping between Stein manifolds

My question is the following:
Let $b:X \rightarrow Y$ be a proper holomorphic mapping between two Stein manifolds $X$ and $Y$. For obvious reasons, $b^{-1}(y)$ are finite subsets of $X$. Is the set
...

**0**

votes

**1**answer

256 views

### Riemann mapping

Let in the complex plane be a bounded Jordan region T (that is a bounded and
simply connected set with the boundary a Jordan curve), containing the origin,
with its Riemann mapping onto the open unit ...

**-3**

votes

**1**answer

176 views

### Can Hartogs' extension theorem be used to prove there's no naked singularity?

Maybe for the first time my question doesn't deal with number theory. Tonight a friend of mine told me about Hartogs' extension theorem and said his work in analytic microlocal analysis was somehow ...

**3**

votes

**1**answer

152 views

### unbounded power series

I want a reference to the literature of a power series convergent in the whole CLOSED
unit disk,but unbounded there.

**0**

votes

**0**answers

224 views

### On uniform convergence of sequences of bounded holomorphic functions with formal convergence

At some point I needed to prove that some formal (iterative) construction yielded an actual convergent power series. To do so I was led to prove the following lemma, where $\mathcal{B}(\Delta)$ is the ...

**0**

votes

**1**answer

338 views

### Dolbeault cohomology

Let $X=S^{2n-1} \times S^1$. I have to compute $H^{(1,0)}_{\bar{\partial}} (X)$ and
$H^{(0,1)}_{\bar{\partial}}(X)$ . I don't know how to do this but if we use Kunnet formula we
have that ...

**2**

votes

**0**answers

200 views

### Analytical continuation of electrostatic potentials

I'm having some trouble figuring out the properties with respect to analytical continuation of functions defined using an integral kernel. More particularly, I am working with the electrostatic ...

**2**

votes

**2**answers

262 views

### Usage of complex moments in complex plane

Suppose $B$ is a bounded region in complex plane. In complex plane, one usually deals with complex moments, i.e. $\int_B {z}dxdy$ where $z \in \mathbb{C}$. What is so special about this complex moment ...

**0**

votes

**1**answer

415 views

### Stein manifolds definiton

There are a few equivalent definitions of Stein manifolds. As far as I know they were initially defined as holomorphically convex complex manifolds, and then the other definitions (e.g. complex ...

**2**

votes

**4**answers

250 views

### polynomial zero within a square

Is there any complex polynomial $p$ of one variable having no zeros within the unit square:
$-1 < \Re(z) , \Im(z) < 1$ such that
$\left|p(0)\right|$ is strictly smaller than $\left|p(z)\right|$ ...

**0**

votes

**1**answer

335 views

### Question on Hartogs's Extension Theorem

Does Hartogs's extension theorem hold if one replaces the word holomorphic by analytic (of course still in several variables)?
For Hartogs's Extension Theorem see here:
...

**1**

vote

**0**answers

102 views

### Question about a oscillatory integrals on manifold

Let $M$ be a compact oriented Riemannian manifold without boundary.
Set $f(x)=a(x)+\sqrt{-1}b(x)$ be a complex-valued function on $M$,
where $a(x),b(x)$ are real-valued function on $M$.
Then, how to ...

**3**

votes

**2**answers

243 views

### j-invariant duplication, triplication and quintuplication formulae… how?

I am interested in finding the derivation of the duplication, triplication and quintuplication formulae for Klein’s j-invariant, which are equations (13) – (24) of the corresponding page (Klein’s ...

**6**

votes

**2**answers

503 views

### Reason for studying coherent sheaves on complex manifolds.

Hello everybody! I would be interested in knowing, what the reason is for investigating coherent sheaves on complex manifolds. By definition a sheaf $F$ on a complex manifold $X$ is coherent, when it ...

**2**

votes

**0**answers

158 views

### Univalent functions with non-negative coefficients

Is anything non-trivial known about univalent functions with non-negative coefficients?
Let $U$ be the unit disc, and $f$ a univalent (=injective) holomorphic function, $f(0)=0$,
$f'(0)>0$. ...

**1**

vote

**0**answers

287 views

### The relation between the weak Lefschetz theorem and the strong Lefschetz theorem

The Weak Lefschetz Theorem states that for a compact Kahler manifold, $Pic(X) \rightarrow H^{1,1}(X, \mathbb{Z})$ is surjective.
The Hard Lefschetz Theorem states that for a compact Kahler manifold ...

**3**

votes

**2**answers

131 views

### Weierstrass factorization with $L^2$ estimates?

Let $\Omega$ be a bounded domain in $\mathbb{C}$. Let $X$ be a discrete set of points whose boundary is in the boundary of $\Omega$. Can I find an $L^2$ holomorphic function which vanishes on $X$? ...

**1**

vote

**2**answers

127 views

### Solution of an infinite differential system

Let $r\in \mathbb N$ and $f$ be an entire function on $\mathbb C$. One assumes that for every $R\in\mathbb C[z]$ there exists polynomials $P_{i,R}\in\mathbb C[z]$ ($0\le i\le r$) not all zero such ...

**3**

votes

**3**answers

806 views

### Can the sum of two roots of unity be a root of unity?

Let $p$ be prime, and $z_0, z_1, ..., z_{p-1}$ be all the $p$-th roots of unity, i.e. solutions of the equation $z^p = 1.$
Is it true or false that a combination of two (or more, in general) of the ...

**1**

vote

**0**answers

71 views

### Volume of complex submanifolds

Let $X$ be a projective algebraic manifold and let $w$ be any Kahler metric on $X$. Then the volume of any complex submanifold of $X$ w.r.t $w$ is bounded below by a uniform positive constant.
My ...

**3**

votes

**9**answers

893 views

### functions of one complex variable: geometric theory

Can someone recommend a good textbook on functions of one complex variable which have good chapters on geometric theory, in English?
When I studied complex analysis, I used two
textbooks:
An ...

**16**

votes

**2**answers

437 views

### A direct proof of the Harer-Zagier recursion enumerating the ways to paste a 2n-gon to get a genus g surface?

In a 1986 paper, Harer and Zagier proved the recursion:
$$(n+1)e(g,n)=(4n-2)e(g,n-1)+(2n-1)(n-1)(2n-3)e(g-1,n-2)$$
where e(g,n) is the number of ways of grouping sides $S_1...S_{2n}$ of a 2n-gon ...

**1**

vote

**1**answer

234 views

### A counterexample to the Polya-Schur master theorems for half-planes

Given an integer $n\ge 1$ we say that $f\in C[z_1,\ldots,z_n]$ is stable if $f(z_1,\ldots,z_n)\neq 0$ whenever $\text{Im}\ z_i>0$ for all $1\leq i\leq n$.
Stable polynomials with all real ...

**2**

votes

**1**answer

115 views

### Entire functions of exponential type with small $L^1$ norm outside a finite real interval

I'm interested in entire functions of exponential type $\sigma$ (Bernstein space $B_\sigma^1$) following
$$\int_{-\infty}^{\infty} |f(x)|dx=1,$$
whose norm is as small as possible outside a range ...

**0**

votes

**0**answers

187 views

### Bounded mappings on unbounded domains

Ahlfors and Beurling (Ahlfors, Lars; Beurling, Arne Conformal invariants and function-theoretic null-sets. Acta Math. 83, (1950). 101–129.) provided an explicit example of a domain in $\mathbb C$ ...

**7**

votes

**3**answers

571 views

### Injectivity bounds for complex analytic functions

Let $$f(z) = z - \sum^\infty_{n=2} a_nz^n.$$
What is the largest ball around $0$ where $f$ is injective?
If we restrict to the case where $a_n \geq 0,$ it seems the radius should be given exactly by ...

**1**

vote

**0**answers

129 views

### The shape operator and an almost contact structure of a real hypersurface in $\mathbb{C}^n$

Let $S$ be an immersed real hypersurface in the Euclidean $\mathbb{C}^n$ with the standard complex structure $J$. Let $A:T(S)\rightarrow T(S)$ be the shape operator of $S$ (e.g. w.r.t. the outer ...

**2**

votes

**1**answer

309 views

### Where i can find the proof of Ostrowski theorem

Where i can find any proof of next theorem?
Theorem of Ostrowski: Let $\sum_{n=0}^{\infty}a_nz^n$ be a power series with radius of convergence 1, which is analytically continuable beyond unit disk. ...

**1**

vote

**1**answer

147 views

### On the set of zero radial limits of bounded analytic functions

Hi,
Let $f$ be a non-identically zero bounded analytic function in the open unit disk $\mathbb{D}$. It is well-known that $f$ has radial limits almost everywhere on the unit circle $\mathbb{T}$. Let ...

**1**

vote

**0**answers

56 views

### Are morphisms of intersection graphs of circle packings harmonic?

Let $P$ and $Q$ be circle packings on compact Riemann surfaces (along with some Riemannian metrics) $X$ and $Y$. Let $f\colon X\to Y$ be a conformal map taking each circle in $P$ to a circle in $Q$. ...

**0**

votes

**0**answers

97 views

### Boundary behavior of Harmonic functions

Assume that $f$ is harmonic in the unit disk $|z|<1$, with boundary function of bounded variation, such that $$\lim_{r\to 1}f(re^{it})= 0$$ for $t\in[0,\pi]\setminus \mathbf{Q}$, where $\mathbf{Q}$ ...

**2**

votes

**2**answers

348 views

### Bolza curve admits no anticonformal fixedpointfree involution

The Bolza curve B double covers the Riemann sphere with branching at the vertices of a regular octahedron. An affine model is given by the locus of $y^2=x^5-x$. How does one show that B does not ...

**5**

votes

**1**answer

326 views

### Branched Regular Cover over 4-times punctured sphere

This is probably trivial but has been bothering me all day.
Suppose $f:\Sigma_g\to \mathbb{S}^2$ is a $g+1$ fold branched conformal map with $\Sigma_g$ a connected genus $g$ surface and $f$ having ...

**3**

votes

**1**answer

268 views

### The right conformal map to make a certain picture

This is a follow-up to a question I asked a year ago, which was helpfully answered by Anton Petrunin:
Fitting a mesh to a density function
I am trying to come up with a way to make a picture of an ...

**8**

votes

**1**answer

504 views

### Polynomial with all zeros on a circle and many real coefficients

On a circle (or a line) $\Omega$ in the complex plane that is not symmetric w.r.t. the real axis, choose $n\ge5$ distinct points $z_1,...,z_n$ and consider the polynomial ...