**0**

votes

**1**answer

101 views

### Comparing two hyperbolic structures on a surface

Let S be a compact hyperbolic surface (a compact riemann surface of genus ≥2). Let S′=S∖P where P⊂S is a finite subset of points. Then S′ is hyperbolic too.
Question: how is related the hyperbolic ...

**8**

votes

**1**answer

294 views

### Local polynomial form of holomorphic functions

It is well-known that a germ of a holomorphic function of $n\geq 2$ variables, with at most an isolated singularity (i.e. the singularity of the analytic variety defined by the zero locus of the ...

**2**

votes

**1**answer

74 views

### How a matrix of C^1 functions on a domain Ω in Cn generates a C^1 distribution in Ω

I am reading the paper Complex foliation generated by (1,1)-forms by M. Klimek. I have a problem understanding why is true a detail in one of the theorems:
Let $\Omega$ be an open connected subset ...

**0**

votes

**1**answer

373 views

### Recomendation of Complex variables book [closed]

I'd like to ask for a book of complex variables that includes a "large" discussion about the Dirichlet problem, Neumann problem, and problems like that, I have now read "basic complex analysis - ...

**3**

votes

**1**answer

192 views

### If $f(x)+f(2x)$ is quasianalytic, is $f(x)$ necessarily quasianalytic?

Assume that $f\in C^{\infty}$ and that $M_n$ is a sequence such that $$\sum_{n=0}^{\infty}\frac{M_n}{(n+1)M_{n+1}}=\infty$$
and for certain compact neighborhood of the origin $U$ of $\mathbb{R}$, ...

**1**

vote

**0**answers

249 views

### Does the property (P) holds true for the derivatives of $L$?

Let $$L(C,s)=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ be the Dirichlet series of the Hasse--Weil L-function of an elliptic curve $C$ over $ℚ$. As $s$ takes on real negative values, there are trivial zeros ...

**1**

vote

**1**answer

85 views

### Normal family and arithmetic progression

It is basic fact that for a holomorphic (or mermorphic) map $f$, the family of iterates $\{f^n\}_{n=1}^{\infty}$, is normal if and only if $\{f^{mn}\}_{n=1}^{\infty}$, is normal $\forall m\geq 1$.
...

**1**

vote

**0**answers

47 views

### A criterion or algorithm for polynomial which admits Markov partition on its Julia set

For a given polynomials $P(z)$, whether there exists general algorithm to check it admits a Markov partition on its Julia sets. (in finite computation time.)
May be it is more difficult for the ...

**3**

votes

**0**answers

140 views

### elementary proof for existence of point with minimal period 2 for entire function

Fatou proved a very interesting result:
for a transcendental entire function $f$, the second itarate
$f^{2}$ has at least has one fixed point.
(Using the technique of Picard theorem)
This result ...

**5**

votes

**1**answer

340 views

### Universal covering map from $\mathcal{H}$ to $\mathbb{C}\setminus \mathbb{Z}\oplus i\mathbb{Z}$ (the countably punctured complex plane)

It's a consequence of the uniformization theorem for simply connected Riemann surfaces that the universal cover of $\mathbb{C}\setminus(\mathbb{Z}\oplus i\mathbb{Z})$ ($\mathbb{C}$ punctured at all ...

**1**

vote

**2**answers

98 views

### general variational principle for the Julia sets of mermorphic function?

I have seen some attempt in considering topological pressure for Julia sets of exponential function, and elliptic function. However, there exists few reference according to my knowledge?
I want to ...

**4**

votes

**2**answers

412 views

### When is $\mathrm{Aut}(X)\times \mathrm{Aut}(Y)$ of finite index in $\mathrm{Aut}(X\times Y)$?

Let $\mathrm{Aut}(X)$ denote the group of biholomorphic autmorphisms of the (non-compact) complex manifold $X$. If $X$ and $Y$ are two (non-compact) complex manifolds, then $\mathrm{Aut}(X)$ and ...

**1**

vote

**1**answer

80 views

### Removing a hyperplane from flag manifolds

It should be known that if we remove a compact complex codimension one submanifold $X$ (hyperplane) of a flag manifold $Z=G/P$, then $Z\setminus X$ is a Stein manifold.
I was wondering if anyone can ...

**1**

vote

**1**answer

235 views

### Composite families of formal power series over $\mathbb C$ as algebraic variety

I was led to prove that the set of composite families $(f_j)_{j \leq k}$ of germs at $0\in \mathbb C^m$ of a holomorphic function (composite = sharing a common divisor belonging to the maximal ideal) ...

**1**

vote

**0**answers

57 views

### irreducible analytic decomposition of sets invariant under a group action

Let $U$ be a complex analytic space with an action of a finitely generated group $\Gamma$. Under what assumptions
is the following true:
Every $\Gamma$-invariant closed analytic subset of $U$ ...

**19**

votes

**5**answers

815 views

### Topology on the set of analytic functions

Let $H(D)$ be the set of all analytic functions in a region $D$ in $C$ or in $C^n$.
Everyone who worked with this set knows that there is only one reasonable topology
on it: the uniform convergence on ...

**3**

votes

**1**answer

176 views

### Fixed points on Riemann surface

It is well known theorem that for a conformal mapping $\phi$ from a bounded and planar domain $\Omega$ to itself has three fixed points , then it must be identity mapping. However, I cannot find a ...

**21**

votes

**4**answers

1k views

### Distribution of roots of complex polynomials

I generated random quadratic and cubic polynomials with coefficients in $\mathbb{C}$
uniformly distributed in the unit disk $|z| \le 1$. The distribution of the roots of 10000
of these polynomials are ...

**5**

votes

**1**answer

140 views

### Ubiquity/scarcity of non-analytically continuable functions

Suppose f(z) is a power series with positive integer coefficients centered at zero and positive radius of convergence. What is the likelihood that f has a dense set of singularities on its circle of ...

**2**

votes

**2**answers

162 views

### Is the Hausdorff dimension $Dim_{H}(J(f))$ of the Julia set less than 2 for quadratic rational map?

Let $f(z)$ be a quadratic rational map with two Siegel disks which can be normalized to be $$f(z)=z\frac{z+e^{2\pi i\alpha}}{e^{2\pi i\beta}z+1}.$$ If one of the ratation numbers $\alpha$ and $\beta$ ...

**1**

vote

**1**answer

139 views

### Forster's Theorem

I am looking for a detailed account, in English or French, of the main result and its proof from this paper by O. Forster
Zur Theorie der Steinschen Algebren und Moduln, Mathematische Zeitschrift 97, ...

**1**

vote

**0**answers

135 views

### Solving a system of rational functions

Given pairwise distinct numbers $c_1, c_2, \dots c_n \in \mathbb{C} \setminus \{0\}$, does the system of equations $$\frac{6}{c_k} + \sum_{i \ne k} \frac{2}{c_k - c_i} = \sum_{i = 1}^n \frac{1}{c_k - ...

**4**

votes

**4**answers

2k views

### Does the inverse Laplace transform of the square root exist?

Does the inverse Laplace transform, defined by the integral,
\begin{equation}
F(t) = \mathscr L_s^{-1}\left[\sqrt s\right](t) = \int_{c - i\infty}^{c + i\infty} \sqrt s ~e^{-st} ds
\end{equation}
...

**0**

votes

**2**answers

177 views

### A Functional Equation concerning analytic functions

Let $P$ be a polynomial and suppose $f : \Bbb{C}\longrightarrow \Bbb{C}$ is a non-constant analytic function such for all $z \in \Bbb{C}, f(z) = f(P(z))$. Clearly when $P$ is linear we can find such ...

**3**

votes

**1**answer

153 views

### The Integral Trick and An Equality in Nakajima's Lecture

In Nekrasov et al's series papers MNS, they calculate such kinds of integral
$$\frac{E_1 E_2}{N(2\pi i)^N(E_1+E_2) }\oint d\phi_1 \wedge d\phi_2\wedge ...\wedge d\phi_N \prod_{i<N} (-\phi_i) ...

**1**

vote

**1**answer

250 views

### Two different forms of Schwarz-Christoffel-Mapping of unit disk to rectangle. Are they identical?

I found two different equations for the Schwarz-Christoffel-mapping of a unit disk to a rectangle (which are the general form of the SC-mapping, I guess). The first, e.g. from Link, page 20, is
...

**2**

votes

**0**answers

123 views

### topology generated by irreducible componets of $\Gamma$-invariant closed sets

For an analytic space $U$ equipped with an action of a group $\Gamma$,
call a subset $Z\subseteq U$ $\Gamma$-closed iff
it is a closed analytic subset and each of its irreducible components
is an ...

**5**

votes

**1**answer

188 views

### Function transformation of exponentials

I came across the following function transformation:
$$
\sum_{j=-\infty}^{\infty} e^{(-j^2\cdot t)} = \sqrt{\frac{\pi}{t}} \cdot \sum_{j=-\infty}^{\infty} e^{(-\frac{\pi^2}{t}\cdot j^2)}
$$
where $ j ...

**2**

votes

**0**answers

82 views

### How can we describe explicitly the “infinitely complex differentiable” complex-valued local martingales?

Let $\mathcal{F}_t$ be a continuous filtration on a probability space, and let $B$ be a standard $\mathbb{C}$-valued $\mathcal{F}_t$-Brownian motion. Let's call a complex-valued process $X$, possibly ...

**1**

vote

**0**answers

55 views

### Is it obvious that the defining conditions to obtain a particular singularity are well defined on the quotient space?

Let $~f:\mathbb{C}^2 \rightarrow \mathbb{C}$ be a holomorphic function
vanishing at the origin, with
the following properties:
$$ f_{00}, ~f_{10}, ~f_{01}, ~f_{20}, ~f_{11} =0,~~f_{20} \neq 0 ...

**0**

votes

**0**answers

243 views

### Local System and Gauss-Manin connection

Fix a complex manifold $X$. Then if we have a line bundle $L=\mathcal{O}(D)$ together with Gauss-Manin connection $\nabla: L \rightarrow L \otimes \Omega^{1}_X$, we get the locally constant sheaf $F$ ...

**1**

vote

**1**answer

77 views

### Finite construction of lacunary functions using algebraic and certain analytic operations

Algebraic functions have a discrete set of singularities. Lacunary functions, e.g. $f(z)=\sum_{n=0}^\infty z^{2^n}$, have a continuum of singularities at every point of the boundary of their disk of ...

**4**

votes

**2**answers

159 views

### Comparing two Delaunay tessellations on a hyperbolic surface

Let $S$ be a closed hyperbolic surface (i.e. a compact Riemann surface of genus $\geq 2$) and let $P=\{p_1,\ldots,p_m\}$ be a non-empty finite subset of $m$ points in $S$. Let $\pi:\mathbb ...

**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

392 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

144 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

734 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

223 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

212 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

386 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

379 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

158 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

301 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

574 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

323 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

446 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

168 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

261 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 ...