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### 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**

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244 views

### on completeness of R_mn, the set of all rational functions of type (m,n)

It is known from finite dimensionality of $P_r$, the space of all polynomials of degree less than or equal to $r$, that $P_r$ is complete with respect to uniform norm.
Considering ...

**2**

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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**

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**1**answer

211 views

### Is the homeomorphism class of a connected open set of C determined by its fundamental group?

Let $U,U'\subseteq\mathbf{C}$ be two connected open sets such that $\pi_1(U)\simeq\pi_1(U')$.
Q: Does this imply that $U$ is homeomorphic to $U'$?
In the case where the $\pi_1$'s are trivial then ...

**6**

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**1**answer

131 views

### Is the dual of $A^1(\Omega)$ known for arbitrary domains ?

Let $\Omega$ be a domain in the complex plane, and $A^1(\Omega)$ be the space of integrable holomorphic functions on $\Omega$ equipped with the $L^1$ norm (it is called the Bergman space).
If $\Delta ...

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81 views

### Pointwise bounds on Hardy space functions with regular boundary behaviour

Let $H^2$ denote the Hardy space on the strip $S:=\{z\in{\mathbb C}\,:\,0<\Im z <1\}$ (or the upper half plane), i.e. $H^2$ consists of all holomorphic functions $f:S\to\mathbb C$ such that for ...

**2**

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**1**answer

355 views

### Do there exist transcendental numbers which are not hypertranscendental?

A complex number is said to be hypertranscendental if the one is not a zero of any entire function with all rational Maclaurin coefficients. Does there exist a transcendental number which is not ...

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262 views

### Any closed form for series like $F(x)=\Sigma_{i=p}^{\infty}x^p$,p is prime$?

Any closed form for series like $$F(x)=\Sigma_{i=2}^{\infty}x^p,\text{p is prime}$$ or $$F(x)=\Sigma_{i=0}^{\infty}x^{i!}$$?
More generally,we can obtain a power series from decimal expansion of a ...

**11**

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711 views

### Anything special (historical?) about surface $x\cdot y\cdot z\ +\ x+y+z=0$?

QUESTION
I wanted to introduce and develop the complex logarithm from scratch. As the result I've arrived a couple of months ago at the following identity after which the road to complex logarithm is ...

**0**

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**1**answer

222 views

### Relations between automorphisms of field of rational functions and Mobius Transfomation

Proposition: If $F$ is a field, let $F[x]$ be the ring of all polynomials whose coefficients are in $F$. The fraction field of $F[x]$, denoted $F(x)$, is defined to be the ratios $r(x) = f(x)/g(x)$ ...

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94 views

### What is the corresponding version in the complex space of this proposition got in the real space real

How can I transform the following proposition that is gotten in $real$ space into the corresponding one used in the $complex$ space,i.e.,$A\in C^{n\times n},x=(x_1,...,x_n)\in C^n$ ?
suppose that ...

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185 views

### An integral with Gamma functions (Part 2)

I was wondering if there is a generalization of the integral discussed here to a case like,
\begin{equation}\int \frac{d^dq}{q^{\nu_1}\vert \vec{q} \pm \vec{k}_1\vert ^{\nu_2}\vert \vec{q} \pm ...

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147 views

### the “three-point” characterization of holomorphy

I want to know the source of the following "folkloric" fact about holomorphic functions.
It seems well described by the phrase:
The three-point characterization of holomorphy.
If F is a ...

**0**

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**0**answers

227 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 ...

**3**

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**1**answer

196 views

### Power series whose partial sums attain only finitely many values

I am wondering if there is a general explanation for the following phenomenon. The partial sums of the geometric series $\sum_{n\geq 0} x^n$ evaluated at a root of unity $\zeta\neq 1$ in the complex ...

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**1**answer

157 views

### A New Analytic Inequality

Consider an analytic function $f : U \longrightarrow \mathbb{C}$ where $U$ is an open subset of the complex numbers which contains the closed unit disk. I have $|f(x)| \geq 1$ for any $ x \in [-1, ...

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271 views

### Convergence at the radius of convergence

Suppose I have (roughly speaking) a multivalued meromorphic function $f(z)$ on all of $\mathbb{C}$ that is single-valued and holomorphic on the open unit disc and has some branch points of finite ...

**2**

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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 ...

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**1**answer

337 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:
...

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**1**answer

127 views

### Trivial Line Bundle-Riemann surfaces

What are the Hermitian metrics in a trivial line bundle on a Riemann surface X?
I read that a Hermitian metric in the trivial line bundle is equivalent to a $\mathcal{C}^{\infty}$ weight function ...

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210 views

### Green's function - Hyperbolic Riemann surface

A Riemann surface is said to be:
-Potential-theoretically hyperbolic if it has a non-constant bounded subharmonic function.
-Poincaré hyperbolic if it is covered by the unid disk.
Are this ...

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404 views

### Hyperbolic Riemann Surface

Let $X$ be a compact Riemann surface and $x\in X$.
Is $X - \overline{D(x,r_x)}$ hyperbolic?

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159 views

### Convergent series, asymptotics and truncation

In regard to the characteristics of certain "explicit formulae" arising in number theory, I am pondering the connection between the rate of convergence of series and the asymptotic order of the ...

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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, $f(0)=0$,
$f'(0)>0$. ...

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181 views

### Boundedness of an Oscillating Integral

Let $g(x):\mathbb{R}_{\geq0}\rightarrow\mathbb{R}$ be real analytic s.t. $g(0)\neq 0$ and $g(x)=O(x^{-2})$ as $x\rightarrow\infty$. I think the following integral should be bounded as ...

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139 views

### Growth of the reciprocal gamma function in the critical strip

I was wondering if there are any results that studied the growth of $\left|\frac{1}{\Gamma(s)}\right|$ where $0 < \Re(s) < 1$ and as $\Im(s) \to \infty$? Any pointers to any results, papers, ...

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79 views

### Monodromy of "complex Schwarz-Christoffel maps

Let:
-- $x_1,\ldots,x_n$ be $n$ distinct points on the complex plane $\mathbb C$.
-- $r_1,\ldots,r_n$ be $n$ real numbers.
Consider the map
$$ z\mapsto u(z)=\int^z \frac{1}{(x-x_1)^{r_1}\cdots ...

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203 views

### Meromorphic Functions as Distributions

For the function $\frac{1}{x}$ on the real line, one can use a modified principal value integral to consider it as a distribution p.f.$(\frac{1}{x}),$ and one can do a similar construction to make ...

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628 views

### A question from complex analysis

Let $n\geq2$. We assume $0<\alpha_n<\cdots<\alpha_2<\alpha_1<1$ and $0<\beta_n<\cdots<\beta_2<\beta_1<1$
, $\alpha_n=\beta_n$, and there exists $1\leq j_0\leq n$ such ...

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341 views

### Polynomial growth of Fourier transforms

I am looking for a theorem that guarantees the polynomial growth of a function $f$ defined by a Fourier integral, that is, when
$$f(x)=\int_{-\infty}^{\infty}F(y)e^{ixy}dy.$$
I am only interested in ...

**2**

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**1**answer

134 views

### A series question related to solution of Laplace equation

Let $u(x,y)$ be the solution of the Laplace equation $\Delta u=0$ on the unit square $(0,1)\times (0,1)$ with boundary condition:
$$ u(x,1)=1, u(x,0)=0, u(0,y)=0, u(1,y)=0$$
The series solution is ...

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854 views

### When does a modular form satisfy a differential equation with rational coefficients?

Given a modular form $f$ of weight $k$ for a congruence subgroup $\Gamma$, and a modular function $t$ with $t(i\infty)=0$, we can form a function $F$ such that $F(t(z))=f(z)$ (at least locally), and ...

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271 views

### is there any algebraic function that has a specific relation to transcendental one?

given transcendental function
$$F(x)=\sum_0^{\infty}a_i x^i,a_i\in \mathcal{N} \bigcup 0,\exists M \space a_i \leq M^i$$.
is there algebraic function $$A(x)=\sum_0^{\infty}b_i x^i,b_i\in \mathcal{N} ...

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291 views

### Variant of the Riemann Mapping Theorem for $Conf(\mathbb H^2)$?

According to the Riemann mapping theorem it is possible to map a simply connected open subset $B \subset \mathbb C$ into any other $B' \subset \mathbb C$ by a (bi-)holomorphic mapping. Moreover, such ...

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253 views

### infimum of the Calabi energy in a given Kahler class

Given a compact Kahler manifold $M^n$ and a Kahler class $\Omega$. We have the Calabi energy (or Calabi functional)
$$\textrm{Ca}(\omega)=\int_Ms^2(\omega)\omega^n,\qquad \forall \omega\in\Omega.$$
...

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444 views

### An integral with Gamma functions

I wanted some insights about the integral in equation A.5 (page 19) in this paper, http://arxiv.org/pdf/1301.7182.pdf
What is the derivation of this?
Is there something more general from where this ...

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661 views

### Analytic continuation of the Dirichlet generating series of the multiplicative partition function

Apologies for the lengthy question, but it seems it's the only way i can convey my thoughts. Consider the Dirichlet series:
...

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174 views

### the general form of entire functions satisfying:

Define $M(r,F)$ as the maximum value of the entire function $F$ on $|z|=r$
Also define the function $log_n = log(log(log(...)))$ n times with base e.
Now,
$ lim_{n\rightarrow ...

**2**

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327 views

### Laurent series expansion for ODE.

OK, then I read Frobenius method in mathworld (I learned when I took ODE 2):
http://mathworld.wolfram.com/FrobeniusMethod.html
My question is:
Are there any ODEs where the solution is given by full ...

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408 views

### Nth root of a matrix as an analytic function?

Let $A$ be a $k \times k$ invertible matrix over complex numbers.
If it possible to write its nth root as an analytic function (i.e. power series in $A$)?
EDIT: Complex coefficients can be functions ...

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190 views

### Class of functions that the Fourier inversion holds

The following is from Stein and Shakarchi's Complex Analysis:
For each $a>0$ we denote by ${\mathcal F}_a$ the class of all functions $f$ that satisfy the following two conditions:
1. The ...

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421 views

### Is this min not less than a min

Let $\mathbf{D}$ be the unit disk, is
$$\inf_{\begin{array}{c}
v_{1},v_{2},v_{3},v_{4}\in\mathbf{D},\\
v_{0}\in\mbox{convexhull}\left(v_{1},v_{2},v_{3},v_{4}\right)
\end{array}}\max_{0\le ...

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1k views

### The Paley-Wiener theorem and exponential decay.

Consider a function whose Fourier transform is supported on a half-ray:
$$
A(t)=\int_0^\infty \omega(E) e^{-iEt}d E,
$$
where I can suppose $\omega(E)\geq 0$ and any suitable regularity conditions on ...

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133 views

### Bounding an integral transform ouside a circle (or inside a strip)

Let $g$ be a symmetric unimodal probability distribution and $H$ be the right half plane.
We call
$$f(z) = \int_{-\infty}^\infty \frac{1}{z-i t}g(t)dt$$
the dispersion function of $g$.
Now, one can ...

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230 views

### Blaschke condition on upper half plane

if f is in $H^{1}$ the Hardy space with integrable norm, and if f is not identically zero, then the zeroes of f (certainly countable in number) satisfy the Blaschke condition.Can any "Blaschke ...

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88 views

### change of variable from `$dz\,dz^*$` to `$d\Re(z) \, d\Im(z)$`

I'm trying to solve a gaussian integral of the form
$$\int dz\,dz^*\exp(zKz^*)$$
I tried to make a change of variables so that $dz \, dz^*\to d\Re(z) \, d\Im(z)$ with the map given by
...

**3**

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405 views

### Undecidability and holomorphic functions (Reference request)

The goal of this question is to recall a certain mathematical fact -not in my field- that I was once briefly told and that I have fogotten, and also to collect similar results.
The fact, I think, ...

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103 views

### Counting complex solutions on a disk.

I am looking for an example of an analytic map $f$ on the complex domain for which there exist $r>0$ such that for $\gamma=\{z:|z|=r\}$, a.s. for $\theta \in [0,2\pi)$ we have that
...

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532 views

### Complexifying a real Banach space and its dual

A standard way to define the "complexification" $E_\mathbb{C}$ of a real Banach space $E$ is to define a complex linear structure on $E\times E$ by (1) $(x,y)+(u,v)=(x+u, y+v)$, (2) ...

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603 views

### On analytic function differentiable on the circle of convergence of its Taylor series

For simplicity, we assume that $f(z)$ is analytic in the unit disk $\Delta: |z|<1$ and continous in the closed disk $\overline \Delta$. Let $f(z)=\sum a_nz^n$ be its Taylor series in $\Delta$ ...