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2
votes
2answers
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
1answer
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
votes
1answer
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 ...
1
vote
0answers
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
votes
1answer
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 ...
1
vote
2answers
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
votes
2answers
708 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
votes
1answer
212 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 $r(x) = f(x)/g(x)$ ...
0
votes
1answer
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 ...
1
vote
0answers
184 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 ...
5
votes
0answers
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
votes
0answers
225 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
votes
1answer
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 ...
2
votes
1answer
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, ...
7
votes
0answers
270 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
votes
0answers
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 ...
0
votes
1answer
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: ...
0
votes
1answer
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 ...
0
votes
2answers
207 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 ...
2
votes
3answers
403 views

Hyperbolic Riemann Surface

Let $X$ be a compact Riemann surface and $x\in X$. Is $X - \overline{D(x,r_x)}$ hyperbolic?
1
vote
0answers
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 ...
2
votes
0answers
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, $f(0)=0$, $f'(0)>0$. ...
2
votes
1answer
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 ...
2
votes
1answer
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, ...
2
votes
1answer
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 ...
0
votes
2answers
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 ...
2
votes
2answers
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 ...
0
votes
2answers
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
votes
1answer
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 ...
17
votes
0answers
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 ...
0
votes
2answers
270 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} ...
2
votes
1answer
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 ...
4
votes
1answer
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) $$\textrm{Ca}(\omega)=\int_Ms^2(\omega)\omega^n,\qquad \forall \omega\in\Omega.$$ ...
2
votes
1answer
442 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 ...
8
votes
0answers
659 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: ...
1
vote
2answers
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
votes
3answers
326 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 ...
2
votes
4answers
406 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 ...
1
vote
3answers
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 ...
1
vote
1answer
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 ...
3
votes
1answer
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 ...
3
votes
0answers
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 ...
0
votes
1answer
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 ...
0
votes
0answers
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
votes
3answers
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, ...
0
votes
1answer
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 ...
9
votes
2answers
531 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) ...
3
votes
1answer
601 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 $\Delta$ ...
1
vote
1answer
149 views

Can a locally defined holomophic function which vanishes on a subvariety $V$ be written in terms of globally defined polynomials vanishing on $V$?

Given a complex algebraic, affine variety $X$ and a subvariety $V$, let $I_V$ indicate the ideal of polynomials on $X$ with vanish on $V$. Given an open subset $U$ of $X$, is it true that the ideal ...
0
votes
1answer
473 views

How to evaluate this complex integral !?

We have the following complex integral : $$\frac{1}{2\pi i}\int_{-i\infty}^{i\infty}e^{-\frac{\pi}{2}\cot\left(\frac{\pi}{s}\right)}\frac{x^{s}}{s}ds$$ Where $x\in\mathbb{R}:x>1$. i tried closing ...