The complex-analysis tag has no wiki summary.

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### 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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### 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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### 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 ...

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### 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 ...

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### 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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### 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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### 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 ...

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### Euler constant transcendality. [closed]

What attempts have been made to prove that $\gamma := \lim_{k\rightarrow \infty} \sum_{k=1}^{n} \frac{1}{k} -\log n$ is transcendental?
Any reference for stuff that has been proved on this constant?
...

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### 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 ...

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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
...

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### 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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### 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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### 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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### 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 $\Delta$ ...

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### 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 ...

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### 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 ...

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### Can infinite polynomials be expressed as a product of its linear factors?

Background:
In the 1700s, Euler solved the Basel Problem, which was to solve $\sum_{n=1}^\infty\frac{1}{n^2}$ in closed-form. Euler showed that it was equal to $\frac{\pi^2}{6}$ by first expressing ...

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### Belyi functions on non-compact surfaces; or: Building Riemann surfaces from equilateral triangles

Some background on (compact) Belyi surfaces
$\newcommand{\Ch}{\hat{\mathbb{C}}}$
A compact Riemann surface $X$ is called a Belyi surface if there exists a branched covering map $f:X\to \Ch$ such that ...

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### Monotonic Increase of the Ratios of Generalized Stirling Functions of the Second Kind

My motivation to the following question stems from the discussion at Complex Zeroes of Stirling functions of the second kind about the location of the complex zeroes of Stirling functions of the ...

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### Complex Zeroes of Stirling functions of the second kind

My motivation to the following question stems from the discussion at Zeros of "exponential" function about the real zeroes of Stirling numbers of the second kind, I am curious in exploring ...

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### A question on the area of the unit disc w.r.t. a complete conformal metric

Question: Let $\Delta$ be the unit disc in $\mathbb{C}$ and $\rho(z)|dz|^2$ be a complete conformal metric on $\Delta$ where $\rho(z)$ is continuous on $\Delta$. Let $a$ be the infimum of $p ...

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### A question about the limit of a sequence of pointwise convergent analytic funtions

Question: Let $\{f_n\}$ be a sequence of analytic functions on the unit disk $\Delta$ and suppose that $f_n$ converges to a continuous function $f$ on $\Delta$ pointwisely. (1) Can we say that $f$ is ...

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### variation of the obstacle in the obstacle problem

Suppose $D \subset \Bbb C$ with smooth boundary. Let $f \in C^{1,1}(D)$. Let $\varphi$ be the supremum of all members in the set
$$\lbrace g \in C^{\infty}(\overline{D})| g \ is \ subharmonic \ and ...