Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.

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0answers
62 views

extension for a complex operator

Let be $\lambda>0$. Put $$ L_{\lambda}=\Big[-\frac{\partial^{2}}{\partial z \partial \overline{z}}+\lambda^{2}|z|^{2} +\lambda\Big(\overline{z}\frac{\partial}{ \partial ...
0
votes
1answer
65 views

Branches of the tetration function

Letting $\eta = e^{1/e}$ where $e$ is Euler's constant, there exists a function $F(z)=\, ^z \eta$ with the following relevant properties. (I won't bother showing the existence of this function, or the ...
1
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0answers
47 views

Zeroes of trigonometric-like function

Consider a function $f(z)=\cos(z)\cosh(az)+\sin(z)\sinh(bz)$ for $z\in \mathbb{Z}, a,b \in \mathbb{R}$. Denote $D\subseteq \mathbb{R}^2$ being the set of such pairs $(a,b)$ of parameters so that NOT ...
7
votes
1answer
234 views

Rotation invariance of an integral

Consider the integral depending on 2 parameters $$f(\tau,x):=\int_{-\infty}^{+\infty}\frac{dp}{\sqrt{p^2+1}}e^{-\sqrt{p^2+1}\tau+ipx},$$ where $\tau >0,x\in \mathbb{R}$. This integral absolutely ...
6
votes
3answers
178 views

Logarithms of matrices in the disk-algebra

It is easy to see that within the disk algebra $A(D)$ $$\Delta(z):= \begin{pmatrix} 1&0\\z&1 \end{pmatrix}\; \begin{pmatrix} 1&1\\0&1 \end{pmatrix}= \begin{pmatrix} ...
2
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0answers
52 views

Functions of form $f(z)/f(z^*)$

I am doing my research in mathematical physics, and in the process I am getting functions of complex variable $z$ of form $F(z) = \frac{f(z)}{f(z^*)}$ In my case $f$ doesn't have any interesting ...
5
votes
0answers
86 views

Complex Stone-Weierstrass Type problem

I have come across this problem which resembles complex Stone-Weierstrass theorem except for a problem that the conjugate of the functions are not necessarily in the sub algebra. Suppose $\Omega$ is ...
2
votes
1answer
150 views

The spectrum and the tangent space of the algebra of holomorphic functions on a Stein manifold

Let $A$ be a Fréchet algebra over ${\mathbb C}$, and let us call the spectrum ${\tt Spec}[A]$ of $A$ the set of all characters, i.e. continuous multiplicative linear functionals $s:A\to{\mathbb C}$, ...
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votes
3answers
495 views

Can the topological algebra of analytic functions be endowed with a norm that defines the natural topology?

Right, so in my research in complex analysis I was puzzled by this question which may have a simple approachable answer that eludes me, but I am truly itching to find out and in need of it so I am ...
1
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0answers
34 views

explicit conformal map of sinus-shaped region

I wonder weather an explicit conformal map of a sinus-shaped region given by $\Re(z) \in [0, 1+\delta \sin(\Im(z)) ]$ onto say a strip or a ball is known (of course $0<\delta < 1$). Thank you.
2
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1answer
87 views

On some curves of real values of a rational function

For given parameters $a_{1},\dots,a_{k}\in\mathbb{R}$, define the rational function $\phi:\mathbb{C}\to\mathbb{C}$ as $$\phi(z)=\frac{1}{z}-a_{1}z-a_{2}z^{2}-\dots-a_{k}z^{k}.$$ The domain of its real ...
2
votes
2answers
88 views

Analytic continuation of a specific integral with respect to a parameter

The following integral absolutely converges for $Re(z)<0$ and is analytic in this domain: $$F(z):=\int_{0}^{+\infty}\frac{\sin r}{r}e^{\sqrt{r^2+m^2}z}dr ,$$ where $m>0$ is fixed. Question. To ...
0
votes
0answers
105 views

Contour integral of non holomorphic but continuous functions [closed]

Let us consider two functions $f(z)$ and $g(z)$, both holomorphic on a domain $U$ (a simply connected subset of $\mathbb{C}$). Thus, because of Cauchy's integral theorem, along any closed rectifiable ...
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0answers
74 views

Poincaré inequality for holomorphic line bundles

Let $M$ be a Riemann surface of genus >1, $g$ be an Hermitian metric on $M$. Let $E$ is a holomorphic negative line bundle over $M$, for example, the holomorhic tangent bundle of $M$. Let $h$ be an ...
2
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0answers
37 views

Estimates for hyperbolic metrics on nested surfaces

Suppose that $X$ and $Y$ are connected hyperbolic Riemann surfaces, with $X\subsetneq Y$. Let $\rho_{X,Y}$ be the density of the hyperbolic metric of $X$ with respect to that of $Y$; then $\rho_{X,Y} ...
4
votes
1answer
260 views

Two similar integrals

Let $n$ be a given even positive integer. We have the following integral \begin{align} ...
1
vote
1answer
53 views

Can I apply Lagrange inversion theorem? [closed]

I want to invert the equation $$\eta = g(x)\sqrt{1+g'(x)^2}$$ to get $x$ as a function of $\eta$. Assume $g(0)=0$, $g'(0)=0$ and $g'(x)>0$ for $x>0$ (Think $g(x) = x^p$ for $p\geq 2$ integer). ...
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0answers
50 views

Complex integration over 1-singular chain

Let $f$ be a continuous function on $\mathbb{C}$ and assume that $\lim_{z\to \infty} zf(z) = \lambda.$ Let us note for all natural $n$ $$C_n = \{z \in \mathbb{C} : |z|=n\}.$$ Then, a usual fact of ...
1
vote
1answer
150 views

find solution of complex number recurrence equation

I have the following recurrence equation: $$(\mu\ n + \nu) f_{n} + J\Phi^{*} \sqrt{n+1}f_{n+1} + J\Phi\ \sqrt{n}f_{n-1} = 0$$ for complex numbers $f_{n}$ where $n = 0,1,2,3,...,\infty$ and complex ...
0
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0answers
31 views

Normality criterion based on Brownian motion

Consider analytic family $\mathcal{F}$ btw domains $U,V\subset \mathbb{C}$. For any $f\in \mathcal{F}$ we have time-changed Brownian motion $f(B_{t})=\widetilde{B}_{\int_{0}^{t}|f(B_{s})|^{2}ds}$. So ...
2
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1answer
130 views

Proof of Euler's reflection formula via rapidly decreasing Fourier series

Story I want to prove Euler's reflection formula by showing that \begin{equation*} f(s) = \sin(\pi s) \Gamma(s) \Gamma(1 - s) \end{equation*} is constant, where $s = \sigma + it$. It's easy to see ...
0
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0answers
41 views

How to get Wiener-Hopf decomposition of this matrix function

I met a matrix function $M(z)$ that I need to get its Wiener-Hopf decomposition that $M(z) = M_+(z) M_-(z)$ with $M_+(z)$ and $M_-(z)$ being analytic inside and outside the unit circle respectively. ...
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0answers
34 views

How to prove that arc segment vanishes

I have this integral: $$iNx^a\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{x^{Ne^{i\theta}}e^{i\theta}}{\Gamma(Ne^{i\theta}+a-m)\zeta(2Ne^{i\theta}+2a+n)\sin (\pi \left(a+Ne^{i\theta}\right))} ...
1
vote
2answers
77 views

Nonlinear PDE for a 2D foliation

I am trying to solve a 4th order nonlinear PDE for a real function $u(x,y)$ of two variables. It is too complicated to reproduce here but it exhibits the following two very nice properties: 1) if ...
1
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0answers
40 views

Modulus of Continuity for an Analytic Function on an Ellipse

This is a question which I stumbled upon while working on Legendre Polynomials, but it is actually a question in complex analysis. Consider: Given $f\in C^{\infty} (E)$, where $E_{\rho} \subseteq ...
4
votes
1answer
161 views

Embed a bordered Riemann surface into punctured Riemann surfaces?

Let $U$ be a bordered Riemann surface of genus $g$ with $n -1$ punctures and one hole (i.e., the border has one connected component). For any punctured Riemann surface $\Sigma$ of genus $g$ with $n$ ...
3
votes
1answer
84 views

Wiener-Hopf factorization of matrices

Given a $2\times2$ matrix, which entries are functions in the complex plane $$\hat{A}(z)=\left(\begin{array}{cc}a(z)&b(z)\\c(z)&d(z)\end{array}\right)$$Where $a(z),b(z),c(z)$ and $d(z)$ are ...
2
votes
0answers
108 views

The boundedness of an entire function along the imaginary axis

I am looking for an answer in the affirmative or the negative concerning the asymptotics of an entire function. The question, relatively simple to state, has proven very taxing to solve and requires ...
7
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0answers
159 views

Biholomorphic neighborhoods of the boundary of Stein domains

Let $(X_1,J_1)$ and $(X_2,J_2)$ be Stein domains with the same contact boundary $(Y,\xi)$. Under what conditions does there exist a biholomorphism between a neighborhood of their respective boundaries ...
3
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1answer
89 views

$L^p$ norm of boundary values of holomorphic function

I am looking for an estimate of the following form: Suppose that $D\subset \mathbb{C}$ is a simply connected domain. Suppose that $F$ is holomorphic and bounded on $D$ and can be holomorphically ...
1
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0answers
62 views

Explicit formula of biholomorphism between the rectangle and unit disk [closed]

From the Riemann mapping theorem we know that there exists a biholomorphism between the rectangle $R$ and the unit disk $D$, can we write this biholomorphic map explicitly?
0
votes
1answer
103 views

How to calculate the expected value of complex-valued random variable? [closed]

Suppose $\theta_1,\theta_2,\cdots, \theta_n$ are independent and identically distributed (i.i.d.) real-valued random variables and here we specifically consider the uniform distribution in the ...
3
votes
0answers
100 views

Siegel's article “The volume of the fundamental domain for some infinite groups”: trouble with understanding computations

This is the article I mentioned. While the idea of what Siegel is doing in order to compute the volume of the fundamental domain described in the article (the very first one, for there are discussed ...
6
votes
1answer
112 views

Convex Hull of univalent functions and Bieberbach Conjecture

Consider the class $S$ of univalent functions on the unit disk $D$ normalized so that $f(0)=0$ and $f'(0)=1$. Each function in $S$ satisfy the Bieberbach conjecture, that is the $n$-th coefficient in ...
-4
votes
1answer
177 views

Proof of formula for $\pi$ in terms of infinite number [closed]

What is the shortest proof of the formula $\pi=\frac{\infty\sqrt[\infty]{-1}-\infty}{\sqrt{-1}}$? Here we choose the principal branch of the root. Hopefully a fairly elementary proof can be provided ...
3
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0answers
83 views

State of the art for univariate complex polynomials factorization with algebraic coefficients

Let $\mathbb{K}:=\overline{\mathbb{Q}}$ be the field of algebraic numbers. We choose to represent an element of $\mathbb{K}$ as its minimal monic polynomial, which is a vector in some $\mathbb{Q}^n$. ...
6
votes
2answers
158 views

Criterion for deciding the conformal class of a metric on a complete surface

For orientable closed Riemannian surfaces $(S,g)$, there is a constant curvature metric $\overline{g}$ on $S$ that is conformal to $g$ in the sense that $\overline{g} = e^ug$ for some smooth function ...
2
votes
0answers
100 views

Asymptotic analysis of generating functions

Let $a_d\!\in\!{\mathbb R}^+$ with $d\!\in\!{\mathbb Z^+}$ be a sequence such that $$\limsup \sqrt[d]{a_d}=1\,.$$ Define $$F(z)=\sum_{d=1}^{\infty}a_d\,{\text{e}}^{d z}\,.$$ Suppose $F(z)$ admits an ...
2
votes
0answers
104 views

Extension to real number system [closed]

Suppose you have equation involving a number $s$ $s^2+ 1 = 0$, to solve it one needs to treat $s$ as complex number $s = \pm i$, and introduce $i$ as imaginary unit. Now suppose you have equation ...
5
votes
2answers
489 views

Holomorphy of a function with values in a Hilbert space

I asked the same question on MathStackExchange. EDIT: This question has now a open bounty worth +50 reputation on MSE. Denote by $\mathbb C^\infty $ the Hilbert space $\ell^2 (\mathbb C)$. Fix $1\leq ...
5
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0answers
87 views

Complex L^1 spaces; reference request

I have been doing a fair amount of research into a complex analytic modified version of the Mellin transform. I have hit a few roadblocks, and am hoping there may already be literature on the subject. ...
3
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1answer
139 views

Do some kind of maximum principle exist on complex manifold?

Consider a holomorphic function $u:C\to C^n$, as $|u|^2 $is sub-harmonic, it satifies a maximum principle. Do some general kind of complex manifold enjoy such property? Say, square of some distance ...
2
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0answers
43 views

Interpolation polynomial smaller than its function?

Let $q$ be a real number such that $q>1$ and $f$ be an entire function on $\mathbb C$ such that $\overline{\lim}_{r\to+\infty}\limits\frac{\ln|f|_r}{\ln^2r}<\frac{1}{2\ln q}$, where ...
3
votes
0answers
112 views

Automorphism groups of elliptic bundles

This is a question in complex geometry, but even for algebraic varieties I don't know the answer: Let $S$ be a smooth compact Kähler surface (for example a smooth complex projective surface) that is ...
3
votes
1answer
129 views

Growth comparision between an entire function and a related function

Let $p$ be a prime number, $\mathbb C_p$ be the completion of an algebraic closure of $\mathbb F_q\left(\left(\frac1T\right)\right)$ for the valution $v(x)=-\deg(x)$. Let $\sum_{n\ge0}a_nz^n$ be a ...
2
votes
1answer
102 views

The motivation and application of Nevanlinna second main theorem for small functions

I once read some books about Nevanlinna theory, most of them will discuss the Nevanlinna main theorem small function theorem under some conditions. While, I know little motivation of small function ...
3
votes
0answers
220 views

Funk-Hecke theorem on the complex sphere

I am interested in paper " Sharp constants in several inequalities on the Heisenberg group " of Rupert L.Frank and Elliott H.Lieb " http://arxiv.org/pdf/1009.1410v2.pdf. In this paper ( page 17 ), ...
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0answers
44 views

Explicit solution of a Cauchy-type singular integral equation with regular part

I am doing research on the Riemann boundary value problem for bi-half-planes, and in a certain case I was able to reduce this problem to a linear singular integral equation of the form ...
0
votes
1answer
71 views

a sum of ratios of quadratic forms

I have the following function that I would like to optimize over the value A $$f(A)=\sum_k \frac{\mathbf{y}_k^H\left[\begin{array}{cc} 1&0\\ 0& A \end{array} ...
1
vote
1answer
87 views

Visualization of non-Smirnov domains

Can one provide a graph of a non-chord arc(non-Lavrentiev) Jordan curve in the plane? That is, more of less, equivalent to a Jordan curve whose interior domain is a non-Smirnov domain.