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3
votes
2answers
382 views

Corona Theorem in several variables

Hallo, I have read about the Corona Theorem (see link:http://en.wikipedia.org/wiki/Corona_theorem). From this one ca deduce that: Let $f_{1}, ..., f_{n}$ be holomorphic bounded functions on the unit ...
1
vote
2answers
186 views

k-Hyperbolic manifolds

A complex manifold $N$ is $k$-hyperbolic ($\dim N \geq k$) if any holomorphic map from $\mathbb C^k$ to $N$ has rank strictly less than k. Brody hyperbolic manifolds are $1$-hyperbolic for example. ...
0
votes
1answer
94 views

Bloch type function

I would like to know whether there exists an analytic function $f$ on the unit disk such that $$\sup_{|z|<1}|f(z)|(1-|z|^2)<\infty$$ and for every $|a|=1$, $$\limsup_{z\to ...
2
votes
1answer
234 views

Schlicht domain

What is a schlicht domain over $\mathbb{C}^n$? How is it different from a domain in $\mathbb{C}^n$? Examples?
6
votes
2answers
636 views

Schwarz type inequality

a) Is true the following statement. Let $h$ be analytic in the unit disk such that $$|h(z)|\le \frac{|z|^2}{1-|z|^2},$$ then $$|h'(z)|\le \frac{2}{(1-|z|^2)^2}.$$ a') Is true the following statement. ...
1
vote
1answer
106 views

About principal values and Wirtinger derivative

Let $K$ be a compact of the plane of Lebesgues measure 0 and $\Omega$ a domain containing $K$. Denote by $E$ the vector space of functions that are holomorphic on $\Omega - K$. I'm interested in ...
0
votes
0answers
151 views

Harmonic Function?

Hi, Let $\varphi : U \rightarrow \mathbb{C}$ be a holomorphic function, where $U \subset \mathbb{C}^{n}$ containing $0$. Is the function $u(x_{1}, ..., x_{n}, y_{1}, ..., y_{n}) := Im( ...
1
vote
0answers
109 views

an infinite series expansion in terms of the polylogarithm function

we have the complex valued function : $$f(z)=\sum_{n=0}^{\infty}a_{n}Li_{-n}(z)$$ we wish to recover the coefficients $a_{n}$ . the only thing i though would work is to try and come up with a function ...
1
vote
1answer
301 views

The Dirichlet series of the Hasse–Weil L-function

I have the following question: Is there is a paper claiming that the Dirichlet series of the Hasse–Weil $L$-function (associated with an elliptic curve over rationals) is of finite order. Thank you in ...
20
votes
1answer
972 views

Provable zero-free region for any entire function that analytically is similar to zeta(s)

Is there an entire function $f:\mathbb C\rightarrow\mathbb C$ such that for some $\delta>0$: $f(z)$ is bounded when $\Re z>1+\delta$ $f(z)$ is unbounded when $\Re z=1$ $f(z)$ grows ...
0
votes
1answer
236 views

Proof that Euler's function cannot be continued beyond the open disc?

It is claimed on it's Wikipedia page that Euler's function, defined by the infinite product $\prod_1^\infty(1-q^n)$ for $|q|<1$, cannot be analytically continued outside the unit disc, that is, the ...
5
votes
1answer
207 views

A converse of the maximum modulus Theorem

W.Rudin in Real and Complex Analysis(262) mentioned that Theorem Suppose $M$ is a vector space of continuous complex functions on the closed unit disc $\bar U$,with the following properites: ...
3
votes
3answers
358 views

Interesting results for open Riemann surfaces

As far as I know, interesting results for open Riemann surfaces are quite rare. One of them is the theorem of Gunning and Narasimhan, which asserts that every connected open Riemann surface admits a ...
1
vote
0answers
203 views

Mellin inverse of the Hadamard product rep. of the Riemann zeta function?

The floor function is given - by Perron's formula - as a Mellin inverse of the zeta function. namely : $$\left \lfloor x \right \rfloor=\frac{1}{2\pi ...
4
votes
2answers
203 views

Dimension of the full automorphism

Let $\mathbb P_1$ be the one dimensional complex projective space. What is the connected component of the full automorphism of $\mathbb C^*\times \mathbb P_1$. Is it a complex Lie group? I mean is it ...
6
votes
4answers
511 views

Seeking a Geometric Proof of a Generalized Alternating Series' Convergence

Let $z \in \mathbb{C} \backslash \lbrace 1 \rbrace$ with $|z| = 1$. We consider the following infinite series, which necessarily converges: $$S(z) := \sum_{n = 1}^{\infty}\frac{z^n}{n}$$ Note that ...
8
votes
3answers
574 views

What's the use of Malgrange preparation theorem?

The Malgrange preparation theorem,which is the $C^{\infty}$ version of the classical Weierstrass preparation theorem,says that if $f(t,x)$ is a $C^{\infty}$ function of $(t,x)\in\mathbb{R}^{n+1}$ near ...
-1
votes
1answer
307 views

holomorphic extension of a function [closed]

hi, I have the following question: let $U \subset \mathbb{C}^{n}$ be some open set containing zero. let $\tilde{U} = U \cap \mathbb{R}^{n}$. assume we have a real-valued analytic function $f : ...
7
votes
3answers
378 views

Is there an explicit formula for the modulus of an annulus given a parameterization of the inner and outer boundries?

Every open set in the complex plane homeomorphic to an annulus is biholomorphic to exactly one annulus whose inner radius is 1 and whose out radius is $r>1$. Each value of $r$ gives a different ...
15
votes
12answers
2k views

2D Problems Which are Easier to Solve in 3D

It sometimes happens that 1D problems are easier to solve by somehow adding a dimension. For example, we convert linear differential equations for a real unknown to a complex unknown (to use complex ...
0
votes
1answer
143 views

If Fatou set has a Multiply connected Fatou component implies every component of F(f) is bounded

Recently when I read a paper about Fatou component, I met the following theorem which cited in Professor Eremenko's paper "on the iteration of entire functions" If Fatou set has a Multiply connected ...
1
vote
1answer
295 views

Almost analytic continuation

Let $f\in S^{\alpha}$ for some $\alpha \in \mathbb{R}$(which means that f is smooth and satisfies $|D^{\beta}f|\leq C(1+|x|)^{\alpha-\beta}$),a function $\tilde{f}$ on $\mathbb{C}$ is called an almost ...
0
votes
1answer
194 views

Lambert $W_{-1}(x)$ as $x\rightarrow 0^-$: Asymptotic behavior

There are well known bounds for $W_0$, the "principal" real-valued branch of the Lambert-W function. For example, $W_0(x)$ lies between $\log x - \log\log x$ and $\log x - \frac{1}{2}\log \log x$, ...
1
vote
3answers
632 views

Is Euler's formula a theorem or a definition? [closed]

The first time I tried to understand Euler's formula was about 2 years ago. I didn't need it, I just randomly run across it, when trying to understand a Fourier transformation. The problem was, that I ...
1
vote
2answers
360 views

Real function to entire functions

Let $g:[0,\infty) \rightarrow \mathbb{R}$ be an increasing function. Is there a way to construct an entire function $f(z)$ such that $f(x)=g(|x|)$ for all real $x$?
1
vote
2answers
246 views

Characteristic Function of a Non-negative Random Variable Evaluated at a Complex Value

Suppose we have a non-negative random variable $X$ with density $p(x)$,and its characteristic function, evaluated at a complex number $z$, being $\phi(z)=E[e^{z X}]=\int_{0}^{\infty}e^{zx}p(x)dx$. It ...
0
votes
2answers
397 views

Diametrically opposite points go to diametrically opposite points under stereographic projection

I asked this question in MSE here: http://math.stackexchange.com/questions/184524/diametrically-opposite-points-go-to-diametrically-opposite-points-under-stereogr but I didn’t get an answer. I really ...
3
votes
0answers
312 views

confusion about rational maps and Fatou components

Dear fellows, I have come to another conclusion which must be wrong. Let $f$ be a rational map and let $U$ be a connected but not simply connected open subset of the Fatou set such that $f(U)$ is ...
4
votes
1answer
823 views

Complex Analysis - Analytic Continuation and Residual Integration

At first an example which I know how to treat. Let's say we have the following integral $\int dx \frac{1}{x^2+a^2}$ Now we do an analytic continuation of the constant $a$ to the complex numbers: $a ...
0
votes
1answer
164 views

parabolic immediate basins always simply connected?

Edit: So, my original question (stated below) was to find an error in my "proof" that immediate parabolic basins for rational maps are always simply connected. Since I have not received any answers as ...
3
votes
2answers
198 views

roots with negative real parts

Under what constraints on the parameters a,b, and c does the transcendental equation $$x+a+be^{-x}+ce^{-kx}=0$$ ($k$ is a constant) have ALL its roots with negative real parts? Alternatively, any ...
2
votes
3answers
530 views

Do approximately the same polynomials have approximately the same roots? [closed]

"If $U$ is an open subset of the complex plane, then matrices $X\in\textrm{M}(n,\mathbb C)$ all of whose eigenvalues belong to $U$ make up an open subset of $\textrm{M}(n,\mathbb C)$." Trying to prove ...
0
votes
1answer
396 views

The zeros of the digamma function

I wonder what work have been done on the zeros of the digamma function and on the values of the gamma function at such points (on the negative real axis). Any help please :)
2
votes
2answers
217 views

Computability of finding roots in holomorphic functions.

Consider a holomorphic function $f: S \to \mathbb{C}$ where $S$ is a path connected open subset of $\mathbb{C}$ (not necessarily simply connected). Is it then possible to determine if $f$ contains a ...
11
votes
2answers
662 views

Determining rational functions by their critical points

Fix an integer $d > 1$ and $2d-2$ points $P_1, \ldots, P_{2d-2}$ in the Riemann sphere (not necessarily distinct). Thanks to the work of Eisenbud and Harris on limit linear series (Inventiones, ...
0
votes
1answer
749 views

In Fourier Transforms: Positive Definite Functions, Bochner's Theorem, and Derivatives

I've been reading about Bochner's Theorem lately, but when I apply it to the derivative of a function, I seem to get a contradiction with the theorem. "Bochner's theorem states that a positive ...
4
votes
2answers
397 views

Zeros of incomplete exponential functions

Let $$ f_N(z) = \sum_{n=N}^\infty \frac{z^n}{n!},$$ where $N$ is a positive integer. Where are the (complex) zeros of those functions located? It would be sufficient for me to know what the ...
12
votes
0answers
582 views

What is the appropriate setting for Cauchy's Integral Formula?

For a $C^1$ function $f:U\to\mathbb{C}$, where $D\subset U\subseteq \mathbb{C}$ with piecewise $C^1$ boundary $\partial D$, we have the following generalized Cauchy integral formula: $$ f(\zeta) = ...
0
votes
0answers
107 views

a question on holomorphic mappings

Could any one give me a hint how to solve this one? Let $\Omega$ be a smoothly bounded pseudoconvex domain with nonconstant automorphism group $G$, Does $G$ contains $\mathbb{R}$?
1
vote
0answers
98 views

Sum of univalent functions

Let $f,g$ be a pair of univalent functions on a proper subdomain $\Omega$ of $\mathbb C$. Does their sum $f+g$ necessarily omit any complex value? Similarly, can all holomorphic function be written as ...
2
votes
2answers
406 views

Positive Fourier coefficients

Hi all, Is there any general way to construct a smooth 2pi periodic function which vanishes on an interval and has positive Fourier coefficients? And if that was too specific I can make this more ...
0
votes
0answers
178 views

Limit of an inverse Mellin transform

In Edwards' very nice book ``Riemann's zeta function'' the following integral comes up in section 1.14. Suppose $\beta = \sigma + i\tau$ with $\sigma > 0$. Suppose $x > 1$. Fix some real number ...
0
votes
1answer
284 views

Uniform convergence of a series to exponent [closed]

I'm trying to prove that in the complex plane $\left(1+\frac{z}{n}\right)^n$ converges uniformly to $e^z$ in every closed disc $|z|\leq c$. I thought about showing the sequence as a logarithm of ...
1
vote
1answer
172 views

( finite ) Blaschke product in higher dimensions ?

Hello, as we know, the (finite) Blaschke product $P$ in $\mathbb{C}$ or in $ \mathbb{R}^2 $ is defined by $\prod_{j=1}^{k} \frac{z-a_j}{1-\bar{a_j}z}, a_j \in \mathbb{D}$. I was wondering whether ...
13
votes
1answer
862 views

Certain functional equations for the Riemann Zeta function?

Referring to this question I asked on math.SE. I am posting a more generalized question here, for answers and further inquiry. For the Riemann zeta function, we know of the standard functional ...
0
votes
1answer
165 views

About an integral transform or generalized Laurent series

We start with a little of context. I needed that a function from $\mathbb{R}^+$ to $\mathbb{R}$ could be represented in the following form, not necessarily uniquely: $$ ...
5
votes
1answer
166 views

How does pseudoconvexity restrict the topology?

A domain of holomorphy in $\mathbb{C}^n$ has vanishing de rham cohomology in real dimensions greater than $n$ - half of it's cohomology is missing. Are there any other restrictions? If I give you a ...
11
votes
4answers
562 views

non-trivial zeros of partial zeta functions

Let $N,a\in\mathbf{Z}_{\geq 1}$. Define a partial $\zeta$-function as $$ \zeta(s;N,a):=\sum_{\substack{n\geq 1\newline n\equiv a\pmod{N}}} \frac{1}{n^s} $$ where $Re(s)>1$. Let $\omega$ be either ...
1
vote
1answer
360 views

How to study the nonregular part of a finite branched holomorphic covering?

A finite branched holomorphic covering is a holomorphic map $f : V \to W$ between holomorphic varieties $V$ and $W$ such that $f$ is a finite branched covering (in the topological sense) There is a ...
0
votes
1answer
268 views

An asymptotic series for the digamma function

As we know, there is an asymptotic series for the digamma function when $z>0$ is a real number. $$ \psi(z)=\ln z+\sum_{n=1}^{\infty}{\frac{B_n}{nz^n}} $$ $B_n$ is the first Bernoulli numbers. How ...