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1
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1answer
518 views

Entire functions of order 1 and type 0

There is no entire function of order 1 and type 0 which is bounded on the real line. This result can for example be found in the book "Entire functions" by Boas. I wonder under which weaker ...
0
votes
1answer
252 views

Fixed norm problem for analytic functions

Hi there, I have the following problems on my hand: Given arbitrary positive sub-harmonic function $l(x,y)$ on plane, for all analytic functions: $f=a(x,y)+ib(x,y)$ on complex plane, consider the ...
0
votes
1answer
137 views

Residue at an integration border in case of a limit?

I am dealing with an integral in a limit of the following shape: $$\lim_{\epsilon \to 0} \int_0^{\frac{\pi}{2}} dx \frac{2 \epsilon}{1-(1-\epsilon^2)\sin^2(x)}$$ Formally, assuming that ...
4
votes
1answer
556 views

Bernoulli number formula involving roots of Taylor polynomial of $\exp-1$

Please prove, give more symbolic or numeric support (counterexample!?), simplify or drop me a reference (or some vague hunch). We have $$B_n = n!\sum_{\lambda} \frac{\lambda^{2n}}{p'_n(\lambda)}$$ ...
2
votes
1answer
315 views

Constructing the imaginary part of a holomorphic function

Hallo, Let $f: U \rightarrow \mathbb{R}$ be a analytic function, where $U \subset \mathbb{C}^{n}$ is a open set (paracompact, starshaped or convex i.e. sufficiently nice). Does there exist a function ...
1
vote
3answers
514 views

Fourier Transform, for entire function

On THIS site, Alexandre used Fourier transform to solve the problem. If we use Fourier transform, how to define it to ensure any entire function has a FT? Classical FT is defined by $$ ...
11
votes
2answers
412 views

A “holomorphic” Peano curve?

A Peano curve is a continuous map $[0,1]\to [0,1]^2$ whose image is the whole square. I would like to know if on can obtain "holomorphic" Peano curves. Namely, is it possible to find a continuous ...
11
votes
1answer
1k views

On equation f(z+1)-f(z)=f'(z)

Original Problem If $f$ is an entire function such that $$ f(z+1)-f(z)=f'(z) $$ for all $z$. Is there a non-trivial solution? ($f(z)=az+b$ is trivial) And here is something uncertainty If we use ...
5
votes
0answers
135 views

Perturbations of zero-dimensional algebraic varieties

Let $P(z,w):\mathbb C^2\to\mathbb C$ be a certain polynomial, and consider $p(s,t)=P(e^{is},e^{it}):\mathbb T^2\to \mathbb C$ its restriction to the real torus. Assume generic situation, so by ...
2
votes
1answer
136 views

Growth in imaginary direction of an entire function with prescribed zeros

Let $\{z_n\}$ be an infinite sequence of complex numbers. Under which conditions on these numbers does there exist an entire function $f$ such that the $z_n$ are the zeros of $f$ and $|f(z)|< C ...
4
votes
1answer
414 views

Similarity between Cauchy-Riemann eqs and Hamilton equations.

I would like to see if this idea has any applications: So CR equations are given by: $$ \frac{\partial u}{\partial x} =\frac{\partial v}{\partial y} ; \ \frac{\partial u}{\partial y}=-\frac{\partial ...
3
votes
2answers
433 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
187 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
95 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
248 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
644 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
109 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
155 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
125 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
311 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
987 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
246 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
216 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
379 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
225 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
204 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 ...
7
votes
4answers
540 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
617 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 ...
-2
votes
1answer
315 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
418 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
153 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
305 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
217 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
747 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 ran across it, when trying to understand a Fourier transformation. The problem was, that I ...
1
vote
2answers
381 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
262 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
419 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
334 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
908 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
175 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
201 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
543 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
429 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 ...
12
votes
2answers
736 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
843 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
405 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 ...
13
votes
0answers
608 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 $\mathbb{R}$?