The complex-analysis tag has no wiki summary.

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### 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 ...

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**1**answer

551 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
$$B_n = n!\sum_{\lambda} \frac{\lambda^{2n}}{p'_n(\lambda)}$$
...

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**1**answer

305 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 ...

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**3**answers

507 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
$$ ...

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**2**answers

410 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 ...

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**1**answer

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 ...

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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 ...

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**1**answer

134 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 ...

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**1**answer

409 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 ...

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422 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 ...

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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. ...

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**1**answer

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 ...

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**1**answer

245 views

### Schlicht domain

What is a schlicht domain over $\mathbb{C}^n$? How is it different from a domain in $\mathbb{C}^n$? Examples?

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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. ...

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**1**answer

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 ...

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154 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( ...

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123 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 ...

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**1**answer

310 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 ...

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**1**answer

983 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 ...

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244 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 ...

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**1**answer

214 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:
...

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377 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 ...

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224 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 ...

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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 ...

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525 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 ...

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**3**answers

606 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 ...

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312 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 : ...

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### 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 ...

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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 ...

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150 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 ...

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303 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 ...

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213 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 $\log x - \frac{1}{2}\log \log x$, ...

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642 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 ...

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376 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 $x$?

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259 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 ...

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415 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 ...

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332 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 ...

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**1**answer

894 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 ...

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170 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 ...

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200 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 ...

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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 ...

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421 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 :)

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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 ...

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### 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, ...

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817 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 ...

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403 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 ...

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601 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) = ...

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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 $\mathbb{R}$?

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### 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 ...

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420 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 ...