The complex-analysis tag has no wiki summary.

**1**

vote

**1**answer

149 views

### 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 ...

**0**

votes

**1**answer

473 views

### 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 ...

**0**

votes

**1**answer

700 views

### 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 ...

**12**

votes

**2**answers

545 views

### 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 ...

**0**

votes

**0**answers

158 views

### 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 ...

**2**

votes

**3**answers

279 views

### 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 ...

**0**

votes

**1**answer

198 views

### 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 ...

**4**

votes

**3**answers

697 views

### 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 ...

**4**

votes

**1**answer

146 views

### 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 ...

**13**

votes

**2**answers

391 views

### Spectra of elements of a Banach algebra and the role played by the Hahn-Banach Theorem.

This problem was posed on Math StackExchange some time ago, but it did not garner any solutions there. I think that it is interesting enough to be posed here on Math Overflow, so here it goes.
Let $ ...

**0**

votes

**0**answers

288 views

### Help with an irregular integral

I am looking for help with doing the following integral :
$$\frac{1}{2\pi i}\int_{1}^{\infty}\ln\left(\frac{1-e^{-2\pi i x}}{1-e^{2\pi i x}} \right )\frac{dx}{x\left(\ln x+z\right)}\;\;\;\;z\in ...

**12**

votes

**3**answers

883 views

### If a formal power series over the complex numbers satisfies a polynomial identity, does it imply that the power series has a radius of convergence?

Let $ P(z) $ be a $\textit{formal}$ power series in $z$ that a priori may not have a non zero radius of convergence. Assume that $P(0) =0$.
Let $\Phi(w,z)$ be a polynomial in two variables, that ...

**3**

votes

**0**answers

359 views

### Real part of a function on a circle in the complex plane

Fix $0 < y <1$.
Assume that $n$ is a large positive integer. Let $H(z):= \sum_{i=1}^n \ln(z^i+i-1)$. Is it true that $\Re(H(e^{\xi+i\theta}))$ has asymptotically (for large $n$) a peak at ...

**1**

vote

**1**answer

533 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

**1**answer

256 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

**1**answer

144 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

**1**answer

559 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

**1**answer

342 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

**3**answers

534 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

**2**answers

418 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

**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 ...

**5**

votes

**0**answers

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

**1**answer

137 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

**1**answer

425 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

**2**answers

450 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

**2**answers

188 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

**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 ...

**2**

votes

**1**answer

265 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

**2**answers

648 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

**1**answer

110 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

**0**answers

156 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

**0**answers

132 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

**1**answer

312 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

**1**answer

994 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

**1**answer

250 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

**1**answer

219 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

**3**answers

391 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

**0**answers

229 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

**2**answers

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

**4**answers

574 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

**3**answers

644 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

**1**answer

322 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

**3**answers

438 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

**12**answers

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

**1**answer

154 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

**1**answer

314 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

**1**answer

226 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

**3**answers

762 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

**2**answers

385 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

**2**answers

268 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 ...