# Tagged Questions

**2**

votes

**1**answer

552 views

### Absolute convergence of logarithm of polynomial with positive coefficient ($\ln G(z) = \sum\limits_{i = 0}^\infty {{q_i}{z^i}} $)

Special problem: Let $G(z)$ be a probability generating function(pgf, the $z$ can be seen as real number or complex number), that is
$$G(z) = \sum\limits_{i = 0}^\infty {{p_i}{z^i}} ,(\left| z ...

**1**

vote

**0**answers

77 views

### Growth of sums of multiplicative functions over Squarefrees

When one looks at the quotient of Euler products $$\prod_p\frac{\sum_{\alpha=0}^{\infty}f(p^{\alpha})p^{-\alpha s}}{1+f(p)p^{-s}}$$
with $|f|\leq 1$, it is observed that the resulting expression ...

**0**

votes

**0**answers

26 views

### A question about the convention for the Plancherel measure on $\mathbb{H}^n$

Say I have to calculate the quantity, $Log Tr [ -\Delta - \frac{1}{4} + m^2]$ on $H^n$. Then looking up the spectral measure $\mu(\lambda)$ and the eigenvalues of the Laplacian ($= -\Delta = - ...

**1**

vote

**0**answers

74 views

### analytic continuation related to Chebyshev functions

Let $\psi$ be the Chebyshev function. I would like to prove that the function
$\sum_{n\ge0}(\sum_{k=0}^n\binom{n}{k}e^{\psi(k)})w^n$ can be analyticaly continued on an open set of $\mathbb C$ ...

**3**

votes

**1**answer

128 views

### Entire solutions of finite difference equations

Let $(E):\ \sum_{k=0}^n P_k(z)g(z+k)=0$ with the $P_k\in\mathbb C[z]$ a finite differences equation in $\mathbb C$.
Is it true that every any entire solution $g$ of (E)
of exponential type $<\pi$ ...

**1**

vote

**1**answer

211 views

### Request for help with two integrals

It would be great if someone can help me do these integrals - using numerical integration on Mathematica it seems that these converge - in what follows $a \in \mathbb{R}$ and $q \in \mathbb{N}$ and $n ...

**10**

votes

**2**answers

609 views

### Is there a holomorphic Morse-Bott lemma?

It asks for a generalization of the question in the post
Normal form for a holomorphic Morse function
Suppose $f$ is a holomorphic function on a complex manifold $M$ which has Bott type critical ...

**5**

votes

**1**answer

215 views

### Fourier expansion of Takagi-function (everywhere non differentiable function).

Let us consider Takagi-function defined by
$T(x) \colon\!= \sum_{n=0}^{\infty}s(2^nx)/2^n$,
where $s(x) \colon\!\!= \underset{n \in {\Bbb Z}}{\mathrm{min}} \,|x-n|$.
$T(x)$ has its period $1$, so ...

**2**

votes

**0**answers

262 views

### Separating the eigenvalues of a Hermitian matrix with a special block structure

I have a square matrix $J \in \mathbb{C}^{2n \times 2n}$ where,
$J=\begin{pmatrix} A&B \\\bar{B} & A \end{pmatrix}$
$A \in \mathbb{R}^{n \times n}$ and is ${\bf diagonal}$.
$B \in ...

**2**

votes

**1**answer

189 views

### almost holomorphic line bundles

Let $(L,\omega_L) \to (M,\omega_M)$ be a symplectic line bundle (symplectic line=real dimension 2) over a symplectic manifold $M$. Each of these objects can be equipped with an almost complex ...

**1**

vote

**0**answers

95 views

### Algebraic set given by sequence of polynomials

When working on some problem, I have end up with a following situation. Suppose
$P(z)=z^d+a_{d-1}z^{d-1}+\ldots+a_1z+a_0$ is a complex polynomial $d\geq2$ and that $\gamma$ and $\delta$ are non-zero ...

**0**

votes

**0**answers

68 views

### Finding the Fractional Derivative of This Function [duplicate]

I've been trying to find an answer to this question, and it seems as though the question has gone unanswered. The question regards the derivative of $f(x)=1+n^{-x}$ where $n$ is a natural number. Is ...

**-1**

votes

**1**answer

143 views

### Order of the zero of a meromorphic function under the action of $Gal(\mathbb{C},\mathbb{Q})$

Let's take $X$ a Riemann surface as an algebraic curve in $\mathbb{P}^n$.
The group $Gal(\mathbb{C},\mathbb{Q})$ (automorphisms of $\mathbb{C}$ which act as the identity on $\mathbb{Q}$) acts on $X$ ...

**11**

votes

**1**answer

294 views

### Minimize norm of a polynomial around a circle (count the solutions)

I already posted this question at MSE here, but as it received no significative feedback for a while I cross-post it here.
I also noticed a related question here on MO (which does not answer my ...

**10**

votes

**1**answer

277 views

### Reverse mathematics of meromorphic functions on Riemann surfaces

Various sources touch briefly on the reverse mathematics of measure theory and complex analysis. But I have found none on the uniformization theorem for Riemann surfaces or the existence of ...

**1**

vote

**1**answer

90 views

### Polynomials are dense in $A_{B(0,1)}$

Let $D(0,1)$ be the disk of center 0 and radius 1 and call $A_{D(0,1)}= \{ f:\overline{D(0,1)} \rightarrow \mathbb{C} : f \text{ is continuous and } f|_{D(0,1)} \text{ is holomorphic} \}$.
Can ...

**0**

votes

**0**answers

149 views

### Fractional Derivatives Of Sums

I have a question regarding the definition of a fractional derivative. I've searched, but I can't find a definition of fractional derivatives that explain the concept in terms of an operator on some ...

**0**

votes

**1**answer

153 views

### On complex exponential sum estimation

Let $c>0$ a real number, let $N$ a large natural number and let $e\left(x\right):=e^{2\pi ix}$. Is it true that $\forall k\in\left[1,\,2N\right]$, $k$ natural number, that ...

**7**

votes

**3**answers

871 views

### Objections to and arguments for the simplicity of all Riemann zeros

It seems to be that the simplicity of all the zeros is quite widely accepted as a working hypotheses, and it is known that a positive proportion are as such.
Titchmarsh explains in the last chapter ...

**2**

votes

**1**answer

106 views

### Singularities of the Remmert reduction of a holomorphic convex manifold

Let $X$ be a holomorphically convex manifold, namely, for any infinite discrete subset of $X$ there exists a holomorphic function on $X$ which is unbounded on this set, then a theorem of Remmert says ...

**2**

votes

**2**answers

467 views

### $ 2|f^{'}(0)| = \sup_{z, w \in D} |f(z)-f(w)|$ if and only if $f$ is linear

I know the following is a well-known result.
Let $D = B(0,1) \subset \mathbb{C} $ a disc, $f$ holomorphic on $D$. Show that $$ 2|f^{'}(0)| \le \sup_{z, w \in D} |f(z)-f(w)|$$
Furthermore, there is ...

**3**

votes

**0**answers

209 views

### What can be said about a function given its asymptotic expansion?

This is probably not a research level question but I honestly don't know how/where to look for techniques to reconstruct a function from its asymptotic expansion.
The expansion I want to know about ...

**4**

votes

**4**answers

300 views

### Extending a function from $\mathbb{Q}$ to the upper half plane $\mathbb{H}\cup\mathbb{Q}\cup\{i\infty\}$

Define the extended upper half plane $$\overline{\mathbb{H}}:=\{z\in\mathbb{C}: \mathrm{Im}(z)>0\} \cup \mathbb{Q} \cup \{i\infty\}.$$
To what extent can an arbitrary function on the rationals ...

**20**

votes

**2**answers

550 views

### Which smooth compactly supported functions are convolutions?

If $f,g$ are smooth functions with support in the interval $[-r,r]$ for some $r>0$, then their convolution $f*g$ is smooth with support in $[-2r,2r]$. My question is about the converse: Given ...

**0**

votes

**1**answer

128 views

### Approximation of analytic functions by Lp functions

Is there any reference where I can find something on approximation of analytic functions on a domain in complex plane by $L^{p}$ analytic functions of the same domain?

**0**

votes

**1**answer

71 views

### Cardinality of Picard exceptional values of holomorphic function without an isolated singularity

Picard's theorem says that if $f\colon D\to \mathbb{C}$ is an entire nonconstant function or holomorphic with an essential isolated singularity, then $\mathbb{C}\setminus f({D})$ is either empty or a ...

**2**

votes

**0**answers

67 views

### riemann mapping theorem for skew-fields of quaternions and beyond

Is there any known generalization of the riemann mapping theorem over skew-fields of quaternions and beyond or at least a conjectured formulation of it?
In a less focused way, how far does the main ...

**2**

votes

**0**answers

218 views

### Convergence of certain L-series

Suppose $|a_{n}| \leq 1$ completely multiplicative function assuming real values.
Suppose further that,
$
L(s)=\sum_{n} \frac{a_{n}}{n^s}
$
may be continued analytically to the left of $s=1$ a bit ...

**0**

votes

**1**answer

89 views

### Looking for methods/results for explicitly bounding iterations of rational functions

In Theorem 2.6.4 of Beardon's book, "Iteration of Rational Functions", he states the values for the first two coefficients of an iterated power series.
That is, suppose that
$$
...

**7**

votes

**3**answers

378 views

### Summation of a series

I would like to sum the series
$$
\sum_{n=0}^\infty \frac{1}{(1+a^2 (n+1/2)^2) ^{3/2}} .
$$
It arose when trying to perform a calculation on superconductivity. In particular I am interested in its ...

**0**

votes

**1**answer

189 views

### The dual space of the Dirac measures on an Abelian group

Let $G$ be a Hausdorff locally-compact Abelian group and $L^2(G)$ the Hilbert space of two-integrable complex functions on the group.
Question. What would be natural vector space $\mathcal{R}$ ...

**4**

votes

**1**answer

171 views

### A homogeneous but slightly asymmetric inequality

I need to prove the following inequality: for any $Z=(z_1,\dots,z_l)\in\mathbb{C}^l$ for any $p\geq 2$ and $l\geq 2$
\begin{equation}
\left|\left|\sum_{j=1}^l ...

**29**

votes

**1**answer

502 views

### Rearrangements of a power series at the boundary of convergence

Take some power series $f(z) = \sum a_n z^n$ with a finite non-zero radius of convergence. I can rearrange the terms of the series, say, to get a different infinite series
$$f_{\sigma}(z) = ...

**4**

votes

**1**answer

241 views

### Analytic continuation of a multiple contour integral

Let $W(t_1,\dotsc,t_n)$ a holomorphic function on some connected open set $U$ of $\mathbb C^n$.
Let $\mathbf t^{(0)}$ a point of $U$.
Assume that there exists a cycle $\gamma$ in $\mathbb C^m$ and a ...

**2**

votes

**1**answer

317 views

### Definition of a complex space

In the definition of a complex space (in the sense of Grauert), one defines a model space (one to which we require a complex space to be locally isomorphic) to be the support of the quotient sheaf ...

**0**

votes

**1**answer

91 views

### quotient of holomorphic functions at a point

Let $0\in U \subset \mathbb{C}$
be a small neighborhood of origin in the complex plane and $f_1,f_2\colon U\to \mathbb{C}$ be two complex valued functions such that
$$f_1(0)=f_2(0)=0$$
...

**0**

votes

**1**answer

515 views

### automorphism groups of unit disk $\mathbf{D}^n $ and unit ball $ B^n $

How does one compute the group of biholomorphisms of $\mathbf{D}^n = \{(z_1, \ldots, z_n) \in \mathbb{C}^n: \forall_i \; |z_i| \leq 1\}$, i.e., the unit polydisk, and of the unit ball $B^n = \{(z_1, ...

**3**

votes

**0**answers

64 views

### Question concerning Mellin transforms

I've recently come across a result I've been trying to generalize.
Say that $\phi(\sigma \pm iy) < Ce^{\frac{\pi}{2}|y|}$ in the strip $a < \sigma < b$
then then the following integral is ...

**3**

votes

**1**answer

179 views

### Local holomorphic equations for symplectic divisors

If $(X,\omega)$ is a symplectic manifold and $J\colon TX \to TX$ is an almost complex structure, we know that $J$ is actually a complex structure if and only if the Nijenhuis tensor $N_J(\cdot,\cdot)$ ...

**3**

votes

**1**answer

320 views

### Series of the inverse quadratic trinomial

Maybe it's a very simple question, but I have a problem with the following series
$$\sum\limits_{n=1}^{\infty}\frac{1}{n^2+pn+q},$$
where $p, q \in \mathbb{R}$. I know about five ways how to calculate ...

**3**

votes

**1**answer

228 views

### Explicit form for hermitian structure $h$ with respect to $\omega$

Let $(M,\omega)$ be a symplectic manifold. and $\pi:L\to M$ be a complex line bundle , we denote $h$ as hermitian structure,i.e. if $s,s'$ are smooth sections of $L$ and if $X$ is a vector field on ...

**12**

votes

**0**answers

214 views

### Analytic contraction of the Stone-Cech compactification of $\mathbb C$

Let $S$ be the Stone-Cech compactification of $\mathbb C$. Then any meromorphic function $f:\mathbb C \to \mathbb{CP}^1$ extends to $S$.
Do the meromorphic functions separate the points of $S$?
...

**0**

votes

**0**answers

93 views

### Arakelian's approximation theorem

I have a difficulty in understanding how one gets relation (2) in the proof of the Theorem, in the nice paper
[Jean-Pierre Rosay and Walter Rudin, Arakelian's Approximation Theorem, The American ...

**0**

votes

**1**answer

118 views

### Use of Jensen's inequality on a Riemann surface

Let $f:\mathbb{C}\to \mathbb{C}$ be entire and consider the composite function $g(z):=f(\sqrt{z^2 - 1})$ on $\mathbb{C}\setminus \big ((-\infty , -1]\cup [1,\infty )\big )$ on the branch of the square ...

**3**

votes

**0**answers

237 views

### The Poisson-kernel in the plane and polynomials

Let
\begin{align*}
p(z) & = \ \prod_{j = 1}^{n} (z - z_j) \, ; \ |z_j| \ = \ 1 \\
& = \ \prod_{j = 1}^{l+1} (z - z_j)^{M_j}
\end{align*}
be a non-constant complex polynomial with ...

**2**

votes

**0**answers

56 views

### Proving convergence of certain Mellin transforms

I am wondering about the following modified mellin transforms and if they are absolutely converging or not.
Let $\phi$ be some holomorphic function such that for all $y \in \mathbb{R}$ we have ...

**1**

vote

**0**answers

115 views

### Automorphism on F_2[[X,S]]

Let us define the automorphism $\sigma$ on ${\Bbb F}_2[[X,S]]$ such that
$\sigma \colon S \mapsto S + S^2 + S^3$
$\sigma \colon X \mapsto X + S$.
It is easy to see that the ideal $(S)$ is stable ...

**3**

votes

**1**answer

307 views

### Automorphisms of complete discrete valuation ring

Let ${\Bbb F}_2[[T]]$ be a c.d.v.r over ${\Bbb F}_2$. We consider the automorphism $\sigma$ of ${\Bbb F}_2[[T]]$ such that $\sigma \colon T \mapsto T + c_2T^2 + c_3T^3 + \cdots$ with $c_i \in {\Bbb ...

**4**

votes

**0**answers

79 views

### status of Invariant subspace problem on Krein Space

What is the status of Invariant subspace problem on Krein Space? What sort of developments have taken place in this area.

**0**

votes

**1**answer

127 views

### smooth holomorphic functions are CR on the boundary? [closed]

Is this true that any holomorphic functions in a domain with smooth boundary, and which is smooth on the boundary is a CR function ?