**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 ...

**0**

votes

**0**answers

135 views

### Fractional Derivative of A specific function

I've tried and tried to do this problem myself, but I've hit some snags on the way.
I'm trying to take the fractional derivative of:
$f(x)=1+n^{-x}$ where n is an integer and $n\geq2$ and $x>1$.
...

**0**

votes

**0**answers

147 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 ...

**7**

votes

**3**answers

860 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

465 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

181 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 ...

**20**

votes

**2**answers

542 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

127 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

66 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 ...

**7**

votes

**3**answers

371 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 ...

**28**

votes

**1**answer

472 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) = ...

**2**

votes

**1**answer

314 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

87 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

461 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

**1**answer

178 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

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

212 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

92 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 ...

**3**

votes

**0**answers

233 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 ...

**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

306 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

78 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

125 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 ?

**2**

votes

**1**answer

185 views

### Can the natural boundary be part of the unit circle?

It is well-known that the function $f(z)=\sum_{n=0}^\infty z^{n!}$ is analytic in the open unit disk and it can not be extended analytically to any proper open superset of the unit disk, i.e., the ...

**3**

votes

**2**answers

344 views

### The holomorphic version of Galois theory

We identify the space of polynomials of degree n with $\mathbb{C}^{n+1}-\mathbb{C}^{n}$, that is an $n+1$ tuple $(a_{n},a_{n-1},\ldots,a_{0})$ with $a_{n} \neq 0$ is identified with $p(z)=a_{n}z^{n} ...

**1**

vote

**1**answer

83 views

### Which rate of growth of the Sobolev norms guarantees analyticity?

Let $u\in C^\infty(\mathbb T^k)$, where $\mathbb T^k$ is the $k$-dimensional torus. (Equivalently, $u\in\mathbb R^k$ and $u$ is $2\pi$-periodic with respect of each argument.)
We define the semi-norm ...

**5**

votes

**1**answer

264 views

### definition of accretive operator

A relation T with domain and range in a Hilbert space is said to be accretive if the
transformation $ (T − \lambda)/(T + \bar \lambda\ ) $
with domain and range in the Hilbert space is contractive for ...

**0**

votes

**0**answers

49 views

### Finding stable ideals of F_3[[X,S]] by group action.

Let k > 1 be a positive integer and define the action σ_k on F_3[[X,S]] by
σ_k: X ---> X + S + X^k
σ_k: S ---> S + S^3.
Then,
Conjecture: There exists a principal ideal (a) other than (S) such ...

**0**

votes

**1**answer

78 views

### A subspace of the algebra of infinitesimal CR automorphisms

Let $(M^{2n+1}, D, J)$ be a strictly pseudoconvex CR sturcture on a compact $2n+1$-dimensional manifold, where $D$ is a nonintegrable distribution of codimension 1. The algebra of the infinitesimal CR ...

**6**

votes

**0**answers

135 views

### Is there an analogue of Mostow-Palais equivariant embedding theorem for noncompact groups

Let $M$ be a (Hausdorff) smooth compact manifold and $G$ a Lie group acting smoothly on $M$. If $G$ is compact then, by Mostow-Palais theorem, there exists an equivariant smooth embedding $M\to ...

**4**

votes

**1**answer

177 views

### Contractibility of connected holomorphic dynamics?

Let $f$ be a function, holomorphic in $\mathbb{C}$, and $K(f)$ its non-escaping set :
$$K(f) = \{ z \in \mathbb{C} : f^{(k)}(z) \nrightarrow_{k \to \infty} \infty \} $$
Question : If $K(f)$ ...

**2**

votes

**1**answer

163 views

### Why do holomorphic maps increase extremal length?

Let $\mathcal{L}$ denote extremal length. The following theorem appears in http://arxiv.org/abs/math/0505191.
Theorem
Let $U$ and $V$ be Riemann surfaces, and let $f:U\rightarrow V$ be a holomorphic ...

**2**

votes

**3**answers

285 views

### Solutions of the $\overline{\partial}$ equation in the upper half-plane

I am looking for references on solutions of the $\overline{\partial}$ equation in the upper half-plane (with conditions on the boundary). But this subject is completely outside my area of expertise ...

**0**

votes

**1**answer

95 views

### Integral and conformal mappings II

Assume that $f_n$ is a sequence of conformal injective mappings of the unit disk $D$ onto the nested smooth Jordan domains $D_n\subset D$, such that $\cup_{n=1}^\infty D_n=D$ and $D_n$ are images of ...

**2**

votes

**1**answer

118 views

### Uniform convergence of conformal mappings

Assume that $D_n\subset D$, where $D$ is the unit disk is an increasing sequence of Jordan domains with smooth boundaries such that $\cup_{n=1}^\infty D_n=D$ and let $f_n: D \to D_n$ be conformal ...

**2**

votes

**2**answers

278 views

### Surjective entire functions without critical points

It is easy to construct surjective locally univalent holomorphic functions $f: {\mathbb D}\to {\mathbb C}$, where ${\mathbb D}$ is the open unit disk.
I am pretty sure that the answer to the ...

**2**

votes

**0**answers

73 views

### Analytic differential equations with estimates

I recently came across the following theorem of Martineau (and others). Let $P$ be a polynomial of $n$ variables with complex coefficients. If we identify naturally powers of partial derivatives ...

**2**

votes

**2**answers

271 views

### A question on Koebe theorem

Assume that $f$ is a conformal mapping of a bounded Jordan domain $\Omega$ onto the unit disk U such that $f(0)=0$. How to prove the following inequality
$(1-|f(z)|)\le K \sqrt{dist(z,\partial ...

**1**

vote

**1**answer

114 views

### Why there are so few conformal mappings in higher dimension? [duplicate]

I think the conformal mapping theory in the plane are quit interesting and useful in physics. I learned that there is very few conformal mappings in higher dimensions, is there any reason for that?

**2**

votes

**1**answer

218 views

### hayman's result for $ A^2(D) $

Consider injective homolomorphic functions $f:\mathbb D\to \mathbb C$ on the unit disk $|z|\leq 1$, normalized by the conditions $f(0)=0$ and $f'(0)=1$.
Thus for $|z|\leq 1$ we have $ ...

**1**

vote

**1**answer

163 views

### ddbar lemma for positive closed (1,1)-currents

This is probably fairly elementary, but does someone know how to prove the following or know a reference.
Let $X$ be a Kaehler manifold. Let $\theta$ be a closed $(1,1)$-form and $T$ be a closed ...

**1**

vote

**0**answers

95 views

### Forcing a set of complex points to be closed under conjugation

I am a PhD student and I have to address the optimization of a real scalar function of complex variables $f(z_1,z_2,\ldots,z_n)$. The function is real valued in my case because each complex $z_i$ ...

**0**

votes

**1**answer

117 views

### Chow lemma for complex spaces

Is there something like the Chow lemma for complex spaces (stating that every compact complex space is birational to a projective variety, or some variant of this, like: every proper morphism of ...

**1**

vote

**0**answers

74 views

### analytical behaviour of holomorphic functions on boundary

Is there any survey about criterions for a holomorphic function to be analytical on the boundary of a pseudo convex domain on which it is defined ?

**20**

votes

**0**answers

366 views

### Decidability of equality of elementary expressions

In the following definition the term expression is to be understood as a finite tree built from formal symbols without any predefined meaning assigned to them.
Define the set $\mathcal{E}$ of ...

**2**

votes

**2**answers

294 views

### Topological information via cohomology of sheaves

On a projective smooth variety $X$ over complex numbers (or rather compact Kahler) we have a specific set of sheaves, namely sheaves of holomorphic forms ${\mathcal \Omega}^p$ of various degrees. The ...

**-1**

votes

**1**answer

145 views

### Stone Cech compactification for exponential map

Recently I met with a problem related to Stone-Cech Compactification theorem
in Furstenberg's famous paper "non-commuting product."
I try my best to understand Stone-Cech compactification theorem by ...

**5**

votes

**1**answer

60 views

### An corona pair (a,b) satisfies (A_2)

Let $b$ be a non-extreme point in the unit ball of $H^\infty$. Let $(a,b)$ be a corona pair, that is $|a|+|b|$ is bounded away from zero in the unit disc. Also let $|a|^2$ satisfy Hunt-Muckenhoupt ...