Holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.
3
votes
0answers
73 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 ...
3
votes
1answer
79 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 ...
-1
votes
0answers
21 views
Contour deformation and inversion [on hold]
I am working on a research problem and would like to know peoples views about what is contour deformation and its inversion. It will be appreciable if one can explain it with the help of an example.
2
votes
1answer
134 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
0answers
63 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
0answers
50 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 ...
10
votes
1answer
227 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 ...
-1
votes
0answers
126 views
Does the Cech cohomology of punctured complex tori vanish? [closed]
I am trying to understand the paper "Osservazioni sui nuclei di Bochner-Martinelli" of Paolo Zappa.
Let $\Gamma$ be an integer lattice in $\mathbb{C}^n$. $\Gamma$-invariant and $ d^{''}$-closed in ...
0
votes
0answers
61 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
0answers
89 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
0answers
18 views
Why is the set of points where a complex polynomial does not vanish is connected? [migrated]
Let p be a complex multivariate polynomial. Let C be the set of those complex tuples where p is nonzero. Then, C is connected.
5
votes
3answers
757 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
0answers
42 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
2answers
410 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 ...
2
votes
0answers
137 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 ...
19
votes
2answers
427 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
1answer
95 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
1answer
49 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
0answers
58 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
3answers
264 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
0answers
101 views
Contour integral (inverse Laplace transform) with arctan
I have what I think is a relatively simple contour integral involving arctan, but it is giving me difficulty. I would really appreciate any help.
The integral itself is, with $\tau$, $\lambda$, and ...
2
votes
1answer
263 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
1answer
72 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
1answer
172 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
0answers
116 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 $X$ admits a complex structure if and only if the Nijenhuis tensor $N_J(\cdot,\cdot)$ of ...
2
votes
0answers
152 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 ...
10
votes
0answers
190 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
0answers
76 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 ...
2
votes
0answers
142 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
0answers
96 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
1answer
284 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 ...
3
votes
0answers
64 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
1answer
92 views
smooth holomorphic functions are CR on the boundary?
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
1answer
134 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 ...
2
votes
2answers
243 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} ...
0
votes
1answer
76 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
1answer
173 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
0answers
47 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
1answer
76 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
0answers
107 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 ...
3
votes
1answer
165 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
1answer
138 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
2answers
181 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
1answer
89 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
1answer
105 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
2answers
214 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
0answers
59 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
2answers
207 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
1answer
102 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?
0
votes
0answers
98 views
On modes of convergence of the Dirichlet series for reciprocal powers of zeta
Let $\mu_k(n)$ denote the nth coefficient in the Dirichlet series for $\zeta^{-k}(s)$. It can be shown that if there exists a $u=u(x,s)$ such that
$$|\sum_{n\leq x}\frac{\mu_k(n)}{n^s} ...
