All Questions
3,560 questions
27
votes
17
answers
9k
views
Using slides in math classroom
I am toying with the idea of using slides (Beamer package) in a third year math course I will teach next semester. As this would be my first attempt at this, I would like to gather ideas about the ...
1
vote
0
answers
173
views
Difference between affine quotient variety and a global quotient orbifold
Given a smooth affine variety $X$ and a finite group $G$ acting by automorphisms on $X$, the quotient space $X/G$ has the structure of an affine variety which is in general not smooth. However, in the ...
- 609
1
vote
0
answers
97
views
Do we have an equivariant Newlander-Nirenberg theorem for finite group action?
Let $X$ be a $2n$-dimensional $C^{\infty}$-manifold, let $G$ be a finite group acting smoothly on $X$. Let $J: TX\to TX$ be a map such that $J^2=-id$ and
$g_*Jg^{-1}_*=J$ for any $g\in G$.
We can ...
- 9,019
6
votes
1
answer
963
views
Is there a, in depth, classification of branch points in complex analysis?
Disclaimer: This question was originally posted in math.stackexchange.com and, after 30 days with no answers, I followed the instructions of this topic.
In complex analysis we have well known results ...
- 480
2
votes
4
answers
3k
views
Prove that the real part of this limit converges to $\frac{1}{2}$
Let $s= 1/3 + 14i$.
Prove that the real part of this limit converges to $\frac{1}{2}$:
$$
\Re\lim_{n \rightarrow \infty}
\left(
\left[
1-
\left(
\sum _{k=1}^n \frac{(-1)^{k-1} \binom{n-1}{...
- 1,183
4
votes
0
answers
72
views
harmonic envelope of holomorphy
Let $D$ be a (real) domain in $\mathbb R^n=\mathbb R^n+i\lbrace 0 \rbrace\subset \mathbb C^n$. Then, due to P. Lelong, there exists a maximal (complex) domain $\tilde D\subset\mathbb C^n$, $D=\tilde D\...
- 246
28
votes
5
answers
5k
views
Why are lacunary series so badly behaved?
Hi!
I just came across the Ostroski-Hadamard gap theorem, and while I can understand the proofs as well as the principle that the series $\sum_{n=0}^\infty z^{2^n}$ ought to have a singularity at ...
- 2,358
27
votes
10
answers
4k
views
What (fun) results in graph theory should undergraduates learn?
I have the task of creating a 3rd year undergraduate course in graph theory (in the UK). Essentially the students will have seen minimal discrete math/combinatorics before this course. Since graph ...
-4
votes
1
answer
213
views
Is the proof in "On Hilbert’s 8th Problem" published on Brazilian Journal of Probability and Statistics correct? [closed]
The article can be freely accessed here. The proof is only five pages. I am quite in doubt.
A new version (2021) of that paper can be found here.
- 263
20
votes
1
answer
747
views
Refinement of mean value conjecture for complex polynomials?
I was playing around with Smale's Mean Value Conjecture and found a curious formulation of it which would be stronger (and which may simply be false). It seems to hold for `generic' random polynomials ...
38
votes
2
answers
13k
views
What, exactly, has Louis de Branges proved about the Riemann Hypothesis?
I know this is a dangerous topic which could attract many cranks and nutters, but:
According to Wikipedia [and probably his own website, but I have a hard time seeing exactly what he's claiming] Louis ...
- 1,990
2
votes
1
answer
238
views
Extension of a Szegő Kernel to the boundary
Let $\Omega\subset\mathbb{C}^n$ be any smooth bounded pseudoconvex domain. Let $S$ denote the Szegő kernel of $\Omega$.
Recall: the Szegő kernel is a kernel of the Szegő projection $P: L^{2}(\partial\...
- 63
23
votes
4
answers
5k
views
Is $\ x\! \cdot\!\tan(x)\ $ integrable in elementary functions?
I'm teaching Calculus and my students asked me to calculate the integral of $\ x\! \cdot\!\tan(x)$.
I spent quite a lot of effort to do this, but I'm now even not sure if the integral could be ...
- 1,437
33
votes
11
answers
13k
views
Lecture notes on representations of finite groups
Next term I am supposed to teach a course on representation of finite groups. This is a third year course for undegrads. I was thinking to use the book of Grodon James and Martin Liebeck "...
0
votes
0
answers
57
views
Integrability of logarithm of a function in Hardy class
Assume that f is in $H^1$ (The Hardy space of holomorphic functions in the unit disk). I need an easy reference to the following fact $ \log |f|$ is in $h^1$, i.e. it belongs to harmonic Hardy space.
- 37
0
votes
0
answers
169
views
On $\sum_{\rho\in D} \text{dist}(\rho)=\frac{1}{2\pi i} \int_{\partial{D}}\log \zeta(s)\ ds$
Let $D$ denote a closed two dimensional figure as: $D=2+iT\to 2\to 2-\delta\to 2-\delta+i(T-\delta)\to \frac{1}{2}+\epsilon+i(T-\delta)\to\frac{1}{2}+\epsilon\to\frac{1}{2}-\epsilon\to \frac{1}{2}-\...
- 11
0
votes
1
answer
129
views
Seeking an integral formulation for an algebraic function
While working with a generating function for the Catalan numbers, I came across the integral representation
$$\frac1{1+\sqrt{1-4x}}=\frac1{2\pi}\int_0^{\infty}\frac{\sqrt{t}}{(t+\frac14)(t-x+\frac14)}\...
- 43.2k
1
vote
0
answers
77
views
Computing some closed trajectories of meromorphic quadratic differentials
I'm learning about meromorphic (!) quadratic differentials on Riemann surfaces, and would like to determine the closed trajectories [EDIT: I mean closed geodesics, not just closed trajectories; ...
- 121
6
votes
0
answers
252
views
Picard-Lefschetz formula for the quotient of a degenerating family of curves by a cyclic group
$\newcommand{\cD}{\mathcal{D}}\newcommand{\cX}{\mathcal{X}}$(This is a slight rephrasing and modification of the original question)
Let $D\subset\mathbb{C}$ be the complex unit disk. Let $X$ be a ...
- 10.7k
16
votes
1
answer
2k
views
What are some of the earliest examples of analytic continuation?
I'm wondering how Riemann knew that $\zeta(z)$ could be extended to a larger domain. In particular, who was the first person to explicitly extend the domain of a complex valued function and what was ...
- 3,699
31
votes
11
answers
13k
views
Uniformization theorem for Riemann surfaces
How does one prove that every simply connected Riemann surface is conformally equivalent to the open unit disk, the complex plane, or the Riemann sphere, and these are not conformally equivalent to ...
14
votes
1
answer
1k
views
What is the analytic continuation of $\varphi(s)=\sum_{n \ge 1} e^{-n^s}?$
My research has lead me to the following function that I'm trying to continue. 3 Months ago I posted this question to MSE, and have placed 3 bounties on the question, but haven't received an answer, ...
- 101
5
votes
0
answers
159
views
Higher Cardano formulae in terms of $\Theta$
Consider a polynomial in one variable with complex coefficient
$$f(x) = x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$$
we are interested in its roots. Babylonian solved for $n = 2$, and Cardano did it ...
- 5,230
1
vote
0
answers
155
views
Function involving argument of the Riemann zeta function
When $t$ is an ordinate of a zero of Riemann zeta function, we define \begin{equation}
f(t):=\frac{t}{2\pi}\log\left(\frac{t}{2\pi e}\right)+S(t)-\frac{1}{8}+\frac{1}{48 \pi t}+\frac{7}{5760 t^3}+...
- 19
1
vote
0
answers
116
views
Converse of transfer theorem : does asymptotic behaviour of coefficients describe the singularity?
I'm interested in Singularity Analysis after reading this Math.Stack post. In the "Analytic Combinatorics" book by P. Flajolet and R. Sedgewick, at p.390, the following transfer theorem (...
- 233
2
votes
1
answer
177
views
On a rigidity question related to spherical derivative of meromorphic functions
The spherical derivative of a meromorphic function is defined as
$$f^\#(z):=\frac{|2f'(z)|}{1+|f(z)|^2}.$$
The motivation is that given a piecewise smooth curve $\gamma$ in the complex plane, the ...
- 1,350
30
votes
4
answers
4k
views
Entire function bounded at every line
I would like to ask about, does there exists an entire function which is bounded on every line parallel to $x$ - axis , but unbounded on the $x$ - axis.
- 469
11
votes
4
answers
2k
views
Is an entire function, with nowhere vanishing derivative, always a covering map?
Assume that $f:\mathbb C\to\mathbb C$ is entire, and also that $f'(z)\ne 0$, for all $z\in\mathbb C$. Does that imply that $f$ is a covering map of $f[\mathbb C]$?
Clearly, $f$ is a local ...
- 2,933
2
votes
0
answers
109
views
Examples of compact non-Kähler complex manifolds with Kodaira dimension zero
Let $X$ be a minimal compact non-Kähler complex manifold. Suppose that Kodaira dimension $\kappa(X)=0$.
Is there a known example where the canonical bundle is not holomorphically torsion?
For ...
- 1,379
23
votes
1
answer
3k
views
More mysteries about the zeros of the Riemann zeta function
Update on 12/26/2020: I added the Appendix at the bottom: simplified formula for $|\zeta(s)|^2$, when $\frac{1}{2}<\Re(s)<1$.
Update on 1/5/2020: I added the section "more interesting ...
- 3,098
33
votes
7
answers
4k
views
Topology on the set of analytic functions
Let $H(D)$ be the set of all analytic functions in a region $D$ in $C$ or in $C^n$.
Everyone who worked with this set knows that there is only one reasonable topology
on it: the uniform convergence on ...
- 91.8k
6
votes
1
answer
570
views
Does it follow that $F^{(1)}(z)=F^{(2)}(z)$ for all $z \in \mathbb H$?
Let $\mathbb H= \{z \in \mathbb C\,:\, \textrm{Re}\,(z)\geq 0\}$ and for $j=1,2$ suppose that $F^{(j)}:\mathbb H\to \mathbb H$ is defined via
$$ F^{(j)}(z) = \sum_{k=1}^{\infty} \frac{a_k^{(j)}}{z+\...
- 4,115
10
votes
2
answers
740
views
Unconditionally convergent series in some functional spaces
Linked with this question and discussion
(Bilinear product of two summable families), I am very
interested in counterexamples/results about the following questions (cf the end).
First, I recall that a
...
- 3,738
52
votes
9
answers
26k
views
Is Galois theory necessary (in a basic graduate algebra course)?
By definition, a basic graduate algebra course in a U.S. (or similar) university with
a Ph.D. program in mathematics lasts part or all of an academic year and is taken
by first (sometimes second) ...
47
votes
3
answers
6k
views
Absolute value inequality for complex numbers
I asked this question on stackexchange, but despite much effort on my part have been unsuccesful in finding a solution.
Does the inequality
$$2(|a|+|b|+|c|) \leq |a+b+c|+|a+b-c|+|a+c-b|+|b+c-a|$$
...
- 953
2
votes
0
answers
68
views
Factorization of the shift of a canonical product
Let an entire function $F : \mathbb C \to \mathbb C$ of order $2$ be given by its canonical product
$$
F(z)=z^me^{az^2+bz+c} \prod_{w \in Z} E_2(z/w), \quad n \in \mathbb N,
$$
where $Z$ is the zero ...
- 190
8
votes
1
answer
640
views
Rate of decrease of the Fourier transform of standard mollifiers
What is the the rate of decrease of $|\widehat{f_p}(t)|$ (as $t\to\infty$), where $p\in(0,\infty)$,
$$\widehat{f_p}(t):=\int_{\mathbb R} e^{itx}f_p(x)\,dx,$$
and
$$f_p(x):=e^{-1/(1-x^2/p)^p}1(|x|<\...
- 128k
3
votes
0
answers
61
views
lower bound for zero multiplicity of function formed from determinant of functions
I have a family of single-variable analytic functions, $D(z)$, formed as follows.
Let $L$, $n$ and $T_{0}$ be positive integers, $\varphi_{1},\ldots, \varphi_{L}$ be analytic functions in ${\mathbb C}$...
- 31
3
votes
2
answers
381
views
Quotients of complex manifolds by symmetric group
$\DeclareMathOperator\Aut{Aut}$Let $X$ and $Y$ be two complex manifolds of dimension $n$, $n\geq 2$. Denote by $\Aut(X)$ and $\Aut(Y)$ the group of bi-holomorphisms of $X$ and $Y$, respectively. ...
- 175
3
votes
2
answers
428
views
Request for recommendation in probability and complex analysis
Could somebody kindly recommend to me some books which deal with the applications of the probabilistic method to problems in real and complex analysis or which consider probabilistic versions of some ...
- 826
1
vote
1
answer
86
views
Discrete uniqueness sets for the two-sided Laplace transform?
Let $f : \mathbb R_+ \to \mathbb C$ be a measurable and integrable function where $\mathbb R_+ = [0,\infty)$. The Laplace transform of $f$ is given by
$$
Lf(s) = \int_0^\infty f(x)e^{-sx} \, dx.
$$
A ...
- 190
42
votes
2
answers
4k
views
Abel and Galois (and Arnold)
Question Is there a connection between Abel and Galois theories of polynomial equations?
Recall that for every polynomial $p(x)\in \mathbb{Q}[x]$ (say, without the free coefficient), Abel considered ...
user6976
0
votes
1
answer
58
views
Analytic continuation of spline wavelets (reference request)
I would like to extend (cubic or higher degrees) spline wavelets to complex domain. First, does this continuation exist? Second, I appreciate it if anyone could point me to some references.
- 350
0
votes
1
answer
166
views
Construction of holomorphic function
I was trying to construct a holomorphic function $f$ on $\mathbb{C}$ such that
$|f|^2(z)=e^{(|z|^2-\frac{1}{2})^2}$.
I will be happy if someone can give me an idea how to do that. I would like also ...
- 35
4
votes
1
answer
191
views
Integral of $\ln(1/|f|)$ for $f$ bandlimited
I came across the following assertion: if $f\in PW_\infty([-a,a])$, i.e. the Bernstein space of functions in $L^\infty(\mathbb{R})$ which are the Fourier transform of a distribution supported on $[-a,...
- 319
4
votes
0
answers
134
views
Converse theorem for zeta universality
Voronin's Universality Theorem for $\zeta(s)$ is that the zeta function can uniformly approximate any non-vanishing holomorphic function to any degree of accuracy in the right-half of the critical ...
- 141
16
votes
5
answers
3k
views
Integrating powers without much calculus
I'll jump into the question and then back off into qualifications and context
Using the definition of a definite integral as the limit of Riemann sums, what is the best way (or the very good ways) to ...
- 30.1k
6
votes
1
answer
204
views
Do we have the Oka coherence theorem for finite group actions?
We first consider the sheaf of holomorphic functions $\mathcal{O}(\mathbb{C}^n)$ on $\mathbb{C}^n$. By Oka coherence theorem, $\mathcal{O}(\mathbb{C}^n)$ is coherent over itself.
Now we consider a ...
- 9,019
6
votes
1
answer
224
views
A cohomological variant of the second Riemann's extension theorem
Let $X$ be a connected compact complex manifold, $U$ an open subset of $X$ such that the complement of $U$ in $X$ is an analytic subset of codimension at least 2 in $X$. Let $O_X$ (resp. $O_U$) be the ...
- 5,050
4
votes
1
answer
151
views
Constructing proper holomorphic self-mappings of the unit disk with a given set of branch points and corresponding ramification degrees
I was trying to solve the following problem:
Let $f: D \longrightarrow D$ be proper holomorphic (so that means it is a Blaschke product with finitely many factors). Suppose $\{ a_1, ..., a_n \} \...
- 53