All Questions
3,560 questions
0
votes
1
answer
163
views
Solutions of complex linear difference equations
I'm wondering what the solutions of complex linear difference equations like
\begin{equation}
f(z+\eta_1)+f(z+\eta_2)+\ldots+f(z+\eta_n)=0,\ \ \ \eta_1 \cdots\eta_n \in \mathbf{C}
\end{equation}
look ...
- 3
7
votes
6
answers
959
views
Intersection of two Jordan curves lying in the rectangle
Given a rectangle $ABCD$.
Let a Jordan curve $L_1$ joins the vertexes $A$ and $C$
and all points of $L_1$ belong the rectangle.
Let a Jordan curve $L_2$ joins the vertexes $B$ and $D$
and all points ...
- 97
2
votes
1
answer
152
views
Reference for asymptotic estimates
In the way of studying an enumerative problem I have found that I have to estimate the Taylor coefficients of functions of the following form. For two polynomials $P(x)$ and $Q(x)$ with $P(0)=Q(0)=1$, ...
- 1,561
0
votes
1
answer
289
views
Can a doubly periodic function be locally univalent?
I am looking for a meromorphic doubly periodic function such that the function is locally univalent.
A standard meromorphic doubly periodic funtion is the Weirestrass $\wp$ function, defined as
$$\wp(...
- 1,350
8
votes
3
answers
1k
views
residue calculation for rational function
A colleague and I are working on a problem and part of it comes down to evaluating the residue of a rational function. In particular,
$$
\mathrm{Res} \left( z^{kn-1} \left( az^{m}+1 \right)^{-k}; r \...
- 91
7
votes
0
answers
204
views
Global generation of $S^n \Omega_X$ for a fake projective plane
Let $X$ be a fake projective plane, namely, a compact complex surface with
$$p_g(X)=q(X)=0, \quad K_X^2=9$$ and $K_X$ ample.
Since $K_X^2=9 \chi(\mathcal{O}_X)$, Yau's celebrated proof of the Calabi ...
- 66.3k
10
votes
0
answers
657
views
“Taylor series” is to “Volterra series” as “Laurent series” is to _________?
Preamble
My question is similar to an earlier MathOverflow question:
“Taylor series” is to “Volterra series” as “Padé approximant” is to _________? which I just answered (hopefully my first ever ...
- 446
4
votes
1
answer
150
views
Quantitative analytic continuation estimate for functions small except on a small set
This question arises as a variation of this question, which was helpfully answered in the negative. It turns out that for my application, a substantially weaker conjecture suffices, which fails to be ...
- 324
1
vote
1
answer
117
views
Resources on blended teaching and flipped classroom in undergraduate mathematics education [closed]
I'd like to learn about the implementation of "blended teaching" in general and "flipped classroom" in particular for the teaching of undergraduate mathematics. Can anyone ...
- 141
0
votes
0
answers
251
views
How to prove that the function $\lambda \mapsto \langle T_{\lambda}f; g\rangle$ is holomorphic?
Let $\langle;\rangle$ be the usual scalar product on $L^2(\Bbb R^2)$.
How to prove that the function $\lambda \mapsto \langle T_{\lambda}f; g\rangle$ is holomorphic on $\Bbb C^+_*=\{z\in\Bbb C:\text{...
- 531
23
votes
5
answers
11k
views
Example of continuous function that is analytic on the interior but cannot be analytically continued?
I am looking for an example of a function $f$ that is 1) continuous on the closed unit disk, 2) analytic in the interior and 3) cannot be extended analytically to any larger set. A concrete example ...
- 757
42
votes
4
answers
3k
views
What is the Krull dimension of the ring of holomorphic functions on a complex manifold?
Consider a connected holomorphic manifold $X$ and its ring of holomorphic functions $\mathcal O(X).$
My general question is simply: in which cases is the Krull dimension $\dim \mathcal O(X)$ known?
...
- 47.5k
0
votes
1
answer
206
views
Proper journal for a preprint in complex geometry
I ran into the very cryptic paper Proper analytic embedding of $\mathbb{C P}^1$ minus a Cantor set into $\mathbb C^2$ by Orekvov on proper holomorphic embedding of the complement of a Cantor set $C$ ...
- 779
17
votes
2
answers
2k
views
Why only $\bar\partial$ but not $\partial$ in Dolbeault cohomology
While I learn about $\partial$ and $\bar{\partial}$ operators, I had some questions about the reason why people prefer $\bar\partial$ over $\partial$. Specifically,
When defining Dolbeault ...
- 419
28
votes
9
answers
5k
views
Applications of algebra to analysis
EDIT: I would like to make a list of modern applications of algebra in analysis. By "modern" I will mean developments since the beginning of the 20th century. It is well known that classical linear ...
12
votes
2
answers
852
views
Visualizing holomorphic differentials on a compact Riemann surface?
It is a classical result that the vector space of holomorphic differentials on a compact Riemann surface of genus $g$ has dimension $g$. I am wondering if there is a way of visualizing this wonderful ...
- 82.7k
4
votes
4
answers
514
views
Does there exist a rational polynomial $P(x)\in{\mathbb Q}[x]{}$ such that $P(\zeta(s))=\zeta(P(s))$?
let $P(x)\in{\mathbb Q}[x]{}$ be a rational polynomial with $P(1) >1$ and $\zeta $ be the Riemann zeta function , I want to know if there exist a rational polynomial such that $P(\zeta(s))=\zeta(P(...
0
votes
1
answer
513
views
When is a product of two two-parameter Mittag-Leffler functions a Mittag-Leffler function?
I am studying properties of the two-parameter Mittag-Leffler function.
$$ E_{\alpha,\beta}(z)=\sum_{k=0}^\infty \dfrac{z^k}{\Gamma(\alpha k+\beta)}.$$
I am particularly interested in recurrences and ...
- 123
8
votes
1
answer
379
views
Is there a real-analytic way to derive the asymptotics of $\int_{-\infty}^\infty e^{ikx} e^{-k^4}\,dk$ as $|x|\to\infty$?
In The Fourier Transform of the quartic Gaussian $\exp(-Ax^4)$: Hypergeometric functions, power series, steepest descent asymptotics and hyperasymptotics and extensions to $\exp(-Ax^{2n})$, Boyd ...
- 922
2
votes
0
answers
79
views
Reference request for literature on the following function--power counting zeta function
I'll start by writing the character of interest, and describing some properties of it, before I get to the meat of the question. Any help is greatly appreciated, even an offhand suggestion/comment/...
- 388
3
votes
1
answer
427
views
Regularity of boundary of a level set of a $C^{1,\alpha}$ function
Let $f:\mathbb{R}^2\to\mathbb{R}$ be a $C^{1,\alpha}$ function. Denote $S_C=\{x\in\mathbb{R}^2\mid f(x)=C \}$ the level set of $f$ with value $C$.
What i want to ask is, if $S_C$ is nonempty for some $...
- 379
7
votes
1
answer
446
views
Improving Cauchy estimates?
Consider an entire function $f:\mathbb C \to \mathbb C$ that is real on the real line and even.
This function has a Taylor series of the form
$$f(z) = \sum_{i=0}^{\infty} a_i z^{2i} \text{ with } a_i \...
- 73
10
votes
1
answer
662
views
Hartogs' theorem for real-analytic subvarieties
One version of Hartogs' extension theorem is the following (see, e.g. [1], Theorem 5B, p. 50).
Theorem. Let $U \subset \mathbb{C}^n$ be open and let $X \subset U$ be a complex-analytic subvariety of ...
- 1,383
17
votes
4
answers
3k
views
Languages beyond enumerable
A language is a set of finite-length strings from some finite alphabet $\Sigma$.
It is no loss of generality (for my purposes) to take $\Sigma=\{0,1\}$; so a language is a set of bit-strings.
...
- 151k
2
votes
1
answer
90
views
Singularity on the boundary of domain of holomorphy
Let $\phi$ be a continuous function on the closed upper half-plane $\{ z\in\mathbb{C}: \operatorname{Im}(z)\ge 0\}$ and holomorphic in the interior.
Suppose that the function $x\phi(x)$ is in $C^1(\...
user473423
12
votes
1
answer
858
views
Is this function concave?
Let
$$h(u):=u^3 \left|\int_u^\infty \frac{e^{-i t}}{t^3} \, dt\right|$$
for $u>0$. Is the function $h$ concave on $(0,\infty)$?
(For context, see Proposition 4.4.4 and formula (4.4.21) in this ...
- 128k
23
votes
12
answers
15k
views
Textbook for undergraduate course in geometry
I've been assigned to teach our undergraduate course in geometry next semester. This course originally was intended for future high-school teachers and focused on axiomatic, Euclid-style geometry (...
-1
votes
1
answer
1k
views
Non trivial zeros of Riemann zeta function [closed]
Question Define $f(z)=(s-1)\zeta(s)$ where $s=\frac{1}{1+z^2}$ and $\zeta$ denotes the Riemann zeta function. Prove that if $\rho$ denotes the non trivial zeros of $\zeta(s)$ then, $$\sum_{|\alpha|&...
user292590
10
votes
1
answer
706
views
Upper bound for maximum modulus of polynomial on unit circle in term of the distribution of its roots
Let $P(z) = \prod_{i = 1}^n (z - z_i) \in \mathbb{C}[z]$ be a monic polynomial having all roots $z_1, \dots, z_n$ on the unit circle $\mathbb{T} := \{z \in \mathbb{C} : |z| = 1\}$.
What is known about ...
- 121
1
vote
1
answer
224
views
Bott-Chern cohomology for singular complex spaces
I'm reading the book 'An Introduction to the Kahler-Ricci Flow' (Lecture Notes in Mathematics 2086). They discuss Bott-Chern cohomology on complex spaces:
Let $X$ be a complex space(i.e. analytic ...
- 361
5
votes
4
answers
956
views
Limit of an integral vs limit of the integrand
I have a simple Fourier transform problem, originating from mathematical physics (system of linear PDEs), which reduces to taking the integral
$$
I(\alpha)\equiv\int_{-\infty}^\infty e^{ikr} \cfrac{\...
71
votes
11
answers
9k
views
How to introduce notions of flat, projective and free modules?
In the coming spring semester I will be teaching for the first time an introductory (graduate) course in Commutative Algebra. As many people know, I have been plugging away for a while at this ...
- 65.4k
6
votes
1
answer
219
views
Fixed points of a function $z\mapsto\overline{P(z)}$ of a complex variable
The equation $z^2=\overline{z}$ has four zeros and this example motivates us to generalize the problem to this form;
How many zeros does the equation $P(z)=\overline{z}$ have if $P(z)$ is a polynomial ...
- 559
4
votes
1
answer
177
views
Restricted Perron-Bremermann envelopes
Consider an upper semicontinuous function $\phi: \Omega \to (-\infty, \infty]$, in the sense that $\phi = \phi^*$, where $\phi^*$ denotes the upper semicontinuous regularization
$$
\phi^*(z) = \...
- 151
9
votes
3
answers
797
views
An integral identity
$\newcommand\la\lambda\newcommand\w{\mathfrak w}\newcommand\R{\mathbb R}$Numerical calculations and other considerations (The min of the mean of iid exponential variables) suggest that
$$\int_\R \frac{...
- 128k
4
votes
2
answers
376
views
$n-1$ quadratic forms for $n$ variables
If we have $n-1$ quadratic forms for $n$ variables $x_i$,
$$p_i(x) = M^{(i)}_{jk} x_j x_k$$
for $1\leq i \leq n-1$ and $1 \leq j,k \leq n$ then the zeros of all $p_i(x)$,
$$p_i(x) = 0$$
is generically ...
- 1,150
5
votes
2
answers
256
views
Can a holomorphic vector field have an attractor homoclinic loop?
It is well known that a holomorphic vector field $z'=f(z), z\in \mathbb{C}$ does not have any limit cycle.See the last paragraph of this post
Orbits space of real-analytic planar foliations
One can ...
- 366
-1
votes
1
answer
250
views
Significance of $N_0(T+1)-N_0(T)\sim \frac{1}{2\pi}\log \frac{T}{2\pi}$
Let $N(T)$ be the number of zeros of Riemann zeta function upto height $T$ in the critical strip and $N_0(T)$ be the number of zeros on the critical line.
What will be the significance of proving ...
user485483
3
votes
0
answers
158
views
Flatness of tensor products of analytic germs
Let $\mathcal{O}(\mathbb{C}^n)_0$ denote the local ring of germs at the origin of holomorphic functions on $\mathbb{C}^n$. Consider the obvious map
$$ \mathcal{O}(\mathbb{C}^n)_0 \otimes_{\mathbb{C}} \...
2
votes
0
answers
119
views
Reverse Sobolev inequality for holomorphic functions
Problem. Let $U \subset \mathbb{C}$ be open and $[0,1] \subset U$. Assume $f(z)$ is holomorphic on $U$. Is it possible to find a constant $C$ (that depends on $f$) such that, for all $0 \leq a < b \...
- 981
11
votes
1
answer
582
views
An extension of the Carlson's theorem in complex analysis
For the statement of Carlson's theorem please see,
https://en.wikipedia.org/wiki/Carlson%27s_theorem.
There is an extension of Carlson's theorem that says that the condition that $f$ needs to vanish ...
- 4,115
9
votes
0
answers
414
views
From holes in the image of peculiar functions to new perspective on the Riemann Hypothesis
I am working with the Dirichlet eta function $\eta(z)$, with $z=\sigma+it$, $\sigma > \frac{1}{2}$, and $t>0$. Let us define
$$\eta_n(z,\gamma)= \sum_{k=1}^n (-1)^{k+1}\lambda_k^{-\sigma} e^{-it\...
- 3,098
1
vote
2
answers
2k
views
A boundary behaviour of holomorphic functions
Let $f\colon\overline{\mathbb{D}}\to\mathbb{C}$ be a continuous function but that $f\colon\mathbb{D}\to\mathbb{C}$ is holomorphic. My question is
Can the restriction of $f$ to $\mathbb{S}$ assume its ...
- 1,453
1
vote
2
answers
2k
views
Fourier transform of a holomorphic function
Question: Is there a simple method for calculating the Fourier transform of a holomorphic complex function ${f{{\left({z}\right)}}}:\Omega\to{\mathbb{{{C}}}}$?
In order for my question to be well-...
- 547
12
votes
2
answers
1k
views
Short research articles
I am a masters student. I am interested in short articles which have counter examples and very few references. I want to write a short and interesting article.
For example; One of the best known ...
34
votes
6
answers
3k
views
Does seeing beyond the course you teach matter? The case of linear algebra and matrices
This question is indeed very important for me. Thus I hope you bear with my subjective explanations for a few minutes. I am an "excellent" lecturer, at least according to course evaluation forms ...
10
votes
1
answer
442
views
A question on the period integral of Rankin-Selberg $L$-function
$\DeclareMathOperator\GL{GL}$Let $\Pi$ and $\pi$ be irreducible automorphic representations of $\GL_{n+1}(\mathbb{A}_F)$ and $\GL_n(\mathbb{A}_F)$ respectively, where $n \geq 2$, $F$ is a number field ...
- 421
45
votes
5
answers
9k
views
Liouville's theorem with your bare hands
Liouville's theorem from complex analysis states that a holomorphic function $f(z)$ on the plane that is bounded in magnitude is constant. The usual proof uses the Cauchy integral formula. But this ...
- 7,062
5
votes
1
answer
660
views
Maximal analytic continuation of $\varphi(s)=\sum_{n\ge1} e^{-n^s}$
About 6 months ago I asked for an analytic continuation of $\varphi(s)=\sum_{n\ge1} e^{-n^s}.$
What's the maximal analytic continuation of $\varphi(s)?$
Doing this will help me better understand how ...
- 101
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