All Questions
3,560 questions
2
votes
1
answer
274
views
Dimension of intersection of real analytic sets
Suppose that $A,B$ are real analytic subsets of $\Omega\subseteq \mathbb{R}^n$ and $p\in A\cap B \neq \emptyset$. Does the intersection inequality from complex analysis still hold, i.e. does the ...
- 513
12
votes
12
answers
2k
views
What are fun elementary subjects in probability?
I have to read several lectures on probability or applications of probability for high school students (of high level). There is no necessary part I must lecture, that is, my aim is just advertisement....
1
vote
2
answers
165
views
Natural boundary with non-zero "thickness"
Many series (in fact, "most" Taylor series) have a natural boundary at the unit circle. This boundary is only 1 dimensional, in the sense that only along the unit circle is it true that ...
- 1,730
4
votes
1
answer
925
views
A question on the use of fractional derivatives in Riemann Hypothesis
We already know that Riemann-zeta function on the critical band is defined as follows:
$$(1-2^{1-\alpha})\zeta(\alpha) = \sum_{k=1}^{\infty} (-1)^{k+1}k^{-\alpha},\quad \Re(\alpha) \in ]0, 1[ $$
Is ...
- 71
5
votes
1
answer
373
views
Inhomogeneous Cauchy–Riemann equation on complex plane with continous right hand side
Consider the equation $\bar{\partial} f=g$ on the complex plane. We may assume $g$ is compactly supported, but we need the case that $g$ is only assumed to be continuous. Is there a solution to this ...
- 527
23
votes
1
answer
5k
views
On equation $f(z+1)-f(z)=f'(z)$
Original Problem
If $f$ is an entire function such that
$$ f(z+1)-f(z)=f'(z) $$
for all $z$.
Is there a non-trivial solution? ($f(z)=az+b$ is trivial)
And here is something uncertainty
If we use ...
- 1,551
5
votes
0
answers
173
views
Counterexamples to the Ahlfors measure conjecture in higher dimensions
Let $\Gamma<SO(3,1)$ be a finitely generated, discrete group of isometries of $\mathbb H^3$. By work of Agol, Calegari, Canary, and Gabai, the limit set of $\Gamma$ is either the entire sphere $S^2\...
- 327
2
votes
1
answer
425
views
Cauchy's integral formula and essential singularities
Let $f$ be holomorphic at $z_0\in\mathbb C$. I would like to compute the integral
$$\oint_{\gamma_{z_0}} f(z)\, e^{\frac{1}{z-z_0}}dz,$$
where $\gamma_{z_0}$ is a small circle around $z_0$.
By ...
- 99
13
votes
5
answers
3k
views
A geometric proof of the Gauss-Lucas theorem
Motivated by a geometric proof of the Fundamental Theorem of Algebra we ask:
Is there a geometric proof for the Gauss-Lucas theorem? Since we are working on a half plane, can one imagine a possible ...
- 366
2
votes
0
answers
227
views
Minimisation and maximisation of the modulus of a complex valued function
I am new to complex analysis and I would be grateful to be guided in the following problem. We know that if $f$ is a function from $\Bbb C \to \Bbb R$, then $|f|$ is a function from from $\Bbb R^2 \to\...
- 826
2
votes
0
answers
211
views
Maximum modulus principle for vector valued functions of several complex variables
In the following paper: Shub and Smale, "On the Existence of Generally Convergent Algorithms", Journal of Complexity 2, 2-11 (1986), trying to understand Lemma 2 on page 4.
Paraphrased, ...
- 75
3
votes
1
answer
174
views
first order quasilinear partial differential equations
I am interested in understanding complex first-order quasilinear partial differential equations. In the real setting there is a huge literature dealing with such equations but in the complex setting, ...
- 35
4
votes
0
answers
241
views
Karamata's Abelian/Tauberian Theorem in the complex plane
The following result is well known (a particular case of Karamata's Tauberian Theorem for Power Series in Corollary 1.7.3 of Regular variation by Bingham, Goldie & Teugels):
Fix $c, \rho>0$. If ...
- 406
6
votes
4
answers
780
views
roots of higher derivatives of exponential
Consider the Gaussian function $f(z)=e^{-z^2}$ which has no zeros on the complex domain. Let $D$ denote derivative w.r.t. the variable $z$.
Question. Is it true that $D^nf(z)=0$ has only real roots ...
- 43.2k
5
votes
0
answers
138
views
The meromorphic continuation of Selberg-like integrals in the symmetric case
Introduction.
In connection with the question (1) (link below), I've been trying to understand the meromorphic continuation in $\alpha,\beta,\gamma$ of the Selberg-like integral $$ S_N(\alpha,\beta,\...
- 301
75
votes
3
answers
9k
views
Does a power series converging everywhere on its circle of convergence define a continuous function?
Consider a complex power series $\sum a_n z^n \in \mathbb C[[z]]$ with radius of convergence $0\lt r\lt\infty$ and suppose that for every $w$ with $\mid w\mid =r$ the series $\sum a_n w^n $ converges ....
- 47.5k
8
votes
2
answers
531
views
Embedding open connected Riemann surfaces in $\mathbb{C}^2$
This question arises in the context of a question asked on MSE: Are concrete Riemann surfaces Riemann domains over $\mathbb{C}$. Part of the answer to that question is the question above which is ...
- 1,566
17
votes
17
answers
3k
views
Readings for an honors liberal art math course
Our university has an Honors section of our "liberal arts mathematics" course. Typically 10-20 students enroll each Fall, with most of them extremely bright, but lacking the interest and/or ...
16
votes
5
answers
2k
views
"Classical" consequences of Bezout's theorem in dimensions $>2$
By Classical I mean something that could have been found before 1900 (say).
A well known consequence of Bezout's theorem for plane curves is Pascal's theorem http://en.wikipedia.org/wiki/Pascal'...
- 14.3k
3
votes
1
answer
329
views
Polynomial and rational approximation of continuous functions in $\mathbb{C}$
I am wondering what the state of the art is for polynomial and rational approximations to continuous/holomorphic functions in $\mathbb{C}$. The particular domains of interest are the closed unit ball $...
- 465
27
votes
5
answers
4k
views
What is the naming reason of poles in complex analysis?
A function $f: \textbf{C} \to \textbf{C}$ has a pole of order $k$ if $f(z) = \frac{g(z)}{(z-z_0)^{k}}$ where $g(z)$ is a nonzero analytic function. Why do we call it poles?
- 279
14
votes
1
answer
3k
views
How is the "conformal prediction" conformal?
The question is clarified by Prof.V.Vovk. See his answer below for discussion.
Recently, early works of Gammerman, Vanpnik and Vovk[4] are rediscovered by Wasserman et.al[1] and proposed it as a ...
- 8,071
13
votes
3
answers
1k
views
Do contact and CR structures have corresponding $G$-structures?
For an $n$-dimensional manifold $M$, almost complex and almost symplectic structures on $M$ correspond to reductions on the structure group of the tangent bundle, introducing a $\operatorname{GL}(n/2,\...
- 131
11
votes
2
answers
1k
views
A problem in additive combinatorics
$\color{red}{\mathrm{Problem:}}$
$n\geq3$ is a given positive integer, and $a_1 ,a_2, a_3, \ldots ,a_n$ are all given integers that aren't multiples of $n$ and $a_1 + \cdots + a_n$ is also not a ...
- 302
6
votes
3
answers
556
views
Acting with all rational rotations on a subset of the circle having positive measure do you fill almost the whole circle?
Set $\Gamma$ for the group of the roots of the identity: $\Gamma=\{z\in \Bbb C | z^n=1$, for some $n\geq 0\}$ and for $E\subset S^1$ set
$\Gamma E=\{z\zeta, z\in \Gamma, \zeta\in E \}$
A trivial but ...
- 441
3
votes
1
answer
267
views
Boundary behavior of an analytic function
Let $f$ be a function holomorphic in a simply-connected domain $D$; for simplicity, assume that the boundary $\partial D$ of $D$ is piece-wise analytic with positive inner angles. Let $0\in \partial D$...
- 702
2
votes
1
answer
173
views
Fixed points of subgroups of $\operatorname{Aut}(\mathbb{C}^n)$ that are connected complex manifolds
Let $\operatorname{Aut}(\mathbb{C}^n)$ be the group of holomorphic automorphisms of $\mathbb{C}^n$. For which subgroups $H$ of $\mathrm{Aut}(\mathbb{C}^n)$ do the fixed points of $H$ acting on $\...
- 87
2
votes
1
answer
672
views
Analytic continuation and convergence of a Riemann zeta related function
The functions in question are
$$L(s)=\sum_{k=1}^\infty \frac{\lambda(k)}{k^s}=\frac{\zeta(2s)}{\zeta(s)} \mbox{ and } L^*(s)=\frac{1}{2}\sum_{k=1}^\infty \frac{\lambda(k)+(-1)^{k+1}}{k^s}=\frac{L(s)+\...
- 3,098
12
votes
4
answers
2k
views
Interesting results for open Riemann surfaces
As far as I know, interesting results for open Riemann surfaces are quite rare. One of them is the theorem of Gunning and Narasimhan, which asserts that every connected open Riemann surface admits a ...
- 643
-4
votes
2
answers
228
views
An elementary-looking integral inequality
This might seem a bit easy but I still like to ask it for pedagogical reasons.
QUESTION. Is this inequality true for non-negative integers $n$?
$$\frac{\pi}2\int_0^1x^n\sin\left(\frac{\pi}2x\right)dx\...
- 43.2k
2
votes
1
answer
145
views
Estimate for an oscillatory integral of the first kind
I am confused in finding the right bound for the following oscillatory integral
$$I = \int_\mathbb{R} (\psi(2^{-k} \xi))^2 e^{i (y \xi - 3 \eta \xi^2 t)} d\xi.$$
Where $\psi(2^{-k} \xi)$ is a smooth ...
- 159
4
votes
1
answer
317
views
Taylor coefficients of Hadamard product
I imagine this to be a very classical question in complex analysis:
Consider the Hadamard product
$$g(\mu) = \prod_{n=1}^{\infty}E_1(\mu z_n),$$
where $E_1(z):=(1-z)e^z$ is the first elementary ...
- 73
3
votes
2
answers
320
views
Holomorphic connectedness in several complex variables
Let $\Omega$ be domain in $\mathbb{C}^n$. Suppose we have taken two distinct points from $\Omega$. Does there exist a domain $U$ in $\mathbb{C}$ such that there is a holomorphic function from $U$ to $\...
- 33
68
votes
1
answer
13k
views
Behaviour of power series on their circle of convergence
I asked myself the following question while preparing a course on power series for 2nd year students. Let $F$ be the set of power series with convergence radius equal to $1$. What subsets $S$ of the ...
- 683
6
votes
1
answer
276
views
How to solve the following ODE with a parameter?
I am considering the following ODE
\begin{equation}
\begin{split}
&\frac{d^2}{dy^2}u + \frac{\alpha}{(1+y^2)^{\frac{r}{2}}}u = \delta(y)\\
&\lim_{|y|\to \infty}u(y) = 0.
\end{split}
\end{...
- 903
0
votes
0
answers
52
views
A query regarding complex vector decomposition
Given a complex vector $V$ of length $n^2$. Each complex entry in the vector is of size (number of digits or bits required to express the complex number) $c$ for some constant $c$.
Is it always ...
- 13
32
votes
1
answer
1k
views
About a claim by Gromov on proper holomorphic maps
At p. 223 of his paper [G03], Mikhail Gromov makes the following claim:
Let $X$, $Y$ be two complex manifolds (not necessarily compact or Kähler) of the same dimension and having the same even Betti ...
- 66.3k
3
votes
0
answers
135
views
Holmgren's theorem on the boundary
Consider $\Omega$ a bounded Lipschitz domain, with $\gamma \subseteq \partial \Omega$ a $C^2$ manifold. I am interested in proving the following.
Let $u: \Omega\times [0,T]\rightarrow \mathbb{R}$ be ...
- 235
3
votes
1
answer
182
views
A question about average deviation of given $n$ complex numbers
This question just came to my mind and I have no idea as to how to approach it. Let $z_1,z_2,\dots,z_n$ be $n$ be any complex numbers in the unit disc $|z| \leq 1.$ Consider a complex function on the ...
- 826
12
votes
1
answer
521
views
Source of a quote by Ferdinand Rudio
I am looking for the source and context of this quote, found e.g. at St Andrews:
Only with the greatest difficulty is one able to follow the writings of any author preceding Euler, because it was ...
- 31.5k
6
votes
1
answer
490
views
A basis of holomorphic differentials on Fermat curves
I am currently reading the paper "Holomorphic Differentials of Generalized Fermat Curves" by Rubén Hidalgo. The case I am interested in is that of a classical Fermat curve $F_k$, which in ...
- 63
1
vote
0
answers
85
views
Holomorphic funtions in infinite dimensional Banach spaces
Let $f \in \mathcal{H}(U)$ a holomorphic function, where $U\subset X$ is an open balanced set in an infinite dimensional Banach space $X$, with power series around $0$
$$f=\sum_{n=0}^\infty P_n,$$
and ...
- 203
0
votes
0
answers
62
views
To find a DFT for complex functions on a semigroup
For a certain commutative semigroup of integer size $n$, $G=(\{1,2,\dots,n\},\circ: x\circ y\mapsto \min(n,x+y))$, consider all complex functions on it, denoted by $\mathbb C[G]$ or $\mathbb CG$. ...
6
votes
3
answers
975
views
For which functions is the (generalized) Riemann hypothesis known?
In [1], Lin Weng shows that the Riemann hypothesis (RH) holds for certain linear combinations of shifted completed Riemann zetas. Further, Deligne's result on the Riemann hypothesis for function ...
user1688
1
vote
1
answer
205
views
Complex polynomial-like functions with conjugate terms
Is there study on polynomial-like functions of the following kind?
$$f(z) = c_0 + a_1z+b_1\bar{z} + a_2z^2+b_2\bar{z}^2 + ...+ a_nz^n+b_n\bar{z}^n$$
My reason for studying it is polynomials are ...
- 175
7
votes
1
answer
378
views
Local optimum for Sendov's conjecture
For Sendov's conjecture, the distance 1 appears in the conjecture is tight, if one consider the polynomials $f_{n}(z) = z^{n} - 1$ for all $n\geq 2$. I wonder if this polynomial is the local optima ...
- 2,215
4
votes
1
answer
982
views
Non-induced analytic structures in complex-analytic case
In Lectures on Analytic Geometry, for complex-analytic geometry, seemingly one only considers maps $(\mathbb C,\mathcal M_{<p})\to(\mathcal A,\mathcal M)$ of analytic rings for $0<p\le1$ where $...
- 2,856
2
votes
0
answers
94
views
Computing $\int_0^{2\pi}\frac{e^{ikt}}{|e^{it}-e^{it_0}|^m}~\text{d}t$, where $k\in\mathbb{Z}$, $t_0\in\mathbb{C}$, and $m=1,3,5,\dots$
I am working on a project on accurate numerical quadrature where I need to compute the following integral in order to find my quadrature weights,
$$
\int_{0}^{2\pi}\frac{{\rm e}^{{\rm i}kt}}{\,\left\...
- 21
0
votes
1
answer
103
views
Measure of preimage of Jordan disk under entire map
Let $f\colon\mathbb{C} \to \mathbb{C}$ be an entire map. For simplicity assume that $f$ is of finite type, i.e., it has finite set $S(f)$ of singular values. $S(f) \subset \mathbb{C}$ is a minimal (...
- 41
1
vote
0
answers
213
views
Convergence of zeta Euler product with additional term
Let's consider the following Euler product ($s=\sigma+it)$:
$$ P(s)=\prod_{p \; \text{prime}} \frac{1}{1-p^{-s}} \; e^{-p^{-s}}$$
So for $\sigma>1$, it is clear the product converges and we have:
$$...
- 1,199