All Questions
3,560 questions
3
votes
1
answer
156
views
Green potential and Hölder continuity
Assume that $U$ is the unit disk and $g\in L^{3/2}(U)$. Define $$f(z) = \int_{U} \log\left|\frac{z-w}{1-z\bar w}\right|g(w)\frac{du \, dv}{\pi}, \ \ w=u+iv.$$ Is there an elementary proof of the fact ...
- 49
1
vote
0
answers
190
views
what belongs in a first university-level geometry course? [closed]
I know this is not really a research question, but I would like to ask it of research mathematicians, to see if there is a consensus. In a recent discussion on this topic, someone suggested that if ...
- 99
3
votes
1
answer
135
views
On well separated circular regions in the Riemann sphere and complex polynomials
It started with a conjecture I had, see A statement on complex polynomials, which was false for $n \geq 3$, as shown by Noam D. Elkies in his answer there. The present post is an attempt to salvage ...
- 5,215
0
votes
0
answers
304
views
Is Baire's theorem stronger than needed for functional analysis?
Many classic theorems in functional analysis involve using Baire's theorem to prove facts about topology that relate to maps between Banach spaces (or, more generally, F-spaces). The application ...
- 109
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
66
views
Vector recurrences (asymptotic property)
Fix $m\in \mathbb{N}.$
For each $n\in \mathbb{N},$ let $A_n\in \mathbb{M}_{m}(\mathbb{C}),$ $X_n\in \mathbb{C}^m,$ and $B_n\in \mathbb{C}^m.$ Suppose that
$$X_{n+1}=A_n X_n+B_n,$$
$$\lim_{n\rightarrow ...
- 3
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
4
votes
0
answers
308
views
Geometric interpretation of Theta functions and the Jacobi inversion problem
A great part of complex geometry and, algebraic geometry has been developed to address the theory of abelian integrals/functions. A very special problem that kept many great mathematicians busy was ...
3
votes
1
answer
427
views
Possible condition for a many variable holomorphic map to be locally surjective
Suppose $a \in \mathbb C^n$, $U$ is a neighbourhood of $a$, and $f: U \to \mathbb C^n$ is analytic. Let $b = f(a)$ and suppose also that $f^{-1}(b) = \{a\}$. Must the image of $f$ contain a ...
- 373
1
vote
0
answers
99
views
Seminorms ported by a compact
Let $X$ be a Banach space, and $U\subset X$ an open balanced set. A seminorm $\rho$ in $\mathcal{H}(U)$ is said to be ported by a compact $K\subset U$ if for all open set $V$ such as $K\subset V \...
- 203
2
votes
1
answer
174
views
Finding the repelling fixed point of an exponential, knowing only its attracting one
This question has been bugging me for a while, I have an answer that is working sufficiently for the program I'm using, but it is a tad slow, and let's say imprecise. It is not an overtly difficult ...
- 388
2
votes
0
answers
129
views
Existence of analytic function in disk algebra [closed]
Does there exist an analytic function $f\in A(\mathbb{D})$, where $A(\mathbb{D})$ is the disk algebra, such that $f(0)=0$ and the real part of $f(z)$ is strictly positive?
- 149
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
4
votes
1
answer
428
views
Ahlfors' proof of Bloch's theorem
In his pioneering paper An extension of Schwarz's lemma, Ahlfors proves the lower bound on the Bloch constant $B \geq \frac{\sqrt{3}}{4}$. The proof of this lower bound proceeds as follows:
Let $W$ be ...
- 137
4
votes
1
answer
333
views
Double sum over zeros of Riemann zeta-function
In a paper by Saffari and Vaughan there appears a complicated-looking double sum
$$\Sigma_1=\sum_{\rho_1}\sum_{\rho_2}\frac{(1+\theta)^{\rho_1}-1}{\rho_1}\cdot \frac{(1+\theta)^{\bar{\rho_2}}-1}{\bar{\...
- 465
3
votes
0
answers
257
views
Complex Hölder space
I already posted this question on math.stackexchange, but got no response and was suggested to post it here.
I came across a space in an ergodic theory paper, which I am calling here a (complex) ...
- 136
25
votes
5
answers
6k
views
When I can safely assume that a function is a Laplace transform of other function?
If I have a function and I want to represent it as being the Laplace transform of another, that is, I want to be sure that there is $\hat{f}(s)$ such that my function $f(x)$ can be written as:
$$f(x) =...
- 923
3
votes
0
answers
186
views
"Circulant-Vandermonde" matrix: in search of a formula
An $n\times n$ circulant matrix $\mathbf{X}_n$ has the form
\begin{align}
\mathbf{X}_n= \begin{bmatrix}
x_1 & x_2 & \cdots & x_{n-1} & x_n \\
x_2 & x_3 & \cdots & x_n&...
- 43.2k
4
votes
1
answer
904
views
The (measurable) Riemann mapping theorem
The Riemann mapping theorem says that a strict, nonempty open subset of the complex plane is conformally equivalent to the unit disk.
The measurable Riemann mapping theorem asserts the existence and ...
- 51
2
votes
1
answer
97
views
For estimation on the integral $g(t)=\int_{-t}^{t}\left\vert\sum_{k=1}^Ne^{ikx}\right\rvert^2dx$ for small $t>0$
For real numbers $t>0$ and $x$, let $f(x)=\sum_{k=1}^Ne^{ikx}$ and $g(t)=\int_{-t}^{t}\lvert f(x)\rvert^2dx$. Then $g(\pi)=\int_{-\pi}^{\pi}\lvert f(x)\rvert^2dx=2\pi N$.
I want to know is there ...
- 622
3
votes
1
answer
253
views
The monodromy in the proof of Little Picard via Klein's $J$
First of all, my question is in the context of formalising proofs in the proof assistant Isabelle/HOL, so my questions are a bit guided by what material we have in the complex analysis library there.
...
- 1,241
1
vote
0
answers
146
views
What can be said about cluster sets for power series of two variables?
I'm still trying to prove the continuity of a function $u$ which can be interpreted as the restriction of a power series of two variables, which I haven't managed to approach the right way yet. To ...
- 63
0
votes
1
answer
354
views
It is obvious for Mahler, not for me!
Let $k,m$ and $\rho$ be positive integers. In "Zur Approximation der Exponentialfunktion und des Logarithmus. Teil II"
Mahler considered the complex integrals $A_k(x)=\frac1{2i\pi}\int_{\...
- 3,998
6
votes
0
answers
395
views
Is there a residue sum formula in quaternionic analysis?
In complex analysis, there is a formula involving the residues of complex functions that one can employ to find the value of certain infinite series.
If the function $f: \mathbb{C} \to \mathbb{C} $ ...
- 4,829
2
votes
1
answer
177
views
Another combinatorial identity
Is it true that
$$\sum_{r=0}^p \sum_{i=0}^r a_{n,p,r,i}=0$$
for all natural $n$ and all natural $p\ge2n$, where
$$a_{n,p,r,i}:=\frac{(-1)^r (n+p-r-1)! (n p-i (r-i))}{i!(r-i)! (n-i)!
(p-r+i)! (n-r+i)! ...
- 128k
3
votes
0
answers
226
views
On an exact expression for the squares of the distances of the critical points to a given zero of a polynomial
Let $p(z) = \prod_{j=1}^{l+1} (z - z_j)^{M_j}$ be a complex polynomial of degree $n$, where the $z_j$ are distinct for $1, \ldots, l+1$. The first $l$ entries in the list $\{z'_1, \ldots, z'_{n-1} \}$ ...
- 429
2
votes
1
answer
651
views
Mellin transform of powers of gamma function
If $a>0$, the Cahen-Mellin integral gives
$$\DeclareMathOperator{\Res}{\operatorname{Res}}
\frac{1}{2\pi i}\int\limits_{a-i\infty}^{a+i \infty}\varGamma(z) u^{-z}dz=e^{-u}=\sum\limits_{m=0}^{\...
- 51
5
votes
0
answers
652
views
Nature of function as $x\rightarrow\infty$
I'm studying the limits and applicability of Abel Plana summation for different test functions (class of functions). In doing so this just pops out and couldn't handle the said integral so asked here (...
- 790
1
vote
1
answer
344
views
Is there a way to tie up even and "newly suggested odd" Riemann zeta values?
Define the sequence
$$a_s=(-1)^{\binom{s-1}2}\left(\frac{\pi}2\right)^s\frac1{2\cdot s!}\begin{cases} s\,E_{s-1}, \qquad \text{if $s$ is odd} \\ 2^{2s}B_s, \qquad \,\,\text{if $s$ is even};\end{cases}$...
- 43.2k
36
votes
6
answers
2k
views
When are some products of gamma functions algebraic numbers?
I want to know when certain expressions of the form
$ {\Gamma(r_1/m) \Gamma(r_2/m) \ldots \Gamma(r_j/m) \over \Gamma(s_1/m) \Gamma(s_2/m) \ldots \Gamma(s_j/m)} $
are algebraic numbers. These ...
- 14k
7
votes
2
answers
343
views
Bezout’s identity for analytic functions of several variables
In (single-variable) complex analysis, given analytic functions $f$ and $g$ with no common zeros, one can find analytic functions $u$ and $v$ such that $uf+vg=1$. I’d like to know if the same holds in ...
- 1,453
0
votes
1
answer
137
views
Zeros of entire functions with parameter
Let $f_w:\mathbb C \to \mathbb C$ be an entire function with $f_w(0)=1$ and at least one root for any choice of $w \in (0,1)$. Assume further that for a dense set of $w$ the function $f_w$ has ...
- 192
1
vote
0
answers
85
views
Holomorphic funtions in infinite dimensional Banach spaces
Let $f \in \mathcal{H}(U)$ a holomorphic function, where $U\subset X$ is an open balanced set in an infinite dimensional Banach space $X$, with power series around $0$
$$f=\sum_{n=0}^\infty P_n,$$
and ...
- 203
6
votes
2
answers
1k
views
Analyzing the decay rate of Taylor series coefficients when high-order derivatives are intractable
This could be a soft question. I am trying to show that the $n$-th Taylor series coefficient of a function is $O(n^{-5/2})$. However, because the function is a function composition of another function ...
- 212
3
votes
1
answer
129
views
Non-holomorphicity of Hecke regularization of weight-2 level-1 Eisenstein series
Originally asked on stackexchange since I figured it was a very elementary and silly error, but since I haven't had any response or interest in a couple days, I thought I'd post it here.
I have a &...
- 2,054
35
votes
2
answers
2k
views
Is it consistent with ZF that $V \to V^{\ast \ast}$ is always an isomorphism?
Let $k$ be a field and $V$ a $k$-vector space. Then there is a map $V \to V^{\ast \ast}$, where $V^{\ast}$ is the dual vector space. If we are in ZFC and $\dim V$ is infinite, then this map is not ...
- 156k
2
votes
1
answer
274
views
Dimension of intersection of real analytic sets
Suppose that $A,B$ are real analytic subsets of $\Omega\subseteq \mathbb{R}^n$ and $p\in A\cap B \neq \emptyset$. Does the intersection inequality from complex analysis still hold, i.e. does the ...
- 513
22
votes
1
answer
1k
views
Hadamard factorization of L-functions
I have already asked this question here in a different form, but really need an answer.
Let $L(s)$ be a "standard" $L$-function, say with Euler product, functional equation, etc...
(Selberg ...
- 13.1k
3
votes
1
answer
385
views
The average value of the modulus $|f(z)|$ of a polynomial with real coefficients
This question just came in my mind and I don't know whether this or similar questions have been considered in the literature. Consider a complex-valued polynomial $$p(z)=a_0+a_1z+\dots+a_nz^n$$ of ...
- 826
1
vote
2
answers
113
views
$\log(t)$ term in the small time expansion of $\mathrm{Tr}( A e^{-tB} )$
Assume $A$ is an operator on a Hilbert space with discrete spectrum. Assume $B$ is a positive operator on the same Hilbert space also with a discrete spectrum. Also assume $A$ and $B$ commute.
I'm ...
- 1,150
3
votes
0
answers
85
views
Can a punctured ball $(B\setminus\{0\})\subset\mathbb{C}^n$ be foliated by complete leaves?
Recently Antonio Alarcón proved that in the case of the unit ball $B\subset\mathbb{C}^n$ for $n\geq 2$ every smooth closed complex submanifold of dimension $q\leq n$, $V\subset\mathbb{C}^n$ defines a ...
- 166
15
votes
1
answer
2k
views
How to interpret Gauss's late fragments on conformal mapping of the interior of an ellipse (to the unit disk) in modern mathematical terms?
My question refers to some not very well known (and unpublished) fragments of Gauss that treat the problem of finding a conformal mapping (angle-preserving mapping) in the complex plane from the ...
- 2,099
11
votes
1
answer
860
views
Is every surjective holomorphic self-map on a compact complex manifold finite-to-one?
I have already asked this question on stack exchange, but I didn’t get any answer.
Let $X$ be a compact connected complex manifold.
Let $f:X \to X$ be a surjective holomorphic map. Is it true that $f$...
- 337
2
votes
1
answer
407
views
Finite generation of certain graded sequences of ideals
Let $U\subset\mathbb{C}^n$ be an open set containing the origin $o$ and $Y\subset U$ a complex analytic subvariety of pure codimension $c$ with ideal sheaf $\mathcal{I}_Y$. Let $\frak{a}_{\bullet}=\{{\...
- 61
10
votes
2
answers
417
views
zeros on the circle of convergence
In this question some experiments were used to conjecture that the zeros of partial sums of a series converging to a function with natural boundary on the unit circle were (weakly) converging to the ...
- 96.4k
6
votes
3
answers
587
views
Truncated Perron - logarithm-free error term?
Let $\{a_n\}_{n\geq 1}$, $|a_n|\leq 1$, be such that $F(s) = \sum_{n=1}^\infty a_n n^{-s}$ can be continued analytically to a neighborhood of the line $\Re s = 1$. (For instance, let $a_n = \mu(n)$.) ...
- 20.2k
51
votes
6
answers
5k
views
What does it take to run a good learning seminar?
I'm thinking about running a graduate student seminar in the summer. Having both organized and participated in such seminars in the past, I have witnessed first-hand that, contrary to what one might ...
1
vote
1
answer
207
views
Is $\log|h|$ BMO when $h$ is analytic?
Let $\mathbb{D}$ be the open unit disk in $\mathbb{C}$. A function $f$ is called a bounded mean oscillation (BMO for short) function on $\mathbb{D}$ if
$$\sup_{disc\,\subset\, \mathbb{D}}\dfrac{1}{|...
- 11
4
votes
1
answer
519
views
Are there any necessary conditions of lacunary functions known?
On the internet, most theorems about lacunary function only give the sufficient conditions. For example, Ostrowski-Hadamard Gap Theorem concerns the asymptotic length of null Taylor coefficients, ...
- 149
5
votes
0
answers
285
views
Is there a geography of Hodge numbers for minimal general type algebraic surfaces?
Let $X$ be a minimal smooth projective surface of general type (over $\mathbb{C}$). Let's call such a surface MSPGT. Such a surface has two chern numbers $c_1^2$ and $c_2$. It is known that they are ...
- 10.7k