All Questions
372 questions
6
votes
0
answers
263
views
roots of a polynomial linked to mock theta function?
The following polynomial (after harmless factors dropped) is found in the paper entitled Mock theta functions and quantum modular forms by Folsom-Ono-Rhoades (see Theorem 1.1)
$$Q_k(z)=\sum_{n=0}^{k-1}...
- 43.2k
6
votes
1
answer
2k
views
Approximation by analytic functions
Dear all.
Let
$$
f(x) = \sum_{k \in \mathbb{Z}} \hat{f}(k) \exp(2\pi \mathrm{i} kx)
$$
be a function given by usual fourier series.
Since my original question hasn't got any answer yet, and I ...
- 3,343
6
votes
1
answer
321
views
Derivatives of norm of vector-valued holomorphic functions
Let $G$ be a connected domain in $\mathbb{C}^{n}$, let $H$ be a Hilbert space and let $f,g:G\to H\backslash \{0\}$ be holomorphic (in my particular situation they are also injective, but I don't think ...
- 5,529
6
votes
2
answers
2k
views
Does there exist a holomorphic function which takes given values on the positive integers?
Inspired of course by What's a natural candidate for an analytic function that interpolates the tower function?
I am minded to ask what looks to me like a more natural question: given a sequence $...
- 41.4k
6
votes
0
answers
1k
views
Evaluating $\iint_{\mathbb{R}\times \mathbb{R}^{+}}e^{-|w-c_{1}|^2-|u-c_{2}|^2}\frac{1}{w_{1}+iw_{2}-u_{1}-iu_{2}}dw_{1}dw_{2}du_{1}du_{2}$
For $c_{1},c_{2}\in \mathbb{H}:=\{Im(z)>0\}$ I want to compute the following integral or prove it doesn't exist:
$$\int_{\mathbb{R}\times \mathbb{R}^{+}}\int_{\mathbb{R}\times \mathbb{R}^{+}}e^{-|...
- 5,474
6
votes
3
answers
394
views
Is there a effective computational criterion to all periodic points of a rational function are repelling.
I came up with a question to know the fatou component of of some types of rational function. In some sense, I may need to give a computational criterion to existence of attracting periodic basin for a ...
- 61
6
votes
1
answer
292
views
Analytic maps on Banach spaces: analyticity upgrade
Consider the following problem.
Let $E,F,G$ be real or complex Banach spaces, such that $F\subset G$ with continuous embedding. Let $U\subset E$ an open set and
$$ f:U\to G $$
an analytic map, such ...
- 1,075
6
votes
3
answers
297
views
Do the bounded isophase lines of a complex polynomial $f$ through the zeroes of $f’$ define a spanning tree?
Let $f: \mathbb{C} \to \mathbb{C}$ be a polynomial and let $\arg(f(z))$ be the phase of $f(z) = | f(z)| \exp(\mathrm{i} \arg(f(z)))$. The zeroes of $f'(z)$ are saddle points of $\arg(f(z))$, i.e. ...
- 4,014
6
votes
1
answer
852
views
Harmonic map proof of Riemann mapping theorem
Is there a way to prove the Riemann mapping theorem using the theory of harmonic maps (in the sense of "Harmonic Mappings of Riemannian Manifolds" by Eells and Sampson)?
- 494
6
votes
0
answers
632
views
Generating functions in countable commutative monoids
Let $f: \mathbb{N}_0 \rightarrow \mathbb{C}$ be a function. The power series of $f$ can be viewed as the function $\mathscr{P}_f : q \mapsto \sum_{n \in \mathbb{N}_0}^{} f(n)q^n$ where $q \in \mathbb{...
- 405
6
votes
2
answers
336
views
On frequency decay of an integral transform of a function
Suppose $f \in C^{\infty}_c((-1,1))$ and assume that there exists constants $a,b>0$ such that
$$
\bigg|\int_{\mathbb R} f(t) \,e^{\tau t^2+i\tau t}\,dt\bigg| \leq a\,e^{-b|\tau|},$$
for all $\tau \...
- 4,115
6
votes
3
answers
974
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
5
votes
0
answers
189
views
Extension of holomorphic maps to smooth family of holomorphic maps
Let $\pi:X \to D^2$ be a family of diffeomorphic (but not isomorphic) complex manifolds. Each fiber is allowed to have boundary but is compact (maybe not Stein) and $D^2 \subset \mathbb{C}$ is a ...
- 1,409
5
votes
2
answers
561
views
$\frac{\partial f}{\partial \overline{z}}=0$ in distributional sense implies $f$ is holomorphic
Let $f=(u,v)\in \mathscr{D}'(U,\mathbb{C})$ be a distribution, where $U\subset\mathbb{C}=\mathbb{R}^2$ is an open set and $u$ and $v$ are the projection of $f$ onto the real and imaginary axis (ie $\...
- 1,149
5
votes
1
answer
291
views
Asymptotics of the Liouville sum at the primes
Let $\lambda$ denote the Liouville function, and let $L(x):=\sum_{n \leq x} \lambda(n)$ be the Liouville sum. Define $c$ to be the supremum of the real parts of the zeros of the Riemann zeta function. ...
- 103
5
votes
1
answer
767
views
Reference for Lindelöf Hypothesis implying finitely many zeros off critical line?
Can anyone give me a reference for the following theorem on the Riemann zeta function?
If the Lindelöf Hypothesis is true (that is $\zeta(\sigma+it)=O(t^\epsilon)$ as $t\rightarrow\infty$), then ...
- 195
5
votes
1
answer
171
views
Existence of Laurent series with zeroes at $𝑒^{2𝑛}$ ($𝑛∈ℕ_0$) and even faster coefficient decay
This is an extension of an earlier question of mine which corresponds to the case $A = 1$. Precisely, my question is as follows:
Given $A > 0$ fixed but arbitrary, is there a non-trivial sequence $...
- 734
5
votes
1
answer
842
views
Hurwitz's automorphisms theorem with deformations
Hurwitz's automorphisms theorem bounds the order of the automorphism group of a negatively curved Riemann in terms of the genus.
Now suppose a finite group $G$ acts faithfully on a Riemann surface $...
- 17.6k
5
votes
1
answer
199
views
An identity for the Lambert $W$ function
Expressing the integral in An integral identity in terms of residues, we come to the following supposed identity:
$$\sum_{k=-\infty}^\infty\frac1{1 + W_k(x)}=\frac12$$
for all $x\in(-1/e,0)$, where $...
- 128k
5
votes
0
answers
245
views
Dimension of highest discriminants of a morphism
Let $f: X\to Y$ be a flat morphism between smooth complex affine varieties. Let $Z$ be the closed set of most singular points of $f$ (in the sense: $p$ is a most singular point of $f$ if the tangent ...
- 1,081
5
votes
1
answer
343
views
harmonic extension of a curve by different parametrization
Let us consider a curve $\gamma :S^1 \rightarrow \mathbb{R}^3$ (or even a planar convex one if it simplifies). Then I look to the harmonic extension to the disc $h:\mathbb{D}\rightarrow \mathbb{R}^3$ (...
- 914
5
votes
0
answers
225
views
Validity of $\ln z=\frac{\pi}{\operatorname{AGP}(\theta_2^2(1/z),\theta_3^2(1/z))}$
Definitions
For the definitions of $\operatorname{AGP}$ and $\operatorname{AGO}$, see here or here. $\theta_2(z)$ and $\theta_3(z)$ are defined as follows:
$$\theta_2(z)=\sum_{n=-\infty}^\infty z^{(n+...
- 83
5
votes
1
answer
167
views
Upper bound for the $n$-th derivative of a rational function $\frac{f}{f+g}$
Let $f$ and $g$ be real polynomials with nonnegative coefficients. Let
$$
h = \frac{f}{f+g}.
$$
I want to prove that the $n$-th derivative of $h$ satisfies:
There exists $C > 0$ such that
$$
|h^{(...
- 187
5
votes
2
answers
1k
views
Cohomology class of a current
Hello,
i still have a question about positive closed currents. In particular i know that if $X$ is a compact complex manifold and $T$ is a positive closed current of bidegree $(1,1)$ such that
its ...
- 75
5
votes
0
answers
248
views
Analytically continuing Euler's partition function
Author's note: This question might be a little hopeless, but maybe someone has some form of good feedback. It's a long one because I tried to be very thorough. I tried to explained all the odds and ...
user78249
5
votes
2
answers
604
views
Sets of vectors related by a rotation
We have a two sets of vectors ($\mathbb{C}^d$), $A=\{ v_1, \ldots v_n\}$ and $B=\{u_1, \ldots u_n\}$.
The question is if there is an efficient solution (polynomial in $n$) for checking whether $A$ ...
- 1,612
5
votes
4
answers
3k
views
Minimizing the modulus of a polynomial around a circle
I'm probably missing something elementary here, but I guess the only way to be sure is to ask here.
Now, I have encountered a situation where given an nth-degree polynomial $p_n(z)$ with complex ...
5
votes
1
answer
833
views
A statement on complex polynomials
I have a feeling the following is true.
Assume that there are $n$ mutually disjoint closed disks $D_i$ in the complex plane and $n$ complex polynomials $p_i(z)$ of degree $n - 1$, with both types of ...
- 5,215
5
votes
1
answer
616
views
On limits of manifolds
This question should be fairly elementary. I’d just like to check I’m not missing anything.
Let $\{M_n\}_{n\ge 0}$ be an inverse system of smooth manifolds with transition maps $f_{t,s} : M_t\to M_s$,...
user131191
5
votes
1
answer
392
views
Is this closed subspace of Fréchet space complemented
In the hope of completing the rich tapestry of complemented (or not) topological vector subspaces, I would like to know (maybe it is immediate for specialists)
whether the space of analytic functions ...
- 3,738
5
votes
0
answers
342
views
Automorphisms of Compact Riemann Surfaces
I read a statement that for a compact Riemann surface $C$ with genus $g\geq 2$, one has
for the Jacobian $J(C)$ of the curve $C$:
$$ Aut (J(C))\sim Aut C$$
when $C$ is hyperelliptic and
$$Aut(J(C))\...
- 51
5
votes
1
answer
444
views
Smoothness in Ecalle's method for fractional iterates
Some four years ago I answered my own question on fractional iteration, concluding that there is a half iterate of sine, that is $f(f(x)) = \sin x,$ which is real analytic for $0 < x < \pi$ but ...
- 25.7k
5
votes
0
answers
159
views
Characterization of the hypergeometric function
One of the definition of the hypergeometric function $_2 F_1$ rely only on its global properties around the singularities (and not on a differential equation or a serie expansion)
In modern language (...
- 263
4
votes
3
answers
685
views
Approximation for complex variables
Approximation theory, which aims to provide the optimal polynomial function approximating the target function in a given domain such as $x\in[-1,1]$, has been well-developed for real variables. In ...
- 149
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
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
4
votes
1
answer
298
views
$L^1$ norm of Littlewood polynomials on the unit circle
A Littlewood polynomial is a polynomial with coefficients from $\{ 1, -1\}$ and the set of Littlewood polynomials with degree $n$ is denoted by $\cal{L}_n$.
I'm interested in a "good" lower bound on ...
- 4,484
4
votes
2
answers
845
views
Are there cusp forms for the full modular group Sp(2,Z) and representations det^3 \otimes Sym^2j(\rho_standard)
What are modular forms or cusps forms, resp. ?
We start with defining their common domains $\mathbb{H}_g$ as the set of symmetric $g \times g$ matrices with positive definite imaginary parts.
The ...
- 85
4
votes
1
answer
401
views
How to correctly renormalize this function at the pole $x=1$? Evaluating: $\sum_{n=1}^{\infty} e^{e^n}$
So I was considering the divergent everywhere but 0 power series
$$ f(x) = \sum_{n=0}^{\infty} e^{e^n} x^n $$
Now one can do the following "questionable" manipulation
$$ f(x) = \sum_{n=0}^{\...
- 2,689
4
votes
0
answers
312
views
Transforming a multivariable integral to make it separable
In the following I will omit requirements of smoothness, extent of domain, finiteness, etc, both to simplify the exposition and because I don't know exactly what the requirements are. Please imagine ...
- 37.7k
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
4
votes
0
answers
294
views
Holomorphic covers pulling back the volume form to any integer multiple
Let $M$ be a closed connected complex manifold with $\mathrm{dim}\:M=n$. Can there exist holomorphic covering maps $\phi_k:M\to M$ for all integers $k\geq 1$ such that $\phi_k^*:H^n(M, \mathbb{Z})\to ...
user145520
4
votes
0
answers
100
views
Generating $H^{\infty}(X)$
Let $X$ be a domain in $\mathbb{C}^d$ and let $\mathbb{D}$ be the open unit disk in $\mathbb{C}$. Consider the Banach algebra $H^{\infty}(X)$ consisting of bounded holomorphic functions on $X$ with ...
- 5,529
4
votes
3
answers
744
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 $l+1$ ...
- 429
4
votes
1
answer
443
views
Riemann $P$-symbol for ODEs
Good afternoon, everyone. Does anyone know what the Riemann $P$-symbols mean when they contain more than three columns (i.e., to what ordinary differential equations they correspond)?
Examples: $P\...
4
votes
1
answer
491
views
Analytic functions with integer coefficients
I would like to ask the following question. This is related to one lemma that I need in a recent research on the arithmetic behavior of transcendental functions with integer coefficients.
Let $\alpha,...
- 81
4
votes
9
answers
4k
views
Functions of one complex variable: geometric theory
Can someone recommend a good textbook on functions of one complex variable which have good chapters on geometric theory, in English?
When I studied complex analysis, I used two
textbooks:
An ...
- 371
4
votes
0
answers
268
views
Four infinite series involving Riemann zeta function
Can you provide a proof for at least one of the claims given below?
It is known that $\pi=\displaystyle\sum_{n=1}^{\infty}\frac{3^n-1}{4^n} \cdot \zeta(n+1)$ where $\zeta$ denotes Riemann zeta ...
- 2,661
4
votes
1
answer
223
views
Asymptotics for 'generalized" Kasteleyn's formula
A follow up on an earlier MO question.
Kasteleyn's formula for the number of domino tilings of a $2n\times 2n$ square
$\prod_{j=1}^n\prod_{k=1}^n \left( 4\cos^2(\pi j/(2n+1))+4\cos^2(\pi k/(2n+1))\...
- 43.2k
4
votes
1
answer
3k
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, \...
- 2,106