All Questions
372 questions
10
votes
2
answers
597
views
How to determine the asymptotics of $\sum_{n=0}^{\infty} e^{-\frac{2^n}{x}}$
I'm generally interested in being able to find an asymptotic expansion of
$$ \sum_{n=0}^{\infty} \left[ e^{- \frac{f(n)}{x}} \right] $$
As $x \rightarrow \infty$ and $f(n)$ is a smooth monotonically ...
- 2,689
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
10
votes
2
answers
728
views
Gauss--Lucas type theorem for tracts and higher derivatives of a polynomial
The Gauss--Lucas Theorem states that all zeros of derivative of a degree $n$ complex polynomial $p(z)$ are contained in the convex hull of the zeros of $p$. By iteration, this implies that the zeros ...
- 1,014
10
votes
2
answers
538
views
Is there an analogue of Mostow-Palais equivariant embedding theorem for noncompact groups
Let $M$ be a (Hausdorff) smooth compact manifold and $G$ a Lie group acting smoothly on $M$. If $G$ is compact then, by Mostow-Palais theorem, there exists an equivariant smooth embedding $M\to {\...
- 31.2k
10
votes
1
answer
1k
views
Is there a holomorphic function on open unit disc with this property?
Let $D=\{z\in \mathbb{C}\mid |z|<1\}$. Is there a holomorphic function $f:D\to \mathbb{C}$ such that for every $n\in \mathbb{N} \cup \{0\},\;f^{(n)}$ has a continuous extension to $\bar D$ but $f$ ...
- 366
10
votes
1
answer
462
views
Reverse mathematics of meromorphic functions on Riemann surfaces
Various sources touch briefly on the reverse mathematics of measure theory and complex analysis. But I have found none on the uniformization theorem for Riemann surfaces or the existence of non-...
- 11.1k
10
votes
0
answers
207
views
Projective tensor squares of uniform algebras
In discussion with a colleague recently (Jan 2017),
$\newcommand{\AD}{A({\bf D})}\newcommand{\CT}{C({\bf T})}$
I was reminded that if $A(D)$ denotes the disc algebra and $\iota: \AD\to \CT$ is the ...
- 25.8k
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
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
$$\eta_n(z,\gamma)= \sum_{k=1}^n (-1)^{k+1}\lambda_k^{-\sigma} e^{-it\...
- 3,098
9
votes
1
answer
2k
views
Why does it suffice to study modular forms for $\Gamma_0(N)$?
Every reference I've seen on modular forms seems to jump from the general definition of a modular form for congruence subgroups to studying modular forms just for $\Gamma_0(N)$.
Once upon a time, I ...
- 7,527
9
votes
1
answer
663
views
Holomorphic Sard's theorem?
I have originally posted this question on math.SE, but it received little attention, so I repost it here.
Let $U\subset \mathbb{C}^{n}$ and $V\subset \mathbb{C}^{m}$ be open and connected. Let $\Phi:...
- 5,529
9
votes
3
answers
1k
views
An analytic proof of the De Franchis theorem
The De Franchis theorem in its simplest form states that given two compact Riemann surfaces $\Sigma_{g_1},\Sigma_{g_2}$ where $g_1,g_2 > 1$, there are only finitely many non-constant holomorphic ...
- 1,169
9
votes
1
answer
741
views
Three questions about three functions similar to $\sin,\cos$
In The Basel problem revisited? a question about the function, similar to sinc, $f(x)$ was asked:
$$f(x) = \prod_{n=1}^\infty \left ( 1+ \frac{x^3}{n^3} \right ) = \prod_{n=1}^\infty \left ( 1+ \frac{...
- 3,074
9
votes
2
answers
8k
views
The Paley-Wiener theorem and exponential decay.
Consider a function whose Fourier transform is supported on a half-ray:
$$
A(t)=\int_0^\infty \omega(E) e^{-iEt}d E,
$$
where I can suppose $\omega(E)\geq 0$ and any suitable regularity conditions on $...
- 2,358
9
votes
1
answer
350
views
Collinear Galois conjugates
This is inspired by this old question, which may provide a bit more background. But the two present questions seem somewhat more fundamental to me.
Let $p$ be an irreducible polynomial with integer ...
- 13.4k
9
votes
0
answers
887
views
How many ways are there to teach class field theory?
I will soon have to teach class field theory (I do not know whether it will be local or global yet:)) to postgraduate students. I wonder, which approaches to this subject(s) exist now.
I definitely ...
- 16.9k
9
votes
1
answer
413
views
Existence of Laurent series with zeroes at $e^{2n}$ ($n \in \Bbb{N}_0$) and extremely fast coefficient decay
I am working on a problem in harmonic analysis, which I converted into the following existence problem concerning Laurent series. I am a bit at a loss concerning this problem, since my knowledge of ...
- 734
9
votes
1
answer
972
views
Non-standard enlargements, $\zeta(s)$ and analytic continuation
Consider an extension of the Riemann zeta function $\zeta(s)$ where $s$ now runs over a non-standard enlargement of the complex plane.
Observe that if $s=\sigma + it$ with $\sigma>1$ real and ...
- 17.6k
9
votes
0
answers
321
views
When exactly is the principal AGM equal to the optimal AGM?
Definitions
Following the terminology of SageMath, let the principal arithmetic-geometric mean, $\operatorname{AGP}$, of $(a,b)\in\mathbb{C}^2$ for $a\ne 0$, $b\ne 0$, $a\ne\pm b$ be defined as ...
- 83
8
votes
1
answer
431
views
Different notions of convergence of complex subvarieties
Let $X$ be a smooth complex algebraic variety (or, better, complex analytic manifold). Let $\{C_i\}$ be a sequence of compact algebraic subvarieties (resp. analytic reduced subspaces) which converges ...
- 21.8k
8
votes
3
answers
2k
views
Riemann mapping for doubly connected regions
Remove the closure of simply connected region from the interior of a simply connected region. Is it true that the resulting domain can be mapped conformally to some annulus?
- 7,442
8
votes
1
answer
387
views
Uniqueness theorem for conformal mapping
Let $f$ and $g$ be analytic functions in the unit disk $D$, continuous in the closed disk and locally univalent, $f'(z)\neq 0,\; g'(z)\neq 0,\; z\in D$.
Assume that each has only finitely many ...
- 91.8k
8
votes
2
answers
496
views
Literature on non-Archimedean analogues of basic complex analysis results
It looks like there is some literature out there on what might be called 'non-Archimedean complex analysis' e.g. Benedetto - An Ahlfors Islands Theorem for non-archimedean meromorphic functions and ...
8
votes
1
answer
607
views
Rotation invariance of an integral
Consider the integral depending on 2 parameters
$$f(\tau,x):=\int_{-\infty}^{+\infty}\frac{dp}{\sqrt{p^2+1}}e^{-\sqrt{p^2+1}\tau+ipx},$$
where $\tau >0,x\in \mathbb{R}$. This integral absolutely ...
- 21.8k
8
votes
2
answers
995
views
Another proof of the bidisc and the ball are biholomorphically inequivalent?
Does this outline of a proof work?
Consider the ball and the bidisc in $\mathbb{C}^2$. Give each space its Bergman metric. To show that the ball and the bidisc are not holomorphic, it is enough to ...
- 12.1k
8
votes
2
answers
392
views
Image of boundary circle under map from punctured elliptic curve to ℂ
Let $E=\mathbb C/\Lambda$ be an elliptic curve,
and let $D\subset E$ be a very small disc.
($D$ is round for the usual flat metric on $E$)
By the main result of [1], there exists a holomorphic ...
- 43.2k
8
votes
3
answers
2k
views
Harmonic level sets and boundary data
This is probably a classic problem, so a good reference book or paper to get me started on this type of question would be great:
Let $\mathbb{D} \subset \mathbb{C}$ be the unit disk with boundary $\...
8
votes
1
answer
1k
views
Beautiful examples of arc-like continua
A continuum is a nonempty compact, connected metric space.
A continuum $X$ is called arc-like if, for every $\varepsilon>0$, there is a continuous and surjective function $f:X\to [0,1]$ such that $...
8
votes
3
answers
1k
views
Behaviour at natural boundary
Suppose I have a holomorphic function $f$ in a domain $\Omega$ with natural boundary $\partial \Omega.$ Let $p \in \partial \Omega.$ Is it true that there is some analogue of Picard's little theorem - ...
- 96.4k
8
votes
1
answer
859
views
Is there an underlying geometric reason for the rigidity of complex geometry?
This question stems from a discussion with a friend about the apparent jump in rigidity of smooth geometry to complex geometry. The word rigidity here is somewhat vague; I mean this in the sense it ...
- 623
8
votes
1
answer
591
views
Defining holomorphic functions in terms of Banach algebras, and similarly for C*-algebras
Let $C$ be the category of commutative Banach algebras and let $U : C \to \text{Set}$ be the usual forgetful functor. The holomorphic functional calculus guarantees that every holomorphic function $f :...
- 118k
8
votes
3
answers
429
views
A density claim
Suppose that $g_k\in C([1,2])$, $k\in \mathbb N$ are continuous functions such that $\|g_k\|_{C([1,2])} \leq \epsilon^k$ for some sufficiently small $\epsilon>0$. Is the following claim true:
If $f\...
- 4,115
7
votes
1
answer
369
views
Looking for holomorphic function on a sector with specified boundary behavior
Fix $h \in (0,\pi/2)$. I am trying to explicitly exhibit a holomorphic function $f\colon \Sigma \to \mathbb{C}$, where $\Sigma$ is the punctured sector
$$\Sigma := \{z \in \mathbb{C} \:|\: z\neq 0, 0\...
- 1,589
7
votes
2
answers
593
views
Dependence of a solution of a linear ODE on parameter
Is the following theorem known, or can be easily derived from known results?
Consider the differential equation
$$w''-kz^{-1}w'=(\lambda+\phi(z))w,$$
where $k>0$ is fixed, $\lambda$ is a large (...
- 91.8k
7
votes
2
answers
661
views
Locus of roots of all convex combinations of two monic polynomials
Let $p,q$ be monic polynomials in $\mathbb{C}[t]$ and for $\alpha \in [0,1]$, let $c_\alpha := \alpha p + (1-\alpha)q \in \mathbb{C}[t]$. Since the roots of a polynomial vary continuously with respect ...
- 1,719
7
votes
1
answer
1k
views
Surjective entire functions
If I have an entire function give as a power series $f(z)=\sum_{i=0}^{\infty}a_iz^i$, is there a way/technique to check if the function is surjective? Weierstarss factorization theorem gives that $f$ ...
- 1,361
7
votes
1
answer
2k
views
Universal covering map from $\mathcal{H}$ to $\mathbb{C}\setminus \mathbb{Z}\oplus i\mathbb{Z}$ (the countably punctured complex plane)
It's a consequence of the uniformization theorem for simply connected Riemann surfaces that the universal cover of $\mathbb{C}\setminus(\mathbb{Z}\oplus i\mathbb{Z})$ ($\mathbb{C}$ punctured at all ...
- 10.7k
7
votes
1
answer
193
views
Greatest lower bound for subordination
Consider the set $X$ of all analytic functions $f$ in the unit disk $U$ satisfying
$f(0)=0, f'(0)\neq 0$. We say that $f\prec g$ if there exists
$\phi\in X$ which maps $U$ into itself, and $f=g\circ\...
- 91.8k
7
votes
0
answers
452
views
Sufficient condition on coefficients for a complex power series to be bounded
Let $f(z)$ be an entire function (on $\mathbb{C}$). Assume it has a power series of the form
$$\displaystyle \sum_{n=0}^\infty (-1)^nc_{2n}z^{2n},$$
where $c_{2n}\geq 0$ for all $n$.
Is there a ...
- 783
7
votes
0
answers
189
views
When is the Locus of Equi-modular points of two monic polynomials with integer coefficients contained in the unit disk?
If $\lambda_{1}(z)$ and $\lambda_{2}(z)$ are two monic polynomials (relatively prime) with integer coefficients and $$\Gamma:=\lbrace z \rm{\ s.t.\ } |\lambda_{1}(z)|=|\lambda_{2}(z)|\rbrace,$$ when ...
- 103
7
votes
1
answer
248
views
Are there such things as non-trivial entire semigroups?
I'll state the theorem I am posing up front, and then explain why I think this theorem appears to be true. I am asking if anyone can prove it, or knows references to where it is proved. Please, ...
user78249
7
votes
2
answers
1k
views
Analog of residue for meromorphic quadratic differentials
Hi I had asked this already on math.stackexchange.com but got no answers.
I was wondering if there was any sort of (natural) analog of the residue of a meromorphic one form that made sense for a ...
- 2,299
7
votes
1
answer
794
views
Non-algebraic curve visualisation
Is there any software which can automatically visualise a non-algebraic
complex curve, I mean the structure of it's ramification points and sheet?
I think a good test example would be the Lambert ...
- 1,343
7
votes
4
answers
2k
views
Visualizing functions with a number of independent variables
I need to graph real valued functions (for exposition and analysis).
The issue is: there are more independent variables so that the conventional graphing methods can't be used, and furthermore I don't ...
- 851
7
votes
1
answer
461
views
How to prove an elementary functional equation for polylogarithms?
Let $Li_s(z)$ denote the usual polylogarithm. The elementary functional equation $$Li_{-n}(z)=(-1)^{n-1}Li_{-n}(1/z)$$ holds for $n\geq 1$. I remember only that the proof used some reproducing ...
- 2,480
7
votes
5
answers
682
views
Bound on sum of complex summands involving binomial coefficients
I am trying to find the asymptotic behavior of the sum:
$$ \sum^n_{i=0} \begin{pmatrix} 2n \\ i \end{pmatrix} x^i y^{2n-i}$$
as $n\rightarrow\infty$. Here $x$, $y$ are complex numbers and I have $|x|...
- 93
6
votes
0
answers
755
views
Discriminant of $\alpha P(u) + (z-u) P'(u)$
I'm trying to find a “closed form” of $\textrm{Discriminant}_u(f(u))$, where $f(u) := \alpha P(u) + (z-u) P'(u)$.
Here $P(u)$ is a monic polynomial of degree $d > 1$ with $u\in\mathbb{C}$, $\alpha$ ...
- 41
6
votes
1
answer
379
views
Complex-doubly periodic function in two variables?
I am looking for a function $f:\mathbb C^2 \rightarrow \mathbb C^2$ that satisfies the two equations
$$\partial_{z_2}f_1(z_1,z_2) + \partial_{z_1} f_2(z_1,z_2)=0 \text{ and }$$
$$\partial_{\bar z_1}...
- 536
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
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