Questions tagged [cv.complex-variables]
Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.
3,168
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Analytic continuation of holomorphic functions
Analytic/meromorphic continuation is a difficult problem in general. For "motivic L-functions", the idea of proving their analytic continuations by first proving their modularity goes back, I guess, ...
- 4,195
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2
answers
774
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Highly connected, compact complex manifolds
Here are four remarks about the homology and homotopy type of a compact, complex manifold $M$:
If $M$ is Kähler, then it is symplectic and thus $H^2(M,\mathbb{R}) \ne 0$. (Also, as explained in a ...
- 56.2k
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1
answer
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Floor of Riemann zeta function
How to show that $$\left\lfloor\zeta\left(1+\frac{1}{n}\right)\right\rfloor=n$$ for every positive integer $n$?
- 537
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1
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What is the "complex third derivative"?
Background
I am including this information about real higher order derivatives because it does not seem to be common knowledge. I also include a review of the complex Hessian.
If $f:\mathbb{R}^n \...
- 12k
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2
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A "holomorphic" Peano curve?
A Peano curve is a continuous map $[0,1]\to [0,1]^2$ whose image is the whole square.
I would like to know if on can obtain "holomorphic" Peano curves. Namely, is it possible to find a continuous ...
- 14k
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4
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A learning roadmap to the Schramm-Loewner evolution (SLE) for the complex analyst
I would like some good references to learn about the Schramm-Loewner evolution (SLE), for a complex analyst with no background in probability.
A quick google search gave a lot of references on SLE ...
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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 ...
- 7,961
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Sums of non-surjective entire functions
Suppose Suppose $A$ and $B$ are two entire, non-surjective, functions. This means
$$
A(z)=e^{f(z)}+c_1
$$
and
$$
B(z)=e^{g(z)}+c_2
$$
for some entire functions $f$ and $g$, and some complex ...
- 1,361
14
votes
3
answers
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Meromorphic 1-form and Picard's theorem
Let $D$ be the open unit disk in the complex plane and $U_1,U_2,\,\ldots\,,U_n$ be an open cover of the puntured disk $D^*= D\setminus\{0\}$. Suppose on each open set $U_j$ there is an injective ...
- 229
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Borel-Écalle re-summation and resurgence: criteria and results
This is about the theory of Borel-Écalle re-summation and resurgence, see Refs below.
This states that the perturbative series (say of the vacuum expectation value of an operator $\mathcal{O}$ in ...
- 10.3k
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2
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Determining rational functions by their critical points
Fix an integer $d > 1$ and $2d-2$ points $P_1, \ldots, P_{2d-2}$ in the Riemann sphere (not necessarily distinct). Thanks to the work of Eisenbud and Harris on limit linear series (Inventiones, ...
- 1,169
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Is analytic Quillen-Suslin simple?
This question is motivated by a sentence on the Wikipedia entry for Quillen-Suslin theorem. This theorem states that every algebraic vector bundle on affine space is trivial. The analogous result is ...
- 151k
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2
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One question on linear combinations of roots of unity
For $n \geq 1$, I want to find all solutions $x_i$ of the equation
\begin{equation}
\begin{array}l
x_i \in \mathbb{Z}, i=0,1,2\dotsc,n-1 \\
x_i^2 = 1, i=0,1,2\dotsc,n-1 \\
\...
- 284
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2
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Zeroes of the Alexander polynomial for achiral knots
Are there some known properties about the position (on the complex plane) of roots of the Alexander polynomial of achiral knots? They are shown as blue points in the following picture of roots for ...
user277578
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votes
1
answer
375
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Regularity of conformal maps
In order to define what it means for a map $f \colon \Omega \subseteq \mathbb R^n \to \mathbb R^n$ to be conformal, it is sufficient to require that $f$ is everywhere differentiable. Does conformality ...
- 1,337
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votes
2
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regular polygon question
Let $a_1,a_2,\ldots,a_n$ be distinct points on the complex plane $\mathbb{C}$ and $L$ be a circle in $\mathbb{C}$ such that
$$f(z):=\sum_{i=1}^n|z-a_i|^{2n-2}$$
is constant on $L.$ Could somebody ...
- 309
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votes
1
answer
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views
Is there a Serre intersection formula in analytic geometry?
There is the famous Serre intersection formula in algebraic geometry using the Tor functor (see for example here). I would like to know if there is such a formula in analytic (i.e. complex) geometry. ...
- 143
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1
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What is the analytic continuation of $\varphi(s)=\sum_{n \ge 1} e^{-n^s}?$
My research has lead me to the following function that I'm trying to continue. 3 Months ago I posted this question to MSE, and have placed 3 bounties on the question, but haven't received an answer, ...
- 101
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On meromorphic continuation of zeta function(s) and special values at negative integers
Euler developped (at least) two different approaches in order to calculate the values $\zeta(-m)$ of the zeta function $$\zeta(s) = \sum_{n\geq 1} \frac{1}{n^s}$$ at non-positive integers.
In one ...
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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 ...
- 1,869
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Analytic continuation of the Dirichlet generating series of the multiplicative partition function
Apologies for the lengthy question, but it seems it's the only way i can convey my thoughts. Consider the Dirichlet series:
$$\kappa(s)=\prod_{m=2}^{\infty}\frac{1}{1-m^{-s}}=\sum_{n=1}^{\infty}\frac{\...
- 271
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5
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A geometric proof of the Gauss-Lucas theorem
Motivated by a geometric proof of the Fundamental Theorem of Algebra we ask:
Is there a geometric proof for the Gauss-Lucas theorem? Since we are working on a half plane, can one imagine a possible ...
- 312
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votes
2
answers
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Fundamental Groups of compact Complex manifolds?
Hi,
are limitations on the fundamental group for compact complex manifolds known?
Can an arbitrary (finite represantable) group be the fundamental group of a compact
complex manifold?
Thanks
- 939
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votes
4
answers
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Complex evaluation of a classical (real) integral
There are several ways to compute the classical integral
$$
\int_{\mathbb R}e^{-x^2}dx=\sqrt{\pi}.
$$
Probably, best known are
(1) squaring the integral with subsequent change
of (now two) variables ...
- 13.4k
13
votes
2
answers
1k
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Is the exponential function the sole solution to these equations?
Let us take the exponential function $\lambda^z$ where $0 < \lambda < 1$. There are many great uniqueness conditions this holomorphic function satisfies. For example, it is the only function ...
user78249
13
votes
3
answers
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Is anything known about the series $\sum_{n=0}^{\infty} x^{\sqrt{n}} $?
It's well known that there are a shocking number of identities for the usual Jacobi theta function $$ \theta_3(x) = \sum_{n=-\infty}^{\infty} x^{n^2}. $$
So I wanted to turn my attention to slowly ...
- 2,283
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4
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"Simple" Kahler manifolds
I have some lecture notes from Demailly on Kahler geometry where he talks about "variétés Kahleriennes simples", which are defined as Kahler manifolds $X$ such that for very generic points $x_0$ in $X$...
- 4,201
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answer
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When are complex polynomial maps almost surjective?
Consider a complex polynomial map $f: \mathbb{C}^n \rightarrow \mathbb{C}^n$.
For $n = 1$, the fundamental theorem of algebra says that, for any $y \in \mathbb{C}$ there exists $x \in \mathbb{C}$ ...
- 133
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2
answers
538
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$f$ real-rooted forbid truncated $\frac1f$ to be so?
Let $f(x)$ be a polynomial in the ring $\mathbb{R}[x]$, the roots are all real and $f(0)=1$. Write the Taylor series of $1/f(x)$ around the origin as
$$\frac1{f(x)}=\sum_{k=0}^{\infty}a_kx^k,$$
and ...
- 41.8k
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2
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Motivation for BMO
At the moment, I don't have access to the early 1960's paper of John and Nirenberg that (from what I understand) introduced the space BMO (bounded mean oscillation). Why were John and Nirenberg ...
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Holomorphic line bundles on a punctured disc
Is every holomorphic line bundle on the - say - punctured unit disc $\dot{\Delta} \subseteq \mathbf{C}$ trivial? Griffiths-Harris (p. 39) prove that $H^{p,q}_{\overline{\partial}}(\Delta) = 0$ (for $q ...
- 1,644
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Is $\pi (x)=\operatorname{R}(x)-\sum_{\rho}\operatorname{R}(x^{\rho})$ correct at all?
It should be the case that, in some appropriate sense
$$\pi (x)\sim \operatorname{Ri}(x)-\sum_{\rho}\operatorname{Ri}(x^{\rho}) \tag*{(4)}$$
with $\operatorname{Ri}$ denoting the Riemann function ...
- 73
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votes
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Sendov's conjecture
It has been more than fifty years for famous Sendov's conjecture which states that if $p(z)$ is a polynomial of degree $n$ having all its zeros in the unit disc $|z|\leq 1$ then each of the n ...
- 559
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votes
3
answers
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views
How bad can a circle domain get?
Let $X$ be a domain in the Riemann sphere $\widehat{\mathbb{C}}$. We say that $X$ is a circle domain if every connected component of its boundary is either a circle or a point.
It was conjectured by ...
- 1,942
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431
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$\pm1$-polynomials with a maximal non-real root
For given $n$, consider a polynomial $\sum_{k=0}^na_kz^k$ with all coefficients $a_k\in\{\pm1\}$. I am interested in the following:
How big can the modulus of a non-real root of such a polynomial ...
- 13.2k
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1
answer
395
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Does the $\overline{\partial}$ operator have closed image?
Let $X$ be a complex-analytic manifold, not necessarily compact.
Does $\overline{\partial} : C^\infty(X) \rightarrow \Omega^{0,1}(X)$ have closed image with respect to the Fréchet topology given by ...
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Surgery in complex geometry
I've been thinking about surgery on complex manifolds. Not very seriously, but just to the point that I think it's odd how there's almost no mention of it in the literature. I figure there's something ...
- 4,201
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3
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Completeness of nonharmonic Fourier Series
I have the following question:
The Exponential System $(\exp(2\pi i n \cdot ))_{n\in \mathbb{Z}}$ constitutes an orthonormal basis of $L^2([-1/2,1/2])$.
Thus, certainly the oversampled system $\Phi:...
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$\pi e$ and an unfamiliar polynomial
Ever since my exposure to this integral involving $\pi e$, I've conjectured and set about evaluating the possible nature of the following integral
$$\int_0^1 x^m \sin(\pi x) x^x (1-x)^{1-x} \ dx, \...
- 1,459
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1
answer
675
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How can one "see" the Hopf fibration in the space of lattices in the plane?
This question is inspired from Etienne Ghys's talk on Knots and Dynamics from ICM 2006.
The map $L \mapsto (G_4(L), G_6(L))$ gives a bijection between all lattices $L\subset \mathbb{C}$ (including ...
- 1,821
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votes
2
answers
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Special values of $\zeta$ outside the real line and the critical strip
The values of Riemann's function at the integers have been extensively studied. I was wondering, is there anything interesting known (or conjectured) to happen arithmetically outside the real line (...
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1
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What does the incidence algebra of the lattices in C tell us about modular forms?
I have two different and probably unrelated questions that can both be superficially described by the title, so I hope you'll forgive me if I ask them together. They both fall under the category of ...
- 115k
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Is it possible that the following integral is $0$?
Given any integer $n\geqslant1$, let $E,F$ be two subsets of $\{\{i,j\}:1\leqslant i<j\leqslant n\}$ such that every two sets in $F$ are disjoint.
It is not difficult to see that
$$\int_{1<|z|&...
- 1,987
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votes
0
answers
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Are the zeros of Tutte polynomials dense in $\mathbb C^2$?
For the chromatic polynomials of graphs we have two nice theorems which describe the behavior of their zeros: Thomassen proved that the set of real zeros of all chromatic polynomials is the union of $\...
- 85.1k
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Meaning of Cauchy integral theorem - the (co)homology viewpoint
I'm not sure what follows is not just a complicated way to deduce a blatant triviality, or if is even correct. Let's try.
In the elementary theory of analytic functions of $1$ complex variable, one ...
- 22.7k
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votes
3
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Can the equation $1+z+z^2=z^n$ for natural $n$ have multiple complex roots $z$?
The question is stated in the title of this post.
It is easy to see that, if $z$ is a multiple root of $p_n(z):=1+z+z^2-z^n$, then $(n-2)z^2+(n-1)z+n=0$, so that we can successively express $z^2,\dots,...
- 118k
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votes
4
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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
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votes
6
answers
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Functions holomorphic on a region minus a Cantor set
Let $X$ and $Y$ be simply connected open regions of $\mathbb{C}$, and let $Z \subset X$ be a Cantor set. Assume we have a homeomorphism $f$ from $X$ to $Y$, which is holomorphic on $X \setminus Z$. Is ...
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Elliptic Curves, Lattices, Lie Algebras
I've recently started to look at elliptic curves and have three basic questions:
Is it correct to say that elliptic curves $E$ in the projective plane are in bijective correspondence with lattices $...
- 1,442
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votes
2
answers
482
views
Behavior of $|f'(z)|/(1+|f(z)|^2)$ as $|z| \rightarrow \infty$?
Let $f(z)$ be an entire holomorphic function in $\mathbb{C}$, and consider the real-valued function
$$g_f(z)=\frac{|f'(z)|}{1+|f(z)|^2}.$$
If $f(z)$ is a polynomial, then it is easy to prove that $\...
- 1,320