Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [cv.complex-variables]

Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.

27
votes
0answers
680 views

“Three great cocycles” in Complex Analysis as cohomology generators

In his lecture notes, C. McMullen discusses "the three great cocycles" in Complex Analysis: the derivative $$f\mapsto\log f',$$ the non-linearity $$f\mapsto (\log f')'dz$$ and the Schwarzian ...
24
votes
0answers
485 views

Decidability of equality of elementary expressions

In the following definition the term expression is to be understood as a finite tree built from formal symbols without any predefined meaning assigned to them. Define the set $\mathcal{E}$ of ...
23
votes
0answers
553 views

Number of real roots of a polynomial

Let $P\in \mathbb{R}[x]$ be a polynomial such that $(P, P') = 1$. Suppose that we want to calculate the number of real roots of $P$ in the interval $[a, b]$ (to simplify, let us assume that $P(a), P(b)...
22
votes
0answers
1k views

Is analytic capacity inner regular?

For a compact set $K$ in the complex plane, define the analytic capacity of $K$ by $$\gamma(K) := \sup |f'(\infty)|$$ where the supremum is taken over all functions $f$ holomorphic and bounded by $1$ ...
19
votes
0answers
530 views

Polynomials with roots in convex position

Let $\mathcal P_n$ denote the set of all monic polynomials of degree $n$ with real or complex coefficients such that $P\in\mathcal P_n$ if for all $k\in\lbrace 0,1,\dots,n-2\rbrace$ the $n-k$ roots of ...
15
votes
0answers
380 views

Aligned roots of irreducible polynomials

It is well known from this famous question that the roots of a random polynomial tend to be close to the unit circle. So I was wondering in a somewhat converse sense: for an irreducible polynomial, is ...
15
votes
0answers
959 views

What is the appropriate setting for Cauchy's Integral Formula?

For a $C^1$ function $f:U\to\mathbb{C}$, where $D\subset U\subseteq \mathbb{C}$ with piecewise $C^1$ boundary $\partial D$, we have the following generalized Cauchy integral formula: $$ f(\zeta) = \...
13
votes
0answers
335 views

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 $\...
13
votes
0answers
2k views

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 ...
12
votes
0answers
417 views

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|&...
12
votes
0answers
291 views

Analytic contraction of the Stone-Cech compactification of $\mathbb C$

Let $S$ be the Stone-Cech compactification of $\mathbb C$. Then any meromorphic function $f:\mathbb C \to \mathbb{CP}^1$ extends to $S$. Do the meromorphic functions separate the points of $S$? I ...
11
votes
0answers
376 views

Inverse Mellin of the exponential of the digamma function

I'm looking for a function $f_p(x)$ with real parameter $p>0$ satisfying $$ \int_0^\infty f_p(x)x^{s-1}dx=e^{-p\psi(s)} $$ where $\psi(s)$ is the usual digamma function. The inverse Mellin formula ...
11
votes
0answers
666 views

Continuous extension of Riemann maps and the Caratheodory-Torhorst Theorem

If $G\subsetneq\mathbb{C}$ is a simply-connected plane domain, then by the Riemann mapping theorem there is a conformal isomorphism $\newcommand{\D}{\mathbb{D}}\varphi:\D\to G$, where $\D$ is the unit ...
11
votes
0answers
863 views

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{\...
11
votes
0answers
1k views

Dolbeault cohomology of complex tori.

Let $T=\mathbb C^n/\Lambda$ a complex torus. It is completely elementary to prove that the de Rham cohomology of $T$ in degree $q$ is isomorphic to the $q$-th exterior power of the dual of $\mathbb C^...
10
votes
0answers
248 views

Riemann–Hilbert-type problem

Let $P$ be a fixed pentagon in the hyperbolic plane $\mathbb H^2$ with all the angles equal to $\pi / 3$. Let $w_1, w_2, \dots, w_5$ be the sides of $P$ going in the counterclockwise order. We are ...
10
votes
0answers
415 views

Witt's proof of Gelfand-Mazur / Ostrowski's Theorem

Previously asked on Math Stackexchange without answers. Background: As sort of a hobby, Ernst Witt gave extremely short proofs for famous theorems. This question is about his six-line proof of the ...
10
votes
0answers
146 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 ...
10
votes
0answers
230 views

How useful is knowing every torsionfree $\mathcal O(D)$ module is flat?

One of the corollaries of Weiertrass' factorization theorem plus the theorem of Mittag Leffler is that $\mathcal O(\Bbb C)$, more generally $\mathcal O(D)$ for some region $D$ is such that every ...
10
votes
0answers
380 views

Cesaro summation of a particular Dirichlet series associated with $\zeta(s)$

If you've investigated the error in Perron's formula in general, you've probably noticed that Cesaro summation $$\lim_{x\rightarrow\infty}\sum_{n\leq x} \left(1-\frac{\log n}{\log x}\right)\frac{a_n}{...
10
votes
0answers
724 views

Convexity of Jacobi's theta function with zero argument

This question may be elementary, I have asked it on math.stackexchange.com but have not received any answer yet. Note that I am not an expert on theta/elliptic functions. Define Jacobi's theta ...
10
votes
0answers
457 views

Parametrisations for Null Temperature Functions: nonuniqueness of solutions to the Heat Equation

Disclaimer I expect this is a highly open problem, but maybe I'm wrong and someone has come up with some answers besides those given here. In any case, all information appreciated, thanks! Definition ...
10
votes
0answers
479 views

Adeles of Holomorphic Functions

In number theory, an adele is a restricted product of elements of the completion at each prime. For function fields, we take (a kind of) product of the completion at each point, and at non-singular ...
9
votes
0answers
231 views

Functional equation or analytic continuation of certain approximations to $\zeta^z(s)$?

Let $z$ be a complex number and $\omega(n)$ denote the number of distinct prime factors of the natural number $n$. I am considering the arithmetic functions $|\mu(n)|z^{\omega(n)}$ and their ...
8
votes
0answers
86 views

Borel-Ecalle re-summation and resurgence: Criteria and Results

This is about the theory of Borel-$\acute{\textrm{E}}$calle re-summation and resurgence, see Refs below. This states that the perturbative series (say of the vacuum expectation value of an operator $\...
8
votes
0answers
232 views

Cohomology of complex manifold vs cohomology of its complex submanifold

Let $X$ be a smooth complex analytic manifold. Let $Z\subset X$ be a smooth compact analytic submanifold. Let $A$ be a holomorphic vector bundle over $X$. Assume that $$H^i(Z, A|_Z)=0 \mbox{ for any } ...
8
votes
0answers
197 views

Analytic space not embeddable in any complex manifold

I am looking for an example of a compact complex analytic space, reduced and irreducible, which does not admit any holomorphic embedding into any (smooth) complex manifold (possibly non-compact). I ...
8
votes
0answers
165 views

Smooth quotients of algebraic spaces that are varieties away from codimension $\ge 2$ subset

This is a question about when a smooth complex algebraic space that is very close to being an algebraic variety is actually an algebraic variety. General question: Let $X$ be a smooth separated ...
8
votes
0answers
369 views

rings of modular functions on the upper half plane

Let $\Gamma_1\le SL_2(\mathbb{Z})$ be a noncongruence subgroup of finite index. Let $\Gamma_2\le SL_2(\mathbb{Z})$ be another subgroup of finite index. Let $M_0(\Gamma_i)$ denote the ring of modular ...
8
votes
0answers
104 views

Extremal length of graphs in surfaces

Given a surface $\Sigma$ with conformal structure $\omega$, the extremal length of a homotopy class $\gamma$ of curves in $\Sigma$ is defined to be $$ \sup_{g \in \omega} \frac{\ell_g(\gamma)^2}{A_g(\...
8
votes
0answers
121 views

What does this number tell me about a convex lattice polygon?

EDIT: I realized I'd tricked myself by working with a too special case of $f$, the question is now updated (boundary lattice points replaced vertices). Suppose I have a convex lattice polygon $P$, ...
8
votes
0answers
521 views

What function space does holomorphic functional calculus give us?

Let $A$ be a unital Banach algebra, $U$ be an open subset of $\mathbb{C}$, and $A_U:=\{x\in A:\sigma(x)\subset U\}$. Holomorphic functional calculus says that any holomorphic function $f:U\rightarrow\...
8
votes
0answers
258 views

Are the Chern numbers of a hyperbolic-type compact complex manifold bounded in terms of the Euler number?

Let $X$ be an $n$-dimensional compact Kahler manifold with negative first Chern class. Are its Chern numbers $\prod_{i=1}^{n-1} c_i^{k_i}$, over $k _i \geq 0$ with $\sum ik_i = n$, bounded in terms ...
8
votes
0answers
446 views

The natural generalization of Euler's derivation of the Basel sum

Euler proved that $$\sum_{n=0}^\infty \frac{1}{n^2} = \frac{{\pi}^2}{6}$$ by comparing the $z^3$ term in the power series expression of $\sin(z)$ given by $$\sin(z) = z - \frac{z^3}{3!} + \frac{z^5}{...
8
votes
0answers
384 views

$C^\infty$ function $f:{\bf C}\mapsto {\bf C}$ such that $f(z)\in\overline{{\bf Q}(z)}$ for all $z\in {\bf C}$

Suppose that $f:{\bf C}\mapsto {\bf C}$ is a $C^\infty$ function such that $f(z)\in\overline{{\bf Q}(z)}$ for all $z\in {\bf C}$, ie $f(z)$ is algebraic over the field ${\bf Q}(z)$ generated by $z$ ...
7
votes
0answers
98 views

What is known about the following series?

For $k\in{\mathbb Z}^2$ write $|k|=\sqrt{k_1^2+k_2^2}$ for the euclidean norm. Then let $g(k)=gcd(k_1,k_2)$. For $s\in\mathbb C$ let $$ D(s)=\sum_{\substack{k\in{\mathbb Z}^2}\\ k\ne 0}\frac{|k|}{g(k)}...
7
votes
0answers
178 views

The inverse of the digamma function

The gamma function is increases on the interval $(x_0, \infty),$ where $x_0$ denotes the unique zero of the digamma function on the positive half line. The inverse function of gamma function defined ...
7
votes
0answers
177 views

Does this bound on an average over character sums have a more direct proof?

A special case of a well known result of Ingham is that $$\sum_{n\leq x} d(n)d(n+1)=\frac{6}{\pi^2}x(\log x)^2+O(x\log x)$$ where $d(n)$ is the number of divisors of $n$. Ingham's results, which ...
7
votes
0answers
209 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 ...
7
votes
0answers
187 views

Biholomorphic neighborhoods of the boundary of Stein domains

Let $(X_1,J_1)$ and $(X_2,J_2)$ be Stein domains with the same contact boundary $(Y,\xi)$. Under what conditions does there exist a biholomorphism between a neighborhood of their respective boundaries ...
7
votes
0answers
124 views

Finite covers in complex analytic geometry

Given a complex manifold or complex analytic space, one has the standard notion of open set. There are two different Grothendieck topologies that one can define using this notion, one where covers ...
7
votes
0answers
247 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 {\...
7
votes
0answers
404 views

Convergence at the radius of convergence

Suppose I have (roughly speaking) a multivalued meromorphic function $f(z)$ on all of $\mathbb{C}$ that is single-valued and holomorphic on the open unit disc and has some branch points of finite ...
7
votes
0answers
216 views

A zeta function using half of the primes

It is well known that the zeta function satisfies the Euler product formula. See this wikipedia article. Enumerate all primes by $p_1, p_2, \ldots $ in ascending order. Set $S$ to be the set of all $...
7
votes
0answers
866 views

Is this Fourier integral well-known?

The following integral is a special case of one that arises in an economics problem: $I(u_{1}, u_{2}) := \displaystyle \int_{z_{1}=-\infty}^{\infty} \int_{z_{2}=-\infty}^{\infty} \frac{ \displaystyle ...
7
votes
0answers
463 views

planar mappings that preserve elliptic measure

Let $D_1$ and $D_2$ be two bounded simply connected Jordan domains in $\mathbb{R}^2$. By Carathéodory's Theorem there exists a homeomorphism $f:\bar{D}_1 \to \bar{D}_2$ such that the restriction $f:...
7
votes
0answers
174 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 ...
6
votes
0answers
144 views

Extremal polynomial majorants of $\log{|f|}$: a multivariate extension of a theorem of Carneiro and Vaaler

Carneiro and Vaaler have proved, as an application of their work on Beurling-Selberg extremal majorants, that for any non-zero complex polynomial $f(z) \in \mathbb{C}[z]$, the infimum value of the ...
6
votes
0answers
168 views

Sobolev space under Mellin transform

The Mellin transform is known to be an isomorphism see wikipedia between $M:L^2(0, \infty) \rightarrow L^2(-\infty, \infty)$ where $$M(f):= \frac{1}{\sqrt{2\pi}}\int_0^{\infty} x^{-\frac{1}{2} + is} ...
6
votes
0answers
148 views

Reference request: normal form of k-differentials and flat surfaces at a puncture

Let $f(z)$ be a holomorphic function defined on a punctured neighborhood of $z=0$ with non-essential singularity of degree $d$ at $0$ (namely, $f(z)=z^dh(z)$, where $h(z)$ is a holomorphic function ...