Questions tagged [cv.complex-variables]

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

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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(\...
Dylan Thurston's user avatar
8 votes
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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 $P$, ...
Ketil Tveiten's user avatar
8 votes
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285 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
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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}{...
Jonah Sinick's user avatar
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Etymology of the O-notation for algebras of holomorphic functions

The notation $O(X)$ seems to be a quite standard notation for the algebra of all holomorphic functions on some connected domain in $\mathbb{C}^n$ (or a complex manifold). I would like to know where ...
ssquidd's user avatar
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7 votes
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214 views

Analytic continuation of Dixon's identity

Many well-known combinatorial identities has an analytic version. For example, the following identities $$ 2^n = \sum_{k=0}^n \binom{n}{k} $$ $$ \binom{2n}{n} = \sum_{k=1}^n \binom{n}{k}^2 $$ can be ...
minhtoan's user avatar
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210 views

Partitions, weights and polynomials with roots on the unit circle

Let us consider the set $[n]=\{1,\ldots,n\}$ and all of its partitions into exactly $m$ blocks, but let us allow each block to be internally ordered. For example, taking $n=6$ and $m=2$, we will ...
Luis Ferroni's user avatar
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297 views

Gottfried Helms' tetra-eta series

Here Gottfried Helms introduces the following fascinating divergent series $$ T_2(x)=- \sum_{n=1}^\infty (-1)^n n^{n^x}$$ The terms don't go to zero, so technically the series does not converge ...
Caleb Briggs's user avatar
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201 views

Global generation of $S^n \Omega_X$ for a fake projective plane

Let $X$ be a fake projective plane, namely, a compact complex surface with $$p_g(X)=q(X)=0, \quad K_X^2=9$$ and $K_X$ ample. Since $K_X^2=9 \chi(\mathcal{O}_X)$, Yau's celebrated proof of the Calabi ...
Francesco Polizzi's user avatar
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156 views

The relation between Wolf's and Teichmüller's parametrization of the Teichmüller space

Let $\mathcal{T}_g$ be the Teichmüller space of Riemannian surface structures on an oriented 2-dimensional manifold of genus $g$. Fix a point $S \in \mathcal{T}_g$. There are two different ways to ...
 V. Rogov's user avatar
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holomorphic functions on Stein manifolds reaching maxima in a given point on the boundary (Shilov boundary)

Let $M$ be a Stein manifold with smooth, strictly pseudoconvex boundary, and $x$ a point on its boundary. Is there a holomorphic function $f$ on $M$, smooth on the boundary, with strict maximum of $|f|...
Misha Verbitsky's user avatar
7 votes
0 answers
196 views

Polynomials having all zeros in the closed left half plane

Let $$P(z) =\sum_{k=0}^n(\alpha_k+e^{i\gamma}\beta_k)z^k=P_1(z)+e^{i\gamma}P_2(z)$$ be a polynomial of degree $n$ with $\alpha_k, \beta_k\geq 0$ for $0\leq k\leq n, $ where $$P_1(z) =\sum_{k=0}^n\...
user159888's user avatar
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299 views

Zero derivative on a connected set

I apologize in advance for this (not really research level) question whose answer should be well known. Complex differentiability of a function $f:A\to\mathbb C$ for $A\subseteq \mathbb C$ without ...
Jochen Wengenroth's user avatar
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159 views

Limiting behavior of a sequence of polynomials

Let $f(z)\in\mathbb{C}[z]$ have all its zeros on the line $\Re(z)=\alpha$ for some $\alpha\in\mathbb{R}$. It is an elementary fact (equivalent to Lemma 9.13 here) that if $u\in\mathbb{C}$ and $|u|=1$, ...
Richard Stanley's user avatar
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600 views

How much differs the category of real-analytic manifolds from $C^\infty$ ones?

I was thinking about the difference between the concept of real-analytic function (for any point the Taylor-series of $f$ converge to the function in a neighborhood of the point) and complex analytic (...
John117's user avatar
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If SO$(3,\mathbb C)$ is isomorphic to PGL$(2,\mathbb C)$, what objects do vectors in $\mathbb C^3$ represent in the context of Möbius geometry?

I hope this question isn't too basic or ambiguous for this site. The following is an explicit isomorphism from $\mathrm{PGL}(2,\mathbb C)$ to $\mathrm{SO}(3,\mathbb C)$: $$\left[\begin{matrix}p & ...
wlad's user avatar
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On a paper of Alain Connes entitled 'Around Wilson's Theorem '

A relatively recent paper Alain Connes - Around Wilson's theorem introduced the function $$ S(n,x ) = \sum_{i=1}^n \sin^2\Bigl(\frac{(i-1)! x}{i}\Bigr). $$ In the same paper, he proved that the ...
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198 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 ...
Kevin Smith's user avatar
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7 votes
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428 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 ...
Deepti's user avatar
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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 ...
PVAL's user avatar
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155 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 ...
Oren Ben-Bassat's user avatar
7 votes
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727 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\...
Rand al'Thor's user avatar
7 votes
0 answers
453 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 ...
Neil Strickland's user avatar
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0 answers
225 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 $...
Khalid Bou-Rabee's user avatar
7 votes
0 answers
892 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 ...
Ian Martin's user avatar
7 votes
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491 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:...
HMPanzo's user avatar
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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 ...
RTodd's user avatar
  • 103
6 votes
0 answers
573 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{...
Tian Vlašić's user avatar
6 votes
0 answers
189 views

Reference request: Automorphisms of $\mathbb C\{x,y\}$ which preserve the equation of the cusp, $x^3 - y^2$

In my research I encountered automorphisms of the ring of convergent power series $$\varphi: \mathbb C\{x,y\} \to \mathbb C\{x,y\},$$ which preserve $f = x^3 - y^2$, i.e. $\varphi(f) = f$. I'm ...
red_trumpet's user avatar
  • 1,071
6 votes
0 answers
140 views

Fourier transform and Hodge-$*$ operator

Suppose I have a full-rank lattice $\Lambda\subset\mathbf{C}$. Then the classical Poisson summation formula says $$\sum_{\lambda\in\Lambda}f(\lambda)=\sum_{\lambda\in\Lambda'}\widehat{f}(\lambda)$$ ...
user avatar
6 votes
0 answers
114 views

Complex beta function $\int_{\mathbb{R}^2} (x^2+y^2)^{\alpha-1}((1-x)^2+y^2)^{\beta-1} \,dx\,dy$

I am interested in showing that the integral \begin{align} & \int_{\mathbb{C}} |z|^{2\alpha-2}|1-z|^{2\beta - 2} \,dA(z) \\[8pt] = {} & \int_{\mathbb{R}^2} (x^2+y^2)^{\alpha-1}((1-x)^2+y^2)^{\...
Hampus Nyberg's user avatar
6 votes
0 answers
168 views

Computing residues at $\infty$

As an initial note, let me show by example what I mean by the terminology 'residue at $\infty$' I use in the title. I assume there is some standard terminology for this stuff, so I'd appreciate it if ...
Caleb Briggs's user avatar
  • 1,662
6 votes
0 answers
365 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} $ ...
Max Muller's user avatar
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6 votes
0 answers
305 views

Are the two-valued homogeneous harmonic functions classified?

Question. Is there a classification of homogeneous two-valued harmonic functions on $\mathbf{R}^n$, valid in dimensions $n \geq 3$? For reference, multi-valued functions are familiar objects in ...
Leo Moos's user avatar
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6 votes
0 answers
74 views

Implications of combinatorial results towards discrete function theory on circle packings

Spurred primarily by a conjecture of Thurston in 1985, there was a series of developments in creating a "discrete analytic function" theory for maps between circle packings of complex ...
Jon Hillery's user avatar
6 votes
0 answers
139 views

What does it mean for the torsion to blow up?

Consider the following theorem which is the main result of the Hermitian Curvature Flow paper by Jeffrey Streets and Gang Tian: Theorem 1.1. Let $(M^{2n}, g_0, J)$ be a complex manifold with Hermitian ...
GradStudent's user avatar
6 votes
0 answers
736 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 $u\in\mathbb{C}$, $\alpha$ ...
Fll'Yissetat's user avatar
6 votes
0 answers
218 views

All complex surfaces embed into a common complex manifold

Is there a closed complex manifold into which every closed complex surface embeds?
user avatar
6 votes
0 answers
294 views

Tsuchiya-Ueno-Yamada's proof that sheaves of conformal blocks are locally free

I'm referring to Tsuchiya-Ueno-Yamada's (TUY hereafter) celebrated paper Conformal Field Theory on Universal Family of Stable Curves with Gauge Symmetries. One of the main goals of their paper is to ...
Bin Gui's user avatar
  • 555
6 votes
0 answers
323 views

Cheap bound on $\zeta'(s)/\zeta(s)$ or $L'(s,\chi)/L(s,\chi)$?

Say you are proving an explicit formula for $L(s,\chi)$ and/or the prime number theorem (in arithmetic progressions or not) in the usual way -- that is, shifting a line of integration from $\Re(s) = 1^...
H A Helfgott's user avatar
  • 19.3k
6 votes
0 answers
279 views

Complex factorization of the angular part of the Laplacian

Some time ago some research led me to the following equality: \begin{equation} \frac{1}{\sin^2 \phi }\frac{\partial^2 }{\partial \theta^2} +\frac{\partial^2 }{\partial \phi^2} +\cot \phi \frac{\...
Daniel Alayón-Solarz's user avatar
6 votes
0 answers
214 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 ...
Vesselin Dimitrov's user avatar
6 votes
0 answers
234 views

Bezout theorem for germs of holomorphic functions

UPDATE. It was pointed out by @Dmitri that two smooth curves given by $f=y$ and $g=y+x^k$ in $\mathbb C^2$ provide a simple counterexample. Let $f_1, \ldots, f_p, g_1, \ldots, g_q$ be germs of ...
Dmitri Zaitsev's user avatar
6 votes
0 answers
159 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 ...
Xin Nie's user avatar
  • 1,764
6 votes
0 answers
151 views

Is there Cauchy-Goursat for $1$-cycles without invoking winding numbers?

Depending on one's approach to Complex Analysis in One Variable, Cauchy's Integral Theorem is one of the first interesting results about holomorphic functions in any course. There are several related ...
M.G.'s user avatar
  • 6,683
6 votes
0 answers
127 views

How big may the maximum set of entire function be?

Let us consider an entire function of several complex variables $f(z_1,\dots,z_n)$ and its modulus maximum $$M(r,f):=\max \{ |f(z_1,\dots,z_n)|: |z_1|\le r,\dots,|z_n| \le r \} $$ with $r\ge 0$. How ...
user64494's user avatar
  • 3,309
6 votes
0 answers
142 views

Evaluate $\sum_{\sigma} (2\pi i)^{-n}\oint \frac{f_{\sigma(1)}(u)\dots f_{\sigma_n(1)}(u)}{(u_2 - u_1)\dots (u_n - u_{n-1})}du_1\dots du_n$

In a probability theory paper I found this rather pleasant result: Theorem 4.1 Let $n \geq 2$ and $f_1, \dots, f_n : \mathbb{C} \to \mathbb{C}$ be meromorphic with possible poles at $\{ \mathfrak{p}...
john mangual's user avatar
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6 votes
0 answers
132 views

Is the map taking a matrix to its semisimple part algebraic (or at least holomorphic)?

Let $\text{Mat}_n(\mathbb{C})$ be the set of $n \times n$ complex matrices. Let $\sigma\colon \text{Mat}_n(\mathbb{C}) \rightarrow \text{Mat}_n(\mathbb{C})$ be the map that takes a matrix to its ...
Helen's user avatar
  • 61
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^{-|...
Thomas Kojar's user avatar
  • 4,414
6 votes
0 answers
262 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}...
T. Amdeberhan's user avatar

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