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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4 votes
0 answers
148 views

Cartan–Remmert reduction of an algebraic variety

Let $V$ be a normal connected algebraic (say, quasi-projective) variety over complex numbers. Assume that underlying complex analytic space $V^\text{an}$ is holomorphically convex, and thus admits the ...
2 votes
0 answers
87 views

Examples of compact non-Kähler complex manifolds with Kodaira dimension zero

Let $X$ be a minimal compact non-Kähler complex manifold. Suppose that Kodaira dimension $\kappa(X)=0$. Is there a known example where the canonical bundle is not holomorphically torsion? For ...
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3 votes
1 answer
81 views

Probability of a number being a bound for roots

Consider the polynomial $p(z)=\sum_0^na_iz^k$ where $a_n=1$ and $a_k \sim N(0,1)$, $k=0,1,2,\dotsc,n-1.$ What is the probability that 2 will be a bound of the roots of the polynomial? How can we find ...
4 votes
0 answers
104 views
+50

Is $(m,n)=(2,3)$ the only solution to $\sum_\limits{k \in\Bbb Z}\frac1{(W_k(x)+1)^m}=\sum_\limits{k \in\Bbb Z}\frac1{(W_k(x)+1)^n}$?

From discussions 1, 2, @HenriCohen wrote a paper on Lambert $W$-Function Branch Identities which includes identities such as $$\sum_\limits{k \in\Bbb Z}\frac1{(W_k(x)+1)^2}=\sum_\limits{k \in\Bbb Z}\...
1 vote
1 answer
33 views

Discrete uniqueness sets for the two-sided Laplace transform?

Let $f : \mathbb R_+ \to \mathbb C$ be a measurable and integrable function where $\mathbb R_+ = [0,\infty)$. The Laplace transform of $f$ is given by $$ Lf(s) = \int_0^\infty f(x)e^{-sx} \, dx. $$ A ...
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0 votes
0 answers
38 views

Laplace operator with respect to complex variable

Assume there is an operator $L$, which acts on a function $P(x,y)$, where $x, y \in \mathbb{R}^n$. Now let's say I make a change of variables $$\mathbb{R}^n \to \mathbb{C}^n$$ $$x \to \tilde{x}$$ such ...
1 vote
0 answers
85 views

Do we have an equivariant Newlander-Nirenberg theorem for finite group action?

Let $X$ be a $2n$-dimensional $C^{\infty}$-manifold, let $G$ be a finite group acting smoothly on $X$. Let $J: TX\to TX$ be a map such that $J^2=-id$ and $g_*Jg^{-1}_*=J$ for any $g\in G$. We can ...
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0 votes
0 answers
28 views

Request for reference about polynomial bounds

I am looking for some good and comprehensive survey about the bounds of the roots of a complex polynomial.Kindly suggest me some books or papers
0 votes
1 answer
98 views

A holomorphic function in the open unit disk satisfying certain properties

Does there exist a function which is holomorphic in $|z|<1,$ continuous in $|z|\leq1$ and such that the series $\sum |a_n|$ is divergent, where $a_n$'s coefficients in the Taylor series expansion ...
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2 votes
1 answer
88 views

References for group of invariance of the Painlevé property

I'm searching for some references about groups of invariance of the Painlevé property other than the book of Robert Conte or more generally birational transformation of Riemann surfaces.
8 votes
0 answers
135 views

Switching the order of a summation and replacing a series by its analytical continuation

Background A useful trick when trying to analyze a series $\sum_{n=0}^\infty f(n)$ is to expand $f(n)$ as some kind of series, swap the order of summation, and then evaluate the inner infinite sum. ...
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-2 votes
1 answer
90 views

Convergence of scrambled product for Dirichlet-$L$ function with modulo 4 character

A Dirichlet-$L$ function is typically defined by its series, and its Euler product is a consequence of the definition. Here my approach is the other way around. I define the function $$ L_4^*(s) = \...
3 votes
0 answers
54 views

lower bound for zero multiplicity of function formed from determinant of functions

I have a family of single-variable analytic functions, $D(z)$, formed as follows. Let $L$, $n$ and $T_{0}$ be positive integers, $\varphi_{1},\ldots, \varphi_{L}$ be analytic functions in ${\mathbb C}$...
0 votes
1 answer
69 views

Solutions of complex linear difference equations

I'm wondering what the solutions of complex linear difference equations like \begin{equation} f(z+\eta_1)+f(z+\eta_2)+\ldots+f(z+\eta_n)=0,\ \ \ \eta_1 \cdots\eta_n \in \mathbf{C} \end{equation} look ...
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0 votes
0 answers
40 views

Gegenbauer polynomial relation with complex argument

Gegenbauer polynomials, $C_j^{\nu}(t)$, are defined to be the coefficient of $h^j$ in the expansion $(1-2ht +h^2)^{-\nu}$. It can be shown using [Higher Transcendental Functions, Vol 1, Harry Bateman, ...
5 votes
1 answer
182 views

An inequality for polynomials

I have been thinking about the validity of the following inequality: if $P(z)=\sum_{k=0}^na_kz^k, a_n\neq 0$ and $P(z)$ is non-zero in $|z|<1, $ then for $\theta \in [0, 2\pi],$ and $p>0$ \...
2 votes
1 answer
137 views

Linear elliptic equation

Let $\Delta:=\partial_z\,\partial_{\overline {z}} $ be the Laplacian operator. I look for a particular non-trivial solution $u$ of $$\Delta u=\frac{a}{1-|z|^2}u$$ where $u\in C^2(\mathbb{D})$ and $a\...
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3 votes
0 answers
40 views

harmonic envelope of holomorphy

Let $D$ be a (real) domain in $\mathbb R^n=\mathbb R^n+i\lbrace 0 \rbrace\subset \mathbb C^n$. Then, due to P. Lelong, there exists a maximal (complex) domain $\tilde D\subset\mathbb C^n$, $D=\tilde D\...
0 votes
0 answers
101 views

Are two notions of generalized solution of Monge-Ampere equation equivalent?

Let $\Omega\subset \mathbb{C}^n$ be an open subset. Let $u\colon \Omega\to \mathbb{R}$ be a continuous plurisubharmonic (psh) function. The theorem of Chern-Levine-Nirenberg defines a non-negative ...
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0 votes
0 answers
42 views

Absolute error in linearly approximating the sum of sum-of-divisors function

The sum-of-divisors function is defined as $\sigma(k):=\sum_{\ell\mid k}\ell.$ It is well-known that $$\sum_{k\le x}\sigma(k)=\frac{\pi^2}{12}x^2+O(x\log x),$$ and therefore it seems natural to study ...
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1 vote
2 answers
100 views

A characterization of plurisubharmonic functions

Let $\Omega\subset \mathbb{C}^n$ be an open subset. Let $u\colon \Omega\to [-\infty,+\infty)$ be an upper semi-continuous function. Recall that $u$ is called plurisubharmonic (psh) if its restriction ...
  • 19.7k
3 votes
0 answers
121 views

Flatness of tensor products of analytic germs

Let $\mathcal{O}(\mathbb{C}^n)_0$ denote the local ring of germs at the origin of holomorphic functions on $\mathbb{C}^n$. Consider the obvious map $$ \mathcal{O}(\mathbb{C}^n)_0 \otimes_{\mathbb{C}} \...
1 vote
1 answer
202 views

Where or what is the general formula for the $n$th derivative of the power-exponential function $x^x$?

It is well-known that the power-exponential function $x^x$ and its first few derivatives are often taught in calculus. Does the general formula for the $n$th derivative of the power-exponential ...
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14 votes
2 answers
903 views

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 \\ \...
10 votes
2 answers
399 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 ...
0 votes
1 answer
44 views

Analytic continuation of spline wavelets (reference request)

I would like to extend (cubic or higher degrees) spline wavelets to complex domain. First, does this continuation exist? Second, I appreciate it if anyone could point me to some references.
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8 votes
1 answer
531 views

A robust version of "a holomorphic function is determined by its modulus"

It is well known that if $f(z)$ and $g(z)$ are both holomorphic on a (path-)connected open set $C$ and $\lvert f(z)\rvert=\lvert g(z)\rvert$ on $C$ then $f(z)=cg(z)$ on $C$ for some constant $c$. Do ...
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4 votes
2 answers
509 views

How to compute $\sin(\frac{d}{dx})f(x)$?

Assuming $f(x)=e^{-x^2}$ for $x$ in $[-10,10]$, I have tried the following: Fourier transform $\mathcal{F}$: $\frac{d}{dx}$ can be diagonalized as $\mathcal{F}^{-1} i\omega \mathcal{F}$. Therefore, $\...
  • 179
1 vote
0 answers
47 views

Factorization of the shift of a canonical product

Let an entire function $F : \mathbb C \to \mathbb C$ of order $2$ be given by its canonical product $$ F(z)=z^me^{az^2+bz+c} \prod_{w \in Z} E_2(z/w), \quad n \in \mathbb N, $$ where $Z$ is the zero ...
  • 180
7 votes
1 answer
354 views

Do non-projective K3 surfaces have rational curves?

Define a compact Kähler surface $X$ to be a K3 surface if $X$ is simply connected, $K_X \simeq \mathcal{O}_X$, and $h^{0,1}=0$. If $X$ is projective, then a theorem typically attributed to Bogomolov ...
  • 1,165
6 votes
0 answers
244 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 ...
  • 1,175
4 votes
2 answers
132 views

Direct proof of the global submean property for $\log |f|$

Given an entire function $f : \mathbb{C} \to \mathbb{C}$, $\log |f|$ is subharmonic. Globally, this means that for any disk $D_r(c)$ we have the submean property $$\log |f(c)| \le \frac{1}{\mu(D_r(c))...
5 votes
1 answer
337 views

Family of functions with prescribed derivatives

Suppose $f: \mathbb C \times (-1,1) \to \mathbb C$ is a smooth function that satisfies $f(0,t)=1$ for all $t\in (-1,1)$. Assume that for any $k\in \mathbb N$, any $z \in \mathbb C$ and any $t \in (-1,...
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6 votes
1 answer
139 views

Is the Gauss hypergeometric function $_2F_1(a,b;c,z)$ univalent in $\left|z - \frac{1}{2} \right| < \frac{1}{2}$?

Consider the Gauss hypergeometric function $$_2F_1(a,b;c,z) = \sum_{n=0}^{+\infty} \frac{(a)_n(b)_n}{(c)_nn!}z^n, \quad |z| < 1, \quad (x)_n = x(x+1)\cdots(x+n-1), \quad (x)_0 = 1$$ The Encylopedia ...
  • 179
5 votes
1 answer
114 views

Hadamard factorization of a function in the Fock space

An entire function $F: \mathbb C \to \mathbb C$ belongs to the Fock space $\mathcal F^2$ if $$ \int_{\mathbb C} |F(z)|^2e^{-|z|^2} \, dA(z) < \infty. $$ It is well-known that every $F \in \mathcal ...
  • 180
6 votes
1 answer
121 views

Are entire functions uniformly bounded from below on a line through the origin?

Let $F : \mathbb C \to \mathbb C$ be an entire function of finite order. Since the zeros of $F$ are countable there exists a constant $c \in \mathbb R$ such that $F$ is zero-free on the line $e^{ic} \...
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1 vote
0 answers
84 views

Converse of transfer theorem : does asymptotic behaviour of coefficients describe the singularity?

I'm interested in Singularity Analysis after reading this Math.Stack post. In the "Analytic Combinatorics" book by P. Flajolet and R. Sedgewick, at p.390, the following transfer theorem (...
  • 179
2 votes
1 answer
129 views

Roots of rational function

Sorry, I asked a similar question yesterday which contained a mistake in the question posed, here is the real question. Let $(x_n)_{n=1}^N$ be a sequence taking values in $[1,2]$ with the property ...
0 votes
1 answer
151 views

Holomorphic function on $\mathbb C^n$ [closed]

I take $F$ from $\Omega\subset \mathbb C^n$ to $\mathbb C^n$ to be a holomorphic function such that $$| \det(J_F)|\leq 1,$$ where $J_F$ is the Jacobian matrix of $F$. My question: Is there any ...
19 votes
5 answers
954 views

What is the relationship between $\sum_{n=0}^\infty f(n) x^n$ and $-\sum_{n=1}^\infty f(-n) x^{-n}$?

Background Taking a relatively arbitrary combination of exponential and polynomial terms, for instance $$\sum_{n=0}^\infty \left(n^{2}\sin\left(n\right)+n\cos\left(3n-2\right)\right)\cos\left(5n+1\...
  • 1,175
2 votes
1 answer
145 views

Resolving complexes of coherent analytic sheaves

Background Throughout, let $X$ be a smooth complex manifold. It is a classical fact that a coherent analytic sheaf admits a local resolution by locally free sheaves (also known as a local syzygy). ...
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0 votes
1 answer
188 views

Proper journal for a preprint in complex geometry

I ran into the very cryptic paper Proper analytic embedding of $\mathbb{C P}^1$ minus a Cantor set into $\mathbb C^2$ by Orekvov on proper holomorphic embedding of the complement of a Cantor set $C$ ...
  • 751
1 vote
0 answers
104 views

Difference between affine quotient variety and a global quotient orbifold

Given a smooth affine variety $X$ and a finite group $G$ acting by automorphisms on $X$, the quotient space $X/G$ has the structure of an affine variety which is in general not smooth. However, in the ...
1 vote
1 answer
199 views

On a lemma of Łojasiewicz in complex analysis of one variable

Context. The question arises from my former question on the remainder of a power series. Precisely, I was trying to understand if the boundary behavior of power series considered by Ricci in his paper ...
8 votes
2 answers
284 views

Holomorphic maps from a Riemann surface of infinite genus

Let $X$ be a Riemann surface of infinite genus and let $n$ be an arbitrarily large natural number. Do there always exist a closed Riemann surface $Y$ of genus greater than $n$ and a nonconstant ...
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4 votes
1 answer
106 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 ...
3 votes
1 answer
131 views

Entire function with almost periodic boundary condition?

Let $v_1 =\lambda_1 \zeta_1$ and $v_2 = \lambda_2 \zeta_2$ with $\zeta_1 = \frac{4\pi i\omega}{3}$ and $\zeta_2 = \frac{4\pi i\omega^2}{3}$ where $\omega = e^{2\pi i/3}$ is the third root of unity and ...
6 votes
1 answer
398 views

a problem in complex-variable inequality

Let $n\ge2$ be a given positive integer, and $z_{1},z_{2},\cdots,z_{n}\in \mathbb{C}$,such $$|z_{1}|^2+|z_{2}|^2+\cdots+|z_{n}|^2\ge n.$$ Prove or disprove $$f_{n}=\sum_{j=1}^{n}\left|\sum_{I\subseteq ...
  • 3,706
3 votes
3 answers
378 views

Quantitative analytic continuation estimate for a function small on a set of positive measure

The following conjecture about analytic functions arose as a way to show the asymptotic growth for certain PDE solutions. As I am unfamiliar with any results of this type, I thought I'd ask here. In ...
3 votes
1 answer
254 views

Bounds on zeros of rational function

Let $(x_n)_{n=1}^N$ be a sequence taking values in $[1,2]$ with the property that $x_1<x_2<...<x_N$ and $$\frac1N \gtrsim \vert x_j-x_{j-1} \vert \gtrsim \frac1N.$$ We then define a function $...

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