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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3 votes
1 answer
449 views

Taylor expansion of Stieltjes Transform

I'm trying to derive a very basic result stated in several books on random matrix theory (e.g. Terry Tao's book and Potters & Bouchaud's book). Given a symmetric matrix $A \in \mathbb{R}^{N \times ...
1 vote
0 answers
200 views

Proving that quotient of orthogonal polynomials is a Padé approximant of Stieltjes transform

This question is reposted from Math Stack Exchange (you can see the original post here). The motivation for reposting is that I feel like the question isn't getting much attention in MSE - if there is ...
5 votes
2 answers
277 views

The constant of the reverse Hölder inequality for polynomials

Denote by $D(r)$ the disc at the origin of radius $r>1$. Denote by $P_m$ the set of polynomials of degree $m$. Since $P_m$ is finite dimensional, there is a constant $C(m,r)$ such that $$ \|p\|_{L^{...
7 votes
0 answers
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 ...
1 vote
0 answers
75 views

Do we have a Grauert-Fischer theorem for non-trivial families?

This question is related to my previous question. Let $X$ be a compact complex manifolds and $\Delta\in \mathbb{C}^n$ be a small neighborhood of $0$. A family of deformations of $X$ over $\Delta$ is a ...
1 vote
0 answers
96 views

Is an isomorphism between holomorphic vector bundles still holomorphic with respect to a deformation parameter?

Let $X$ be a compact complex manifold and $E$ be a finite dimensional holomorphic vector bundle on $X$ with a fixed $\bar{ \partial}$-connection $\bar{\partial}_E$. Now we consider a small ...
2 votes
1 answer
172 views

Inverse of Bochner–Martinelli formula

Suppose that $f$ is a holomorphic function on a domain $D$ in $\mathbb{C}^n$, $\partial D$ is smooth, and $f$ is $C^1$ on $\partial D$. Then, the Bochner-Martinelli formula states that $f(z) = \int_{\...
3 votes
0 answers
161 views

Topology of level sets for meromorphic function

Let $F$ be a meromorphic function on $\mathbb{C}$. I consider the "level set" $$E_\varepsilon=\{z:|F(z)|\leq\varepsilon\}.$$ My objective is to find conditions under which $E_\varepsilon$ ...
2 votes
1 answer
160 views

A specific question on the Griffiths' paper: the reduction of the pole order

If someone has gone through the Griffiths' paper ``On the periods of certain rational integrals: I,'' could you help me to understand Lemma 8.10? I don't get why $\eta\in Z^{q,k+1}(l-1)$; although $\...
6 votes
3 answers
504 views

A need for analytic continuation of a finite sum function

Let $\varphi(n):=(-1)^{n+1}(n+1)2^{2n}$. I am able to prove the following identity (${\color{red}{\mathbf{LHS}}}$=infinite series, ${\color{blue}{\mathbf{RHS}}}$=finite sum) \begin{align*} {\color{red}...
2 votes
0 answers
82 views

Does the reduction of the pole order to compute the Poincare residue work?

I am trying to understand the Poincare residue and referring to On Computing Picard-Fuchs Equations, which is cited by Wikipedia's page on the Poincare residue. On pp. 5--6, he gives a way to compute ...
-1 votes
1 answer
139 views

On the bound for $\int_{x}^{x+i\infty} (\cot(\pi z)+ i)z^{-s} \, \mathrm{d}z$

I'm reading Titchmarsh's "The theory of the Riemann zeta function", and on p.81 it is claimed that $$ \int_{x}^{x+i\infty} (\cot(\pi z)+i)z^{-s} \, \mathrm{d}z \ll \frac{x^{-\sigma}}{2(n+1)\...
3 votes
1 answer
150 views

A limit arising from Mellin Inversion: How to compute a specific term of an asymptotic series?

So I am wondering if there exists a general procedure for the following problem: given a monotonically increasing function $f(n)$ which is nonegative on the interval $[0,\infty)$ and grows faster than ...
3 votes
0 answers
72 views

Separate holomorphicity implies holomorphicity on analytic varieties

Suppose that $M$ and $N$ are two complex analytic varities and suppose that $f\colon M\times N \to \mathbb{C}$ is a map. Further assume that $f$ is such that for every $p\in M$ the map $f(p,\cdot)\...
1 vote
0 answers
97 views

Contour integral with two essential singularity

I'm solving problems on the Gamma random variables and there is this question where it wants me to calculate the Mellin transform of sum of two independent Gamma variables from their moment generating ...
0 votes
0 answers
75 views

Completeness of a normed space

We consider the set $\mathcal{PC}([-r,0],X)$ $$\mathcal{PC}([-r,0],X):=\{\varphi:[-r, 0] \rightarrow X: \varphi \text{ is continuous everywhere except for a finite number of points } t_* \text{ ...
5 votes
1 answer
323 views

Boundary zeros of a holomorphic function $f: \Omega \to \Bbb C$

My question stems from the following result about holomorphic functions on the unit disc: "A function, continuous on the closed unit disc, holomorphic inside, and vanishing on an open subset of ...
7 votes
1 answer
508 views

Help finding an analytic continuation

I am looking for the analytic continuation of \begin{align*} & f_m(v,w) := \sum\limits_{k,l=0}^\infty v^k w^l {k+l+m \choose k} {k+l+m \choose l} \ , \end{align*} where $m \in \{1,2,...\}$ is ...
2 votes
1 answer
178 views

The sum of $q^{-2}$ over nonzero Gaussian integers

I'm reading about the Weierstrass zeta function. In this context, $\phi(z)=\zeta(z)-\pi\bar{z}$ is periodic over the lattice $$\mathcal{L}=\{a+bi\mid a,b\in\mathbb{Z}\}.$$ If we take $w\in\mathcal{L}\...
4 votes
0 answers
115 views

Matrix product of entire functions

Suppose I have two $d \times d$ entire matrix functions $F, G$ defined on $\mathbb{C}$ with the the property that $\|FG^*\|_{L^\infty(\mathbb{C})} < \infty$. Can anything be said about $F$ and $G$, ...
2 votes
1 answer
185 views

Local equality of functions implies global equality?

The following question arised in my research, and I was unable to settle it after playing with it for sometime. Let $\{a^k_i\}_{i\geq 1}$ (for $k\in \{1,2,3,4\}$) be four sequences of real numbers. ...
0 votes
0 answers
32 views

Resolvent estimates for Laplacians on directed graphs

Let us consider a directed and weighted graph $G$ with $N = |G|$ nodes. Denote the corresponding (weighted) adjacency matrix by $W \in \mathbb{R}^{N\times N}_{\geq 0}$ and let $D$ be the diagonal in-...
0 votes
0 answers
95 views

Asking a reference about the $p$-Laplacian of $|\nabla u|^p$

It is well-known that for a harmonic function $u$, i.e. $$ \Delta u=0, $$ the quantity $|\nabla u|^2$ is subharmonic, i.e. $$\Delta (|\nabla u|^2) \geq 0. $$ Reason: $$\Delta (|\nabla u|^2)= 2 \nabla (...
4 votes
0 answers
251 views

Order of growth of $\left|\frac{1}{\zeta’(\rho)}\right|$ as $\Im(\rho)\rightarrow\infty$?

Let $\zeta$ denote the Riemann zeta function, and let $\rho\in\mathbb{C}$ be a variable that takes its values among the zeros of the zeta function, so that $\zeta(\rho)=0$, and write $\rho=\sigma+it$. ...
7 votes
1 answer
150 views

Plane curve with continuously increasing Hausdorff dimension

In a recent paper, we required the following fact. Proposition 1. There exists a simple closed curve $\gamma\subset\mathbb{C}$ with the following property. If $\phi$ is a biholomorphic map, defined on ...
23 votes
1 answer
1k views

Can we just use the linear term of exponential sums to sum divergent series

Suppose you want to compute the sum $\sum_{n=0}^{\infty} a_n $ You could consider the expression $f(x) = \sum_{n=0}^{\infty} e^{a_n x}$ and try to compute the coefficient of an $x^1$ term in the ...
4 votes
1 answer
177 views

Is there a complete Kahler metric on a bounded domain?

Let $\Omega\subset\mathbb{C}^n$ be a bounded domain. By the theorems of Cheng-Yau and Mok-Yau, $\Omega$ admits a complete Kahler-Einstein metric if and only if $\Omega$ is a pseudoconvex domain. My ...
9 votes
1 answer
1k views

Proving the Replica Trick works

The replica trick attempts to calculate the expectation of the logarithm $X=\log(Z)$ of a random variable $Z$. The wikipedia article describes the logarithm as the limit $$ \log(Z) = \lim_{n\to 0}\...
2 votes
2 answers
165 views

Hardy space inclusion in the right-half plane

I'm looking for an example of a function $u \in H_2$ such that $u \notin H_\infty$, where $H_p$ is the Hardy space on the right-half plane. Since this notation is perhaps not standard, here is a ...
5 votes
1 answer
890 views

Mapping the doubly connected domain to an annulus

Is there a direct proof, using the Riemann mapping theorem for the Jordan domain, than every doubly connected domain in the complex plane can be mapped conformaly onto a round annulus.
2 votes
0 answers
63 views

Differentiable functions on analytic varieties

Let $\iota\colon X\to \Omega\subseteq \mathbb{C}^n$ be a complex analytic variety $X$ in an open subset $\Omega$ of $\mathbb{C}^n$. If $N$ is a smooth manifold and $h\colon M\to X$ is a continuous map,...
1 vote
1 answer
141 views

Bounds for the logarithmic derivative in the Selberg Class

Let $F \in \mathcal{S}$, where $\mathcal{S}$ is the set of $L-$functions in the Selberg Class. Are there established upper and lower bounds for $$\left|\frac{F^{'}(s)}{F(s)}\right|,$$ where $s = \...
5 votes
1 answer
344 views

Is there a meaningful interpretation of an $L^i$-space?

Do complex-normed spaces exist? Is there an extension of $p$-norms to $p\in\Bbb C\setminus\Bbb R$? A while ago I thought of extending $L^p$-spaces to the complex-normed setting. After some discussions,...
1 vote
2 answers
138 views

Location of the negative real roots of certain integer-valued polynomials

The following question on polynomials arose as a potentially helpful intermediate step on a proof of a Theorem that I want to demonstrate. Its statement is quite elementary, and I can think of a ...
2 votes
0 answers
212 views

Zeta function associated with a function $f$

Let the function $f(t) = \cos(at)$, where ($0 < a < 1$). Let us define $$\zeta(z, f) = \frac{1}{\Gamma(z)} \int_0^{+\infty} \frac{t^{z-1}\cos(at)}{e^t-1}\, dt. $$ Is there a general formula that ...
1 vote
0 answers
69 views

Vanishing components of Kähler metric

Let $(X, \omega) $ be a $n$-dimensional complex Kähler manifold such that $\omega^{n-1}=d\alpha $. Does $\partial\alpha^{n-1,n-2} =0$ (resp. $\bar\partial\alpha^{n-2,n-1} =0$) Where $\alpha^{n-1,n-2}$ ...
1 vote
0 answers
139 views

Top cohomology of the canonical class of a compact non-Kähler manifold

Let $X$ be a complex compact manifold of complex dimension $n$. Let $K_X$ denote its canonical class. Is it true that the cohomology group $$H^n(X,K_X)$$ is one dimensional? Remark. If $X$ is Kähler ...
7 votes
1 answer
285 views

When $H^{p,q}_{\bar\partial}(X)$ can be seen as a subspace of $H^k(X,\mathbb C)$?

It is known that for a $\partial\bar\partial$-manifold $X$ (a compact complex manifold satisfies the $\partial\bar\partial$-lemma), the Bott-Chern cohomology $H_{BC}^{\bullet,\bullet}:=\frac{\ker \...
14 votes
1 answer
658 views

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 ...
0 votes
0 answers
137 views

A mistake on a Barnes beta like integral

Consider the Barnes beta like integral: Where Re$(a)$ ,Re$(b)$ is greater than $0$ and $c$ is not a negative integer, where $|z|<1$ and $|$arg$(-z)$|$<\pi$, $$\frac{1}{2\pi i}\int_{-i\infty}^{i\...
2 votes
1 answer
199 views

An integral transform computation

In Erdelyi, Tables of Integral Transforms, p. 344 Section 7.2. they note that \begin{align} \frac{1}{2 \pi i} \int_{c-i\infty}^{c+i\infty} s^{\nu} e^{\alpha s^2} x^{-s} \, ds = 2^{-\nu/2} \pi^{-...
4 votes
1 answer
327 views

Riemann mapping theorem with smooth boundary

This is closely related to the question here. The setup is that $U\subset\mathbb{C}$ is an open bounded simply connected domain with $C^\infty$ boundary. If $\phi:U\rightarrow\mathbb{D}$ is a ...
0 votes
0 answers
37 views

lower bound on an ellipse

Is there anyway to estimate the lower bound for $Re(zP'(z)/P(z))=\sum_{k=1}^n\frac{z}{z-z_k}$ on an ellipse where $P(z)=\prod_{k=1}^n(z-z_k)$ with $z_k, 1\leq k\leq n$ lying entirely within the ...
16 votes
3 answers
3k views

When is a holomorphic submersion with isomorphic fibers locally trivial?

A justly celebrated theorem by Ehresmann states that a proper smooth submersion $\pi: X\to S$ between smooth manifolds is locally trivial in the sense that every point $s\in S$ downstairs has a ...
2 votes
1 answer
86 views

Pair of positive harmonic functions with negative inner product in Drury-Arveson space

Define a reproducing kernel on the Euclidean ball in $\mathbb{C}^d$ by $$k(z,w)=\frac{1}{1-\langle z,w\rangle}+\frac{1}{1-\langle w,z\rangle}-1.$$ Call the corresponding real reproducing kernel ...
2 votes
1 answer
152 views

Bounding minimal absolute value of a point on a complex algebraic variety

Given a system of complex polynomial equations in $n$ variables, giving rise to an affine variety $V \subseteq \mathbb C^n$, is there a bound $b \in \mathbb R$ such that if $V(\mathbb C) \neq \...
8 votes
2 answers
343 views

Real analytic subvariety in complex manifold which is complex outside of its singular set

Let $M$ be a complex manifold, and $Z \subset M$ a closed real analytic subvariety. Suppose that the set of smooth points in $Z$ is complex analytic in $M$. Will it follow that $Z$ is complex analytic?...
8 votes
2 answers
460 views

Embedding open connected Riemann surfaces in $\mathbb{C}^2$

This question arises in the context of a question asked on MSE: Are concrete Riemann surfaces Riemann domains over $\mathbb{C}$. Part of the answer to that question is the question above which is ...
4 votes
0 answers
57 views

Is there a dense set of Lipschitz functions in $H^\infty(U)$, each of which maps $(1,0,\ldots,0)$ to 1, where $U$ is the unit ball in $\mathbb{C}^N$?

Let $U$ be the open unit ball in $\mathbb{C}^N$, let $A(U)$ be the algebra of functions analytic on $U$ and continuous on $\bar U$, and let $u=(1,0,\ldots,0)$. Let $\mathcal{B}=\{f\in H^\infty (U): \|...
2 votes
0 answers
163 views

Can the equation $1+z^p+z^q+z^r=z^n$ have multiple complex roots $z$?

The math overflow post asks whether the equation $1+z^p+z^q=z^n$ can have multiple complex roots where $p<q<n$ (On the irreducibility of certain trinomials and quadrinomials). Q. Let us ...

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