Questions tagged [cv.complex-variables]
Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.
3,168
questions
7
votes
1
answer
193
views
Greatest lower bound for subordination
Consider the set $X$ of all analytic functions $f$ in the unit disk $U$ satisfying
$f(0)=0, f'(0)\neq 0$. We say that $f\prec g$ if there exists
$\phi\in X$ which maps $U$ into itself, and $f=g\circ\...
-6
votes
1
answer
338
views
On gaps between consecutive zeros of the Riemann zeta function
Let $\gamma$ denote the imaginary part of a non-trivial zero of the Riemann zeta function. Do there exist some function $f$ such that $\gamma_{n+1} - \gamma_n > f(n)>0$ for all large $n$? To be ...
9
votes
2
answers
1k
views
On the error term of the Riemann explicit formula
Let: $\rho$ be a non-trivial zero of the Riemann zeta function, $\Lambda$ be the von-Mangoldt function and $\psi(x) =\sum_{n \leq x} \Lambda(n)$. What is the best known upper bound for
$$f(x, T) := \...
1
vote
2
answers
151
views
Transcendental functions with two prescribed values
Let $\alpha$ and $\beta$ two algebraic numbers lying in unit ball. Let $T:=(t_k)_k$ be an increasing sequence of positive integers such that $t_{k+1}/t_k$ tends to $1$ as $k\to \infty$.
I would like ...
0
votes
0
answers
43
views
Solving the Global Relationship of the Dampened String $q_{tt} + \frac{\rho}{2} q_{t} - q_{xx}=0$ with the Method of Fokas
I was interested in learning more about the Fokas method for solving partial
differential equations after hearing the impressive claim that it could resolve
any linear partial differential equation. I ...
0
votes
0
answers
70
views
A coradius of convergence - biggest open disk contained in the image of a power series?
Let $f \in \mathbb{C}\{z_1,\dots,z_n\}$ be non-constant with $f(0) = 0$, where $n \geq 1$, and let $D$ be its domain of convergence. Recall that for $n=1$ this is just some open disk $\mathbb{D}_r(0)$ ...
19
votes
6
answers
1k
views
What is the relationship between $\sum_{n=0}^\infty f(n) x^n$ and $-\sum_{n=1}^\infty f(-n) x^{-n}$?
Background
Taking a relatively arbitrary combination of exponential and polynomial terms, for instance
$$\sum_{n=0}^\infty \left(n^{2}\sin\left(n\right)+n\cos\left(3n-2\right)\right)\cos\left(5n+1\...
6
votes
1
answer
226
views
Fractional integrals and $\sum f(n) n^x$
Preamble
The following is a rather unrigorous way to obtain the Euler-Maclaurin formula. Consider some $\sum_{n=1}^\infty f(n)$. We may rewrite this as
$$\sum_{n=1}^\infty f(n)=\sum_{n=1}^\infty \sum_{...
9
votes
1
answer
397
views
How to prove that $\phi'(z)<0$ for $\theta\in (0,\pi)$?
Let $a_1=(1,0), a_2$ be two points on the unit circle $T$ of the complex space $\Bbb C$. Assume that the angle between $a_1$ and $a_2$ are $\theta$ (see the image below):
Define the function
$$
r(z)=\...
15
votes
3
answers
1k
views
Pointers for direct proof of extension of the Descartes Rule of Signs to complex polynomials?
The following describes an extension of the Descartes Rule of Signs to polynomials with complex coefficients.
First, I need to define the notion of a "sweep"... Given a complex polynomial p(z) := c0 ...
1
vote
2
answers
238
views
Reference request and clarification for Central Limit Theorem for complex random variables
I'm looking for a reference and a proof of the following version (or eventually a more general version) of the Central Limit Theorem for complex random variables.
Theorem. Let $Z_1, Z_2, \dots, Z_n$ ...
0
votes
0
answers
78
views
Non-triviality of the sum of simple rational functions
Recently, in the study of unicity problems in complex analysis, I met a problem that can be stated in the following way,
Let $\{m_i\}_{i=0}^{3}$ and $\{n_i\}_{i=0}^{3}$ be eight integers in $\mathbb{Z}...
17
votes
2
answers
2k
views
Is this entire function a square?
Let $f$ be the entire function on $\mathbb C$ defined by
$$
f(z)=\frac{z-\sin z}{z}.
\tag{1}\label{1}$$
It is easy to see that $f$ is positive on $\mathbb R^*$ and has a zero of order 2 at 0.
Does ...
2
votes
1
answer
238
views
On the upper bound for $|\zeta(s)|$ near the zeta zeros
Let $T \in \mathbb{R}$ be large and $\rho$ be a non-trivial zero of the Riemann zeta function. Assume that $|\rho|=|\rho_T| \approx T$ and let $\varepsilon_T \approx \frac{\log \log T}{\log T}$. Is it ...
21
votes
4
answers
6k
views
When I can safely assume that a function is a Laplace transform of other function?
If I have a function and I want to represent it as being the Laplace transform of another, that is, I want to be sure that there is $\hat{f}(s)$ such that my function $f(x)$ can be written as:
$$f(x) =...
14
votes
1
answer
2k
views
How to interpret Gauss's late fragments on conformal mapping of the interior of an ellipse (to the unit disk) in modern mathematical terms?
My question refers to some not very well known (and unpublished) fragments of Gauss that treat the problem of finding a conformal mapping (angle-preserving mapping) in the complex plane from the ...
3
votes
0
answers
114
views
An open problem of Hardy and Littlewood on $p$-integral means
In Duren's book "Theory of $H^p$ spaces" (MSN) in the comment section after Section 4, it is mentioned that Littlewood and Hardy proved in Some properties of conjugate functions that if $u$ ...
0
votes
0
answers
51
views
Optimization in complex vector space
I am trying to solve the following question
$$ \max_{ \mathbf{w}\in\mathbb{C}^N } \ f(\gamma)$$
$$\textrm{s.t.} \ \gamma=a|\mathbf{h}^H\mathbf{w}|^2$$
$$\|\mathbf{w}\|^2\leq b,$$
$$|\mathbf{g}_{k}^H\...
4
votes
1
answer
458
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
201
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
280
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
$$
\|p\|_{L^{...
7
votes
0
answers
210
views
Partitions, weights and polynomials with roots on the unit circle
Let us consider the set $[n]=\{1,\ldots,n\}$ and all of its partitions into exactly $m$ blocks, but let us allow each block to be internally ordered. For example, taking $n=6$ and $m=2$, we will ...
1
vote
0
answers
75
views
Do we have a Grauert-Fischer theorem for non-trivial families?
This question is related to my previous question. Let $X$ be a compact complex manifolds and $\Delta\in \mathbb{C}^n$ be a small neighborhood of $0$. A family of deformations of $X$ over $\Delta$ is a ...
1
vote
0
answers
97
views
Is an isomorphism between holomorphic vector bundles still holomorphic with respect to a deformation parameter?
Let $X$ be a compact complex manifold and $E$ be a finite dimensional holomorphic vector bundle on $X$ with a fixed $\bar{
\partial}$-connection $\bar{\partial}_E$.
Now we consider a small ...
2
votes
1
answer
173
views
Inverse of Bochner–Martinelli formula
Suppose that $f$ is a holomorphic function on a domain $D$ in $\mathbb{C}^n$, $\partial D$ is smooth, and $f$ is $C^1$ on $\partial D$. Then, the Bochner-Martinelli formula states that
$f(z) = \int_{\...
3
votes
0
answers
161
views
Topology of level sets for meromorphic function
Let $F$ be a meromorphic function on $\mathbb{C}$.
I consider the "level set" $$E_\varepsilon=\{z:|F(z)|\leq\varepsilon\}.$$ My objective is to find conditions under which $E_\varepsilon$ ...
2
votes
1
answer
160
views
A specific question on the Griffiths' paper: the reduction of the pole order
If someone has gone through the Griffiths' paper ``On the periods of certain rational integrals: I,'' could you help me to understand Lemma 8.10?
I don't get why $\eta\in Z^{q,k+1}(l-1)$; although $\...
6
votes
3
answers
509
views
A need for analytic continuation of a finite sum function
Let $\varphi(n):=(-1)^{n+1}(n+1)2^{2n}$.
I am able to prove the following identity (${\color{red}{\mathbf{LHS}}}$=infinite series, ${\color{blue}{\mathbf{RHS}}}$=finite sum)
\begin{align*}
{\color{red}...
2
votes
0
answers
84
views
Does the reduction of the pole order to compute the Poincare residue work?
I am trying to understand the Poincare residue and referring to On Computing Picard-Fuchs Equations, which is cited by Wikipedia's page on the Poincare residue.
On pp. 5--6, he gives a way to compute ...
-1
votes
1
answer
141
views
On the bound for $\int_{x}^{x+i\infty} (\cot(\pi z)+ i)z^{-s} \, \mathrm{d}z$
I'm reading Titchmarsh's "The theory of the Riemann zeta function", and on p.81 it is claimed that
$$ \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
151
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
103
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
76
views
Completeness of a normed space
We consider the set $\mathcal{PC}([-r,0],X)$
$$\mathcal{PC}([-r,0],X):=\{\varphi:[-r, 0] \rightarrow X: \varphi \text{ is continuous everywhere except
for a finite number
of points } t_* \text{ ...
5
votes
1
answer
335
views
Boundary zeros of a holomorphic function $f: \Omega \to \Bbb C$
My question stems from the following result about holomorphic functions on the unit disc:
"A function, continuous on the closed unit disc, holomorphic inside, and vanishing on an open subset of ...
7
votes
1
answer
509
views
Help finding an analytic continuation
I am looking for the analytic continuation of
\begin{align*}
& f_m(v,w) := \sum\limits_{k,l=0}^\infty v^k w^l {k+l+m \choose k} {k+l+m \choose l} \ ,
\end{align*}
where $m \in \{1,2,...\}$ is ...
2
votes
1
answer
181
views
The sum of $q^{-2}$ over nonzero Gaussian integers
I'm reading about the Weierstrass zeta function. In this context,
$\phi(z)=\zeta(z)-\pi\bar{z}$
is periodic over the lattice
$$\mathcal{L}=\{a+bi\mid a,b\in\mathbb{Z}\}.$$
If we take $w\in\mathcal{L}\...
4
votes
0
answers
116
views
Matrix product of entire functions
Suppose I have two $d \times d$ entire matrix functions $F, G$ defined on $\mathbb{C}$ with the the property that $\|FG^*\|_{L^\infty(\mathbb{C})} < \infty$. Can anything be said about $F$ and $G$, ...
2
votes
1
answer
185
views
Local equality of functions implies global equality?
The following question arised in my research, and I was unable to settle it after playing with it for sometime. Let $\{a^k_i\}_{i\geq 1}$ (for $k\in \{1,2,3,4\}$) be four sequences of real numbers. ...
0
votes
0
answers
32
views
Resolvent estimates for Laplacians on directed graphs
Let us consider a directed and weighted graph $G$ with $N = |G|$ nodes. Denote the corresponding (weighted) adjacency matrix by $W \in \mathbb{R}^{N\times N}_{\geq 0}$ and let $D$ be the diagonal in-...
0
votes
0
answers
96
views
Asking a reference about the $p$-Laplacian of $|\nabla u|^p$
It is well-known that for a harmonic function $u$, i.e.
$$ \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
151
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 ...
24
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
$$
\log(Z) = \lim_{n\to 0}\...
2
votes
2
answers
170
views
Hardy space inclusion in the right-half plane
I'm looking for an example of a function $u \in H_2$ such that $u \notin H_\infty$, where $H_p$ is the Hardy space on the right-half plane. Since this notation is perhaps not standard, here is a ...
5
votes
1
answer
904
views
Mapping the doubly connected domain to an annulus
Is there a direct proof, using the Riemann mapping theorem for the Jordan domain, than every doubly connected domain in the complex plane can be mapped conformaly onto a round annulus.
2
votes
0
answers
64
views
Differentiable functions on analytic varieties
Let $\iota\colon X\to \Omega\subseteq \mathbb{C}^n$ be a complex analytic variety $X$ in an open subset $\Omega$ of $\mathbb{C}^n$. If $N$ is a smooth manifold and $h\colon M\to X$ is a continuous map,...
1
vote
1
answer
147
views
Bounds for the logarithmic derivative in the Selberg Class
Let $F \in \mathcal{S}$, where $\mathcal{S}$ is the set of $L-$functions in the Selberg Class. Are there established upper and lower bounds for $$\left|\frac{F^{'}(s)}{F(s)}\right|,$$ where $s = \...