Questions tagged [cv.complex-variables]
Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.
5
votes
0answers
80 views
Do analytic functionals form a cosheaf?
Let $X$ be a complex-analytic manifold.
Consider the sheaf of holomorphic functions $\mathcal{O}_X$ as a sheaf with values in the category of locally convex vector spaces. For $U\subseteq X$ open, we ...
-5
votes
0answers
42 views
What is the Laurent series of z+(1/z)? [on hold]
What is the Laurent series of z+(1/z)? Is it just the series itself?
3
votes
0answers
53 views
Metric with singularities on Riemann Surfaces and the associated Laplacians
I have asked this question on Math Stack Exchange
Metric with singularities and associated Laplacian
but I have not got any answers/comments, therefore I post this question on the MO.
Suppose $M$ ...
0
votes
0answers
30 views
Is there a way to categorise the valleys of a holomorphic function (potentially of $\geq 2$ variables) (multidimensional steepest descent)
More specifically, I am particularly interested in the question: given some $f:\mathbb{C}^n \to \mathbb{C}$, can we categorise $\mathbb{C}^n$ by which valley the steepest descent curves of a point (...
5
votes
1answer
184 views
Dependence of a solution of a linear ODE on parameter
Is the following theorem known, or can be easily derived from known results?
Consider the differential equation
$$w''-kz^{-1}w'=(\lambda+\phi(z))w,$$
where $k>0$ is fixed, $\lambda$ is a large (...
2
votes
0answers
106 views
Is there any neat way to calculate the Fourier transform for an inverse Vandermonde determinant?
$$\mathrm{PV}\int_{\mathbb R^n} \frac{e^{-i\langle w, x\rangle}}{\prod_{j<k}(x_k-x_j)}dx=?$$
Other than integrate this term by term (which might look crazy)?
Let $f(x)=1/\prod_{j<k}(x_k-x_j)$, ...
3
votes
2answers
360 views
Reference request: Oldest complex analysis books with (unsolved) exercises?
Per the title, what are some of the oldest complex analysis books out there with (unsolved) exercises? Maybe there are some hidden gems from before the 20th century out there. I am aware of the ...
1
vote
1answer
116 views
An equation with Gamma Euler function in critical strip
Let
$$
D=\{z \in \mathbb{C} : 0<\Re(z)<\frac{1}{2} \text{ or } \frac{1}{2}<\Re(z)<1 \}
$$
that is the critical strip without critical line.
I have to find if the following equation, with ...
3
votes
0answers
103 views
Injective resolution of the ring of entire functions
Let $R$ be the ring of entire functions on $\mathbb{C}$. I heard that the concrete value of the global dimension of the ring depends on continuum hypothesis.
I would think that the injective dimension ...
3
votes
1answer
48 views
Rational approximation on rotation invariant compact subsets of complex plane
What does the Vitushkin's theorem say about the equality $A(K) = R(K)$ in the special case when $K$ is rotation invariant? More precisely, what are necessary and/or sufficient conditions on $\{|k|: k \...
2
votes
2answers
104 views
Coefficients of entire functions with specified zero set
Let $Z \subseteq \mathbb{C}$ without limit point. By the Weierstrass factorization theorem there is an entire function $h$ those zero set is $Z$. Let $a_n > 0$ be a sequence where $\lim_n \sqrt[n]{...
3
votes
1answer
156 views
Local phase statistics of the nontrivial Riemann zeros
(The question is inspired by Owen Maresh's post)
The local phase of a nontrivial zero $s$ of the Riemann $\zeta$ is the argument of $\zeta'(s)$.
Numerical results on the first 10000 zeros suggest ...
1
vote
0answers
101 views
Defining integrals by residue theorem
I have always been interested in alternative definitions of mathematical objects. I wonder if one can craft an useful definition of definite integral by using the Residue Theorem from complex analysis....
0
votes
0answers
141 views
An (unusual) uniqueness theorem for analytic functions
Let $L$ be the class of analytic functions in $\mathbf{C^*}$
with positive Laurent
coefficients:
$$f(z)=\sum_{-\infty}^\infty c_nz^n,\quad 0<|z|<\infty,\quad c_n\geq 0.$$
Each $f\in L$ has a ...
0
votes
1answer
74 views
Solving or bounding the real part of the integral $\int_0^{2 \pi i m} \frac{e^{-t}}{t-a} dt$
I would be interested in finding a closed form or, at least, bounding (in terms of $m$ as it becomes larger) the real part of the following itnegral:
$$f(m,a):=\int_0^{2 \pi i m} \frac{e^{-t}}{t-a} ...
3
votes
0answers
37 views
Modulus of image of a curve family in a rectangle
I don't expect to get a positive answer to this question but I may as well try.
Let $R$ be the rectangle in $\mathbb{C}$ given by $\{z=x+iy: 0\leq x \leq l, 0 \leq y \leq h\}$ for some $l,h>0$. ...
0
votes
1answer
69 views
meromorphic extension of dirichlet series
Suppose $\{a(n)\}_{n\ge 1}$ is a bounded complex sequence. Let $\phi(s)=\sum_{n\ge 1} \frac{a(n)}{n^s}$. Obviously, the Dirichlet series $\phi(s)$ is absolutely convergent for $\mathcal{R}(s)>1$. I ...
2
votes
1answer
119 views
Must $q$ be analytic?
I have a continuous function $q:\mathbb{R}^+ \to \mathbb{R}^+$. An interesting property of this function is that
$$F(s) = \frac{e^{-q(s)}q(s+1)}{1-e^{s-q(s)}}$$
which also takes $\mathbb{R}^+ \to \...
2
votes
1answer
142 views
Differences of $\omega$-plurisubharmonic functions
Let $X$ be a complex manifold, and $\omega$ a Kähler form on $X$.
A smooth function $\phi$ on $X$ is $\omega$-plurisubharmonic ($\omega$-psh for short) if the form $\omega+\sqrt{-1}\partial\bar{\...
5
votes
0answers
158 views
Regular functions vs holomorphic functions
Let $X$ be an affine smooth variety over the complex numbers, $X^{an}$ its associated smooth complex analytic space, and $\mathcal{O}$, resp. $\mathcal{O}^{an}$ the respective structure sheaves.
Is ...
4
votes
1answer
68 views
Existence of Laurent series with zeroes at $𝑒^2𝑛$ (𝑛∈ℕ0 ) and even faster coefficient decay
This is an extension of an earlier question of mine which corresponds to the case $A = 1$. Precisely, my question is as follows:
Given $A > 0$ fixed but arbitrary, is there a non-trivial ...
4
votes
1answer
160 views
Zeros of derivatives of Dirichlet Eta function
Let
$$
\eta^{(d)}(z) =
\sum_{n=1}^\infty
\dfrac
{(-1)^d(-1)^{n-1}\ln(n)^d}
{n^z}
$$
be the derivative of Dirichlet Eta function of order $d$.
Does it exist any known or not known zero of $\eta^{(d)}...
0
votes
1answer
106 views
What are the holomorphic automorphism groups of unit ball, polydisc, and Hartogs domain in C^n ( n>1)? [closed]
What are the holomorphic automorphism groups of unit ball, polydisc, and Hartogs domain in $\mathbb{C}^n$ ($n>1$) ? I would be pleased if you tell me.
9
votes
1answer
147 views
Existence of Laurent series with zeroes at $e^{2n}$ ($n \in \Bbb{N}_0$) and extremely fast coefficient decay
I am working on a problem in harmonic analysis, which I converted into the following existence problem concerning Laurent series. I am a bit at a loss concerning this problem, since my knowledge of ...
10
votes
0answers
275 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 ...
3
votes
1answer
95 views
Decay of the binomial expansion of $f^{\circ k}$
Suppose $f$ is a holomorphic function in a neighborhood of zero fixing zero. Suppose $f'(0) = \lambda$ and $0<\lambda < 1$. It's not so hard to prove that $f^{\circ k}(z) = f(f(\ldots\text{$k$ ...
3
votes
0answers
36 views
Extremal metric for image of a curve family
Let $U\subset \mathbb{C}$ be a domain and $\Gamma$ some family of curves in $U$ with $\textrm{mod}(\Gamma)<\infty$ and such that $\rho$ is an extremal metric for the modulus. Suppose we are given a ...
27
votes
2answers
1k views
A sum involving roots of unity
Let $n$ be a positive integer and $\zeta$ be a primitive $n$th root of unity. It is not hard to show that
\begin{align*}
\sum_{k=1}^{n-1}\frac{\zeta^k}{1-\zeta^k}=\frac{1-n}{2}.
\end{align*}
Since $\...
1
vote
0answers
86 views
Fourier inversion formula for compactly supported distributions
I know that the Fourier transform of a compactly support distribution $u\in \mathscr{E}'(\mathbb{R}^{n})$ is smooth and also satisfies
$$
|\hat{u}(\xi)|\leqslant C_{N}(1+|\xi|)^N,\label{1}\tag{1}
$$
...
0
votes
0answers
84 views
A certain ratio condition for polynomials with real coefficients
Let $p:\mathbb{C} \longrightarrow \mathbb{C}$ be a polynomial with real coefficients and suppose that $p$ satisfies
\begin{equation}
\frac{p(y)}{y} \le \frac{p(x)}{x} \tag{*} \label{ratcond}
\end{...
0
votes
1answer
81 views
Generalized Lambert W Function
I am looking for inverse functions for the following family of functions:
$
\begin{aligned}
f_0(z) &= z+e^z \\
f_1(z) &= ze^z \\
f_2(z) &= z^z \\
&\cdots \\
f_{n+1}(z) &=...
5
votes
1answer
111 views
Failure of Borel-Schmid quasi-unipotency theorem in non-algebraic case
Let $\mathcal{X} \to \Delta^{\times}$ be a smooth family of compact Kähler manifolds over a punctured disc.
When this family is algebraic, the celebrated quasi-unipotency theorem (I think, due to ...
2
votes
0answers
64 views
How to compute expansion factors for hyperbolic rational maps?
It is a commonly-referenced result about certain rational maps acting on $\mathbb{\hat{C}}$ that they are expanding on a neighborhood of their Julia sets. A sufficient condition to be expanding is ...
0
votes
2answers
150 views
Dominant root of a family of polynomials
Let $f(x)=x^5-x^4-x^3-x^2-x-c$, where $c>2$ is a real number. It is easy to prove that there exists a positive real root $\alpha>2$ of $f(x)$ and all the other roots are non real.
Also, by ...
2
votes
1answer
93 views
Modulus bounded by Nevanlinna characteristic in several variables
Let $f:\mathbb{C}^n\to\mathbb{C}$ be an entire holomorphic function of $n$ complex variables. Then its Nevannlinna characteristic equals
$$
m_f(r)=\int_{\partial B(r)}\log^+|f(z)|d\eta(z),\quad\forall ...
4
votes
0answers
163 views
When is entire function bounded on a ray?
Let $f(z)=\sum_{n=1}^\infty c_n z^n$ be entire function on the complex plane. May we express the property $\int_0^\infty |f(x)|^2 /x dx<\infty$ or some other property controlling the behavior for ...
4
votes
1answer
170 views
On the roots of Bernoulli polynomials
Consider the Bernoulli polynomials denoted by $B_n(z)$. Now, start plotting the set of all (combined) complex roots $\mathcal{A}_N$ of $B_n(z)$, say for $n=1,2,\dots,N$ for some enough large $N$. It ...
0
votes
1answer
91 views
Compatible solution of PDE
Let $c=c(z, \bar z)$ be a complex function satisfying $\partial_{z} \bar c=\partial_{\bar z} c$. It follows that there exists a real function $f$ such that $\partial_{\bar z} f=-c$. Would it be ...
12
votes
1answer
665 views
Zeros of an infinite series
Let $\sum_{j=1}^{\infty}a_{j}$ be a convergent series of positive numbers and $\{z_{j}\}_{j=1}^\infty$ a closed discrete subset of the open unit disc $\mathbb{D}$. Then $h(z):=\sum_{j=1}^{\infty}\frac{...
3
votes
0answers
130 views
Chern number of projection-Topological magic in physics
I enclosed a computation from a well-known paper in the field of mathematical physics where the Chern number of the first Landau level is computed (the result claimed is $-1$) and the full paper can ...
2
votes
0answers
45 views
Entire analytic functions with entire analytic Fourier transform, and corresponding distributions
I'm interested in the Fourier transform on a space of distributions that includes more than the usual tempered distributions, and in particular allows for $\delta$-distributions supported at complex ...
4
votes
2answers
145 views
Density of Lacunary Functions
I'm curious whether lacunary functions (functions in the complex plane that are holomorphic in some open ball about the origin, but cannot be analytically extended past that ball) are typical or the ...
0
votes
1answer
68 views
Bringing a Heun equation into canonical form
It is a well known fact that any second order Fuchsian differential equation on the complex plane $$u''(x) + p(x)u'(x) + q(x)u(x)=0$$ with exactly $4$ regular singular points may be suitably ...
5
votes
3answers
199 views
Origin of term Ahlfors-David regular
Much of the literature on analysis in metric spaces makes use of an assumption called Ahlfors regularity or Ahlfors-David regularity. Let $q>0$. A metric space $(X,d)$ is Ahlfors(-David) $q$-...
4
votes
1answer
131 views
Compilation of representations of holomorphic functions
Holomorphic functions are my muse. As my muse, I love drawing them different ways. Allow me to frame this as though an artist talking about his muse.
A holomorphic function $f$ on the unit disk $\...
2
votes
1answer
106 views
Extended Abel-Jacobi theorem
Let $X$ be a compact Riemann surface, $u$ a meromorphic function on $X$ with divisor supported on a set of points $\{P_1, ..., P_n\}$, and $f$ a meromorphic function on $X$ such that $f$ has no pole ...
3
votes
2answers
264 views
On finite extensions of the field of meromorphic functions
Let $\mathcal{M}$ be the field of meromorphic functions of one (complex) variable and $w = w(z)$ an analytic function satisfying a polynomial equation
$P(w; z) := w^n + a_{n-1}(z) w^{n-1} + \cdots + ...
-1
votes
1answer
212 views
On a certain representation of the Riemann zeta function
Let $\zeta$ denote the Riemann zeta function. In this answer: https://mathoverflow.net/a/314066/133634, @Paul Garret considers the representation
$$\frac{\zeta(s)}{s} = \int_1^\infty (\sum_{1 \le n \...
4
votes
0answers
75 views
LlogL and Hardy space on the upper half plane
Set $\mathbb{T}$ the unit circle, $dm$ the Lebesgue measure on $\mathbb{T}$ and $\mathbb{C}^+=\left\{z\in \mathbb{C},s.t.\,\Im(z)>0 \right\}$ the upper half plane.
It is well-known that the Cauchy ...
1
vote
1answer
567 views
An integral involving the argument of the Gamma function and the Riemann Hypothesis
Evaluate $$I=\int_{0}^{\infty} \frac{t\arg
\Gamma(\frac{1}{4}+\frac{it}{2})}{(\frac{1}{4}+t^2)^2}\mathrm{d}t$$
where $\Gamma(s)=\int_{0}^{\infty}e^{-x}x^{s-1}\mathrm{d}x.$
Note that $I$ converges ...
