Questions tagged [cv.complex-variables]
Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.
3,146
questions
6
votes
1
answer
374
views
Tight error terms for partial sums $\sum_{n\leq x} 1/n^s$
(a) Let $s>1$, $x>0$ be real. Then it is not hard to see that
$$\sum_{n\leq x} \frac{1}{n^s} \leq \zeta(s) - \frac{1}{(s-1) x^{s-1}} + \frac{1}{2 x^s},$$
basically because $x\mapsto 1/x^s$ is ...
5
votes
1
answer
257
views
Find a function $F$ on $[0,1]$ with moments decaying as $(\ln n)^{-n}$
Let $F:[0,1]\to\mathbb{R}$ be a measurable function such that
$$
\mu_n(F)=\int_0^1F(t)t^ndt\sim\frac1{(\ln n)^n}\quad\mbox{as}\quad n\to\infty.
$$
More precisely,
$$
0<c<|\mu_n(F)|(\ln n)^n<...
2
votes
1
answer
200
views
Euclidean length of hyperbolic geodesics for annuli with bounded geometry
I am wondering whether there are estimates for the Euclidean length of vertical hyperbolic geodesics for annuli with good geometry.
More precisely:
Take an annulus $A$, whose outer boundary $\gamma_{...
2
votes
1
answer
202
views
families of Riemann mappings
Let $U\subset \mathbb R^n$ be an open.
Let $f: S^1 \times U \to \mathbb C$ be a smooth map (i.e. $C^\infty$) s.t. for every $x\in U$ the map $f_x:=f|_{S^1\times \{x\}}:S^1\to \mathbb C$ is a smooth ...
2
votes
0
answers
91
views
Sub Banach spaces (Banach algebras) of the disc algebra which are invariant under the differentiation operator
In this question the disc algebra $\mathcal{A}(\mathbb{D})$ is the Banach algebra of all holomorphic functions on the unit open disc $\mathbb{D} \subset \mathbb{C}$ which have a ...
1
vote
0
answers
87
views
An oscillatory integral estimate
Let $n \geq 3$ and consider two sequences of strictly monotone functions $\{\mu_l(t)\}_{l=1}^{n}$ and $\{\lambda_l(t)\}_{l=1}^n$ on the interval $[-1,1]$ with $\mu_l(0)=0$ and $\lambda_l(0)=1$ for all ...
8
votes
1
answer
283
views
Notational question about quadratic differentials in Strebel's book "Quadratic differentials"
In Kurt Strebel's book "Quadratic Differentials", in Chapter 2, $\S4$, he begins by saying:
"Every analytic function $\varphi$ is a domain $G$ of the $z$-plane defines, in a natural way, a field of ...
2
votes
1
answer
107
views
Intersection of superlevel set of polynomials
Let $P_1$ and $P_2$ be complex polynomials with complex coefficients and $c > 0$. Can we find polynomial $P_3$ and $c’>0$ such that
$\{z \in \mathbb C : |P_1(z)| \geq c\} \cap \{ z \in \...
10
votes
1
answer
1k
views
Is there a holomorphic function on open unit disc with this property?
Let $D=\{z\in \mathbb{C}\mid |z|<1\}$. Is there a holomorphic function $f:D\to \mathbb{C}$ such that for every $n\in \mathbb{N} \cup \{0\},\;f^{(n)}$ has a continuous extension to $\bar D$ but $f$ ...
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 ...
2
votes
0
answers
118
views
Covariant derivative of the Monge-Ampere equation on Kähler manifolds
I am reading D. Joyce book "Compact manifolds with special holonomy" and I have some problems of understanding some computation on page 111, the first line in the proof of Proposition 5.4.6. More ...
1
vote
0
answers
75
views
Identities for beta functions and twisted cohomology
This is a question about notation, I apologize if it is too basic. In the paper
Cho, Koji; Matsumoto, Keiji, Intersection theory for twisted cohomologies and twisted Riemann’s period relations. I, ...
3
votes
1
answer
162
views
An integral inequality for diffeomorphisms
Assume that $F(e^{it})=e^{if(t)}$ is a diffeomorphism of the unit circle onto itself and let $A=\left|\int_0^{2\pi}(1-F^2)\,dt\right|$ and $B=\left|\int_0^{2\pi} F^2(1-F^2) \,dt\right|$. It seems that ...
4
votes
1
answer
202
views
Is the disk algebra a complemented subspace of the algebra of bounded analytic functions?
It is well known that the disk algebra (viewed as an algebra on the circle) is uncomplemented in $C(\mathbb T)$. What can be said about the pair
$(A(\mathbb D), H^\infty(\mathbb D))$?
0
votes
0
answers
41
views
integrating multivariable rational function over a product of disks
Suppose I have a rational function of $k$ complex variables:
$$ R(x_1,...,x_k) = P(x_1,...,x_k)/Q(x_1,...,x_k) $$
where $P$ and $Q$ are polynomials. Now I'd like to compute the integral of this ...
11
votes
0
answers
199
views
Holomorphically convex manifolds and Bergman complete manifolds
Suppose $X$ is a complex manifold which admits the Bergman metric (for definitions, see for instance Kobayashi's book "Hyperbolic Complex Spaces"). Suppose moreover that the Bergman metric of $X$ is ...
1
vote
1
answer
531
views
A question about the proof of Riesz-Thorin interpolation theorem
I was reading the proof of Riesz-Thorin interpolation theorem in http://www.math.kit.edu/iana3/lehre/fourierana2014w/media/rieszthorinproof.pdf
and get stuck at the last step. We construct the complex ...
2
votes
0
answers
112
views
Calculus over Function Fields of Characteristic Zero
Having done some cursory searching of the internet, it is clear to me that there is a very well-developed theory of how to do calculus over function fields, such as fields of Laurent series in a ...
3
votes
1
answer
380
views
Why are Poincare series defined as they are?
We know the Poincare series are defined as the following:
The $m^{th}$ Poincare series of weight $k$ for $\Gamma$ is:
$$
P_{m}^{k} (z) = \sum_{(c,d)=1} (cz+d)^{-k} e^{2 \pi in(\tau z)}.
$$
The ...
1
vote
0
answers
142
views
Affine algebraic variety as a set of common zeroes of holomorphic functions on ${\mathbb C}^n$
Let $V$ be an affine algebraic variety in ${\mathbb C}^n$, i.e. a set of common zeroes of some set $S$ of polynomials on ${\mathbb C}^n$:
$$
V=\{z\in {\mathbb C}^n:\ \forall p\in S\quad p(z)=0\}.
$$
...
4
votes
2
answers
390
views
Sums of entire surjective functions
Suppose $(f_n)_n$ is a countable family of entire, surjective functions, each $f_n:\mathbb{C}\to\mathbb{C}$. Can one always find complex scalars $(a_n)_n$, not all zero, such that $\sum_{n=1}^{\...
1
vote
3
answers
207
views
Existence of solution to linear fractional equation
We consider the equation
$$ \sum_{j=1}^n \frac{\lambda_j}{x-x_j} =i$$
where $\lambda_j>0$ and $x_j$ are real distinct numbers.
I want to show that if $\lambda_k$ is small compared to the ...
6
votes
1
answer
252
views
Do analytic functionals form a cosheaf?
Let $X$ be a complex-analytic manifold.
Consider the sheaf of holomorphic functions $\mathcal{O}_X$ as a sheaf with values in the category of locally convex vector spaces. For $U\subseteq X$ open, we ...
3
votes
0
answers
85
views
Metric with singularities on Riemann Surfaces and the associated Laplacians
I have asked this question on Math Stack Exchange
Metric with singularities and associated Laplacian
but I have not got any answers/comments, therefore I post this question on the MO.
Suppose $M$ ...
7
votes
2
answers
546
views
Dependence of a solution of a linear ODE on parameter
Is the following theorem known, or can be easily derived from known results?
Consider the differential equation
$$w''-kz^{-1}w'=(\lambda+\phi(z))w,$$
where $k>0$ is fixed, $\lambda$ is a large (...
4
votes
2
answers
914
views
Reference request: Oldest complex analysis books with (unsolved) exercises?
Per the title, what are some of the oldest complex analysis books out there with (unsolved) exercises? Maybe there are some hidden gems from before the 20th century out there. I am aware of the ...
1
vote
1
answer
150
views
An equation with Gamma Euler function in critical strip
Let
$$
D=\{z \in \mathbb{C} : 0<\Re(z)<\frac{1}{2} \text{ or } \frac{1}{2}<\Re(z)<1 \}
$$
that is the critical strip without critical line.
I have to find if the following equation, with ...
4
votes
0
answers
130
views
Injective resolution of the ring of entire functions
Let $R$ be the ring of entire functions on $\mathbb{C}$. I heard that the concrete value of the global dimension of the ring depends on continuum hypothesis.
I would think that the injective dimension ...
3
votes
1
answer
59
views
Rational approximation on rotation invariant compact subsets of complex plane
What does the Vitushkin's theorem say about the equality $A(K) = R(K)$ in the special case when $K$ is rotation invariant? More precisely, what are necessary and/or sufficient conditions on $\{|k|: k \...
2
votes
2
answers
398
views
Coefficients of entire functions with specified zero set
Let $Z \subseteq \mathbb{C}$ without limit point. By the Weierstrass factorization theorem there is an entire function $h$ those zero set is $Z$. Let $a_n > 0$ be a sequence where $\lim_n \sqrt[n]{...
5
votes
2
answers
805
views
Local phase statistics of the nontrivial Riemann zeros
(The question is inspired by Owen Maresh's post)
The local phase of a nontrivial zero $s$ of the Riemann $\zeta$ is the argument of $\zeta'(s)$.
Numerical results on the first 10000 zeros suggest ...
2
votes
1
answer
333
views
Defining integrals by residue theorem
I have always been interested in alternative definitions of mathematical objects. I wonder if one can craft an useful definition of definite integral by using the Residue Theorem from complex analysis....
0
votes
1
answer
235
views
Solving or bounding the real part of the integral $\int_0^{2 \pi i m} \frac{e^{-t}}{t-a} dt$
I would be interested in finding a closed form or, at least, bounding (in terms of $m$ as it becomes larger) the real part of the following itnegral:
$$f(m,a):=\int_0^{2 \pi i m} \frac{e^{-t}}{t-a} ...
3
votes
0
answers
109
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 $l,h>0$. ...
2
votes
1
answer
159
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
1
answer
140
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
1
answer
272
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{\...
4
votes
0
answers
332
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 ...
5
votes
1
answer
165
views
Existence of Laurent series with zeroes at $𝑒^{2𝑛}$ ($𝑛∈ℕ_0$) 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 sequence $...
4
votes
1
answer
371
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)}...
1
vote
1
answer
1k
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
1
answer
389
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
0
answers
349
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 ...
4
votes
1
answer
125
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
0
answers
53
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
2
answers
2k
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
0
answers
136
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
1
answer
202
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
1
answer
244
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
0
answers
105
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 ...