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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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 ...
H A Helfgott's user avatar
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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<...
Bedovlat's user avatar
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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_{...
user44316's user avatar
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2 votes
1 answer
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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 ...
André Henriques's user avatar
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 ...
Ali Taghavi's user avatar
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 ...
Ali's user avatar
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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 ...
stupid_question_bot's user avatar
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 \...
Mayuresh L's user avatar
10 votes
1 answer
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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 $f$ ...
Ali Taghavi's user avatar
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 ...
user2554's user avatar
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2 votes
0 answers
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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 ...
BenjaminRaj's user avatar
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, ...
Kostya_I's user avatar
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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 ...
Lira's user avatar
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4 votes
1 answer
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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 $(A(\mathbb D), H^\infty(\mathbb D))$?
ray's user avatar
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0 answers
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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 ...
user6013's user avatar
  • 169
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 ...
diverietti's user avatar
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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 ...
aurora_borealis's user avatar
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 ...
MCS's user avatar
  • 1,256
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 ...
Robert's user avatar
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1 vote
0 answers
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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 ${\mathbb C}^n$: $$ V=\{z\in {\mathbb C}^n:\ \forall p\in S\quad p(z)=0\}. $$ ...
Sergei Akbarov's user avatar
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}^{\...
user137377's user avatar
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 ...
user avatar
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 ...
Daniel Bruegmann's user avatar
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 $M$ ...
Wenzhe's user avatar
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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 (...
Alexandre Eremenko's user avatar
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 ...
Squid with Black Bean Sauce's user avatar
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 ...
Matey Math's user avatar
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 ...
Mare's user avatar
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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 \...
Arkady Kitover's user avatar
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]{...
tj_'s user avatar
  • 2,160
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 ...
LeechLattice's user avatar
  • 9,411
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....
Penchez's user avatar
  • 341
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} ...
user3141592's user avatar
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 $l,h>0$. ...
user470881's user avatar
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 ...
user119197's user avatar
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 \...
Richard Diagram's user avatar
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{\...
user135826's user avatar
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 ...
Ari's user avatar
  • 181
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 $...
PhoemueX's user avatar
  • 754
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)}...
Matey Math's user avatar
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.
user135645's user avatar
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 ...
PhoemueX's user avatar
  • 754
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 ...
Misha's user avatar
  • 121
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$ ...
Richard Diagram's user avatar
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 ...
user470881's user avatar
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 $\...
Chitsai Liu's user avatar
  • 2,143
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 $$ |\hat{u}(\xi)|\leqslant C_{N}(1+|\xi|)^N,\label{1}\tag{1} $$ ...
Gabriel Palau's user avatar
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) &=...
Ismael Ghalimi's user avatar
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 ...
 V. Rogov's user avatar
  • 1,115
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 ...
Justin Lanier's user avatar

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