Skip to main content

Questions tagged [cv.complex-variables]

Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.

20 questions from the last 30 days
Filter by
Sorted by
Tagged with
-2 votes
1 answer
60 views

when does $h$ exist?

Let $\zeta(s)$ denote the Reimann zeta function in the critical strip. It is easy to see that $$ \zeta(s) = 0 \Longleftrightarrow \Re(\zeta(s))+\Im(\zeta(s)) = 0 ~~~~ \text{and} ~~~~~~ \Re(\zeta(s)) \...
Roy Burson's user avatar
-2 votes
0 answers
48 views

Is the real and imaginary part of the Dirichlet eta function closest to its partial sums when trigonometric function changes signs?

To grasp the question we are concerned with three Theorems 1,2, and 3 in bold font below. First let us consider the Dirichlet eta function $\eta: \mathbb{C}\rightarrow \mathbb{R}$ $$ \eta(s) = \sum_{n=...
Roy Burson's user avatar
2 votes
0 answers
48 views

Convergence of finite-difference method for Cauchy-Riemann equations

Let $I\subseteq \mathbb{R}$ an open interval. Let $f:I\rightarrow \mathbb{C}$ real analytic. Suppose we want to numerically compute an analytic extension of $f$. We will assume the following: we are ...
Plemath's user avatar
  • 312
1 vote
0 answers
53 views

Description of all biholomorphic maps from annulus [duplicate]

Consider the collection $\mathcal{C}$ of all maps $f \colon B_1 -\overline{B}_\beta \to \mathbb{C}$ such that $f$ is biholomorphic onto its image. Is this collection $\mathcal{C}$ path-connected? In ...
Jinyang wu's user avatar
0 votes
0 answers
77 views

Singular behavior of zeros of incomplete zeta function

I've been looking at the zeros of the incomplete zeta function $\zeta_{lower}(s, z)$ recently. $$ \zeta_{\mathrm{lower}}(s,z)=-\frac{{\Gamma(1-s)}}{2\pi i}\int_{z}^{\infty}\frac{{(-t)^{s-1}}}{e^{t}-1}...
Zhobbyist's user avatar
1 vote
0 answers
39 views

Currents with logarithmic poles compared with those with no poles

I am learning Deligne homology via U. Jannsen, "Deligne homology, Hodge-$\mathscr{D}$-conjecture, and motives." There, the currents with logarithmic poles are given in Definition 1.4 by $$ '\...
neander's user avatar
  • 161
3 votes
2 answers
145 views

Vector bundles over a Stein space are projective

It is a "well known" fact that locally free sheaves over a Stein space $X$ are projective as $\mathcal{O}_X$-modules (see e.g. just after Lemma 1.6 in O'Brian-Toledo-Tong's "The trace ...
Tim's user avatar
  • 1,109
0 votes
1 answer
124 views

Holomorphic functions of certain blow up at origin

Suppose that $D=\{z\in \mathbb C\,:\, |z|\leq 1\}$ and let $f$ be holomorphic on $D\setminus\{0\}$ such that $|f(z)|\leq e^{\frac{1}{|z|}}$ for all $0<|z|\leq 1$ and assume additionally that $\lim\...
Ali's user avatar
  • 4,135
1 vote
0 answers
146 views

integral over the unit sphere of $\Bbb C^n$

Please, is there a way to calculate this integral $$\int_{S_{2n-1}} \frac{e^{a \langle z, \zeta \rangle}}{|z - \zeta|^{\beta}} \, d\sigma(\zeta)$$ where $ z $ is a fixed point in the complex unit ball ...
zoran  Vicovic's user avatar
2 votes
1 answer
199 views

Section 3 of Atiyah's "On analytic surfaces with double points" — some questions

I have some questions about section 3 of Atiyah's "On analytic surfaces with double points," a short 9 page paper. Section 3 is all dedicated to proving lemma 4. Near the end of section 3, ...
maxo's user avatar
  • 129
0 votes
0 answers
57 views

Complexity of evaluation of analytic functions

Given an analytic function $f(x)$ (say as combination of elementary functions and operators), is it possible to compute $n$ first bits of the value of the function on the whole interval $[a, b]$ ...
roignoirewg's user avatar
1 vote
0 answers
46 views

Can one explicitly define a right inverse for a convolution operator on the space of entire functions?

A result of Meise and Taylor in 1988 shows that every non-zero convolution operator on the Frechet space $H(\mathbb{C})$ of all entire functions on $\mathbb{C}$ has a continuous linear right inverse $...
David Walmsley's user avatar
5 votes
1 answer
273 views

Why "no wandering domain" fails in parabolic basin?

Theorem (Sullivan). Every Fatou component $U$ of $f$ rational map is eventually periodic, that is, there exist $n > m > 0$ such that $f^n(U) = f^m(U)$ I am familiar with the proof: spread around ...
Ricky Simanjuntak's user avatar
0 votes
0 answers
79 views

Alternative proof of parabolic implosion

I am working on an alternative proof of parabolic implosion from complex dynamics, but only allowing hyperbolic perturbation. Theorem (Parabolic Implosion) Let $f(z)=z^2+z$ and $U_f$ be parabolic ...
Ricky Simanjuntak's user avatar
-3 votes
1 answer
194 views

Bounding a number-theoretic integral

Find a good upper bound on $$\int_1^T\frac{\zeta'(s)}{\zeta(s)\zeta(1-s)}X^sdt,$$ where $s=c+it$ for a constant $c>1$ and $X>0$ is a parameter. If needed, we can assume RH. My attempt here is ...
charlie_beck's user avatar
2 votes
0 answers
70 views

Is the hypothesis "uniformly equivalent" needed?

I am reading S. Shimorin's paper titled Complete Nevanlinna-Pick property of Dirichlet-type spaces. My question concerns Lemma 2.3. which is as follows: Assume $\mathscr{H}$ is a Hilbert space of ...
ash's user avatar
  • 151
0 votes
1 answer
119 views

Nonstationary phase method for oscillatory integral

I want to approximate an integral of the form $$\int_a^bf(t)e^{ig(t)}dt,$$where $f(t)$ is smooth, $g(t)$ is real-valued and smooth. The stationary phase method says that if $t_0\in [a,b]$ is such that ...
charlie_beck's user avatar
1 vote
1 answer
63 views

Need bound for absolute value of complex-valued special functions (Taylor coefficients of Faddeeva's w(z))

To guarantee accuracy for code [1] that computes Faddeeva's w(z) [2] using Taylor expansions around different centers, I would need upper bounds for the absolute values $|w_n(z)|$ of the coefficients $...
Joachim W's user avatar
  • 111
1 vote
2 answers
225 views

Bounds of zeta function near $\Re(s)=1$

Richert proved in https://link.springer.com/article/10.1007/BF01399533 that $$ \zeta(s) =O\left( |\Im(s)|^{100(1-\Re(s))^{3/2}} (\log |\Im(s)|)^{2/3}\right)$$ uniformly in the region $\Re(s)\in [1/2,1]...
Dr. Pi's user avatar
  • 3,062
2 votes
1 answer
214 views

How can one test whether a given analytic curve in the plane is algebraic or not?

Suppose $\Gamma$ is an analytic Jordan curve in the complex plane $\mathbb{C}$. What are ways to test whether $\Gamma$ is (contained in) an algebraic curve, i.e., whether there exists a real ...
Malik Younsi's user avatar
  • 2,154