Questions tagged [cv.complex-variables]
Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.
3,298 questions
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Monstrous moonshine, Dedekind eta function, and the hypergeometric function
I. Monstrous Moonshine
Let $q = e^{2\pi i\tau}$ and $\tau = \sqrt{-d}$ or $\tau = \frac{1+\sqrt{-d}}2$ for positive integer $d$. Given the Dedekind eta function $\eta(\tau)$, consider the known ...
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When inclusion between two Kobayshi hyperbolic manifolds is distance decreasing?
Suppose that $X$ and $Y$ are two Kobayshi hyperbolic complex-analytic manifolds such that $X \subset Y$. It is known $d_Y(x_1, x_2) \leq d_X(x_1, x_2)$ for all $x_1, x_2 \in X$. In other words, the ...
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Some questions on Hardy's spaces
In the paper http://www.numdam.org/item/CM_1976__33_3_261_0.pdf, the authors have asked in Page 285 whether the Hardy space $H^p$ embeds isometrically into the Hardy space $H^q$ for $1\leq q<p<...
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Minimum of a subharmonic function
Let for $j=1,\dots, m$, $z_j$ be distinct points from the unit disk $|z|<1$ and let $$g(z)=-\sum_{k=1}^m \log \frac{(1-|z|^2)(1-|z_k|^2)}{|1-z\overline{z_k}|^2}.$$ It seems that $g$ has a unique ...
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Dirichlet Series that fail to be L-functions
For $\sigma \in \mathbb{R}$, let each $\mathbb{C}_\sigma = \{s \in \mathbb{C} : \Re(s) > \sigma\}$. For a sequence $a_n \in \mathbb{C}$, consider the Dirichlet series $D(s) = \sum_{n\ge 0} a_n n^{-...
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Characterizing the integral as a function of $n$
Let $\alpha \in [0,3], \beta \geq 1, \lambda \geq 1$ and fix $n \in \mathbb{N}$. Consider the function $f(x;\alpha, \beta, \lambda) = x^{\alpha}\exp(-\lambda x^\beta)$. Let $I(n; \alpha,\beta,\lambda) ...
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Uniqueness of Neumann series
Let $f$ be an entire function. Then there exist numbers $a_0,a_1,\ldots$, independent of $z$, such that
$$f(z)=\sum_{n=0}^\infty a_n J_n(z),\quad \forall z\in\mathbb{C}$$
where $J_n$ is the Bessel ...
- 317
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Higher-dimensional analogue of the relation between stable Higgs bundles and constant curvature metrics
In Hitchin's famous paper[1] on the self-dual Yang-Mills equations, he discussed the relation between the stable Higgs bundles and the Teichmüller space for a compact Riemann surface. Namely, through ...
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Can doubly parabolic Blaschke product (BP) contained in another doubly parabolic BP?
Let $f:\mathbb{D}\rightarrow\mathbb{D}$ be a degree $d$ doubly parabolic Blaschke product with Denjoy-Wolff point at $z=1$. That is, $f(1) = 1$, $f'(1)=1$ and $f''(1)=0$.
Let $U \subset \mathbb{D}$ be ...
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What does analytic uniformly in $s$ mean?
Suppose I have a complex vector space $V$ with finite basis $\{e_{1},...,e_{s}\}$. Then, I can consider the algebra $\mathcal{U}$ of formal polynomials on the variables $e_{1},...,e_{s}$. Suppose ...
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Reconstruction of Riemann surface from a germ of holomorphic function
Let $\Sigma$ be a compact Riemann surface of genus $g$, and $f: \Sigma \to \mathbb{C}$ a
meromorphic function. Take $U \subset \Sigma$ an open disk in $\Sigma$ biholomorphic to a disk
in $\mathbb{C}$, ...
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What's the asymptotic behaviour of $_1F_1(a,b,az)$ when $a\to\infty$?
I'm working towards the solution to a problem about involving the Landau-Zener transition, but I'm finding some difficulties. I need to estimate $ \,_1 F_1\left(\frac{\mathrm i}{4\epsilon},\frac12;\...
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Is $(n!^{-d})_{n\geq 0}$ a Pólya frequency sequence?
Fix a positive integer $d$. Is the sequence $(n!^{-d})_{n\geq 0}$ a
Pólya frequency (PF) sequence? Equivalently, is the Toeplitz matrix
$A=[a_{ij}]_{i,j\geq 0}$, where $a_{ij}=0$ if $i>j$ and
$a_{...
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Analytic continuation to the Mittag-Leffler star using Mittag-Leffler summation
This is a reference request for a theorem I thought I had read in a book by Steven Krantz, but I can no longer find it.
Searching for Mittag-Leffler star, I can find references to the following result....
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An integral containing modified Bessel functions
During my studies I am facing the following problem. Let $I_\nu(x)$ be the modified Bessel function for $\nu\in(0,1/2]$.
I want to compute the following integral (it is are resolvent)
$$
R(z) = \frac{...
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3
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312
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Exponent of convergence of the sequence of zeros of $e^z+z$
Question: How to calculate the exponent of convergence of sequence of zeros of the function $f(z)=e^z+z$?
I know the formula (given below) to calculate the exponent of convergence but for this, I need ...
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Reference dual Dirichlet space $D^1$
Let $\mathbb{D} = \{ z \in \mathbb{C} : |z| < 1 \}$ be the unit disk. The Bergman space $A^1 = A^1(\mathbb{D})$ is the Banach space of holomorphic functions on $\mathbb{D}$ such that
$$
\|f\|_{A^1} ...
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Can we strengthen this exercise in Forster's book on Riemann surfaces?
Exercise 2.5 in Otto Forster's Lectures on Riemann Surfaces states
Suppose $p_1,\ldots,p_n$ are points on the compact Riemann surface $X$ and $X':=X\setminus\{p_1,\ldots,p_n\}$. Suppose $$f:X'\to\...
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Derivate involving Bessel function of second type
Let.
$$f := (x, y) \mapsto \text{BesselK}(1, c \cdot (a - b \cdot (x + y))) \cdot \exp(c \cdot b \cdot (y - x))$$
Is there a close formula for this $$\frac{\partial^{m+n}}{\partial y^m \partial x^n} f(...
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Special function in the Hardy space
Let $H^2(\mathbb{D})$ denote the complex Hardy space, this is: analytic functions defined unit disc $\mathbb{D}$ whose coefficients form a sequence in $\ell^2$. Functions in $H^2(\mathbb{D})$ have a ...
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Is the map $G^g/G \to \operatorname{Bun}_G X$ locally an isomorphism in good cases?
$\DeclareMathOperator\Bun{Bun}$Suppose you have a closed Riemann surface $X$ constructed by cutting out $2g$ holes into a sphere and sewing pairs of holes together. Given elements $g_1, \dotsc g_{g}$ ...
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Integral of $\log|e^{it}-1|$
Does there exist an elegant proof of
$$
\int_0^{2\pi}\log|e^{it}-1|\,dt=0 \ ? \label{1}\tag1
$$
Of course, one can do some $\varepsilon$-$\delta$ stuff and get it, but I look for a nice proof. In the ...
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Inclusion in Hardy-Smirnov spaces for the analytic continuation of a Cauchy-Type integral with a continuous boundary function
Let $D$ be a bounded simply connected domain in the complex plane $\mathbb{C}=\{z=x+iy\}$ with a Jordan rectifiable boundary $C=\partial D$. Let $P_1$ and $P_2$ be two distinct points on $C$, and let ...
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Explicit bounds on gaps between zeros of $\zeta^\prime(s)$
In $\S$9.1 of "Theory of the Riemann Zeta Function", Titchmarsh uses Borel-Carathéodory and Hadamard Three Circles to show that every circle of radius 6 and center $3+iT$ contains a zeros of ...
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Conformally mapping between the upper half complex plane, and the plane with a tree on spatial points removed
A stochastic process such as SLE$_{\kappa}$ can be defined by taking the scaling limit of a curve in the upper half complex plane: put simply, one removes a line segment, then another, $n$ times, each ...
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What are necessary and/or sufficient conditions for a Dirichlet series to admit analytic continuation?
Let $A = \{a(n)\}_{n \geq 1}$ be a sequence of complex numbers. By normalizing, we may as well assume that $|a(n)| \leq 1$ for all $n \geq 1$. Under this assumption, the Dirichlet series
$\...
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How to calculate an integral over the complex unit sphere
We want to calculate the following integral over the complex unit sphere $S^{2n-1}$:
$$\int_{S^{2n-1}} \frac{1 }{|1 - \langle z, \zeta \rangle|^2} \, d\sigma(\zeta),$$
where $ z $ is a fixed point in ...
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Which proofs of the fundamental theorem of algebra are "essentially the same" vs. "essentially different"?
The classic MO thread Ways to prove the fundamental theorem of algebra contains $60$ proofs of FTA, and I'm sure there are many more in the literature. It would be nice to have some way to organize ...
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Nontrivial extension of the action of complex hyperbolic group $H$ on $\mathbb{C}$
Inspired by this question about conjugation of reql analytic maps to a holomorphic function and with a group action view point we ask the following question.
The complex Lie group $H=\...
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On a functional equation of Mahler?
Recently, I was trying to introduce the concept of natural boundaries to a fellow math student, and what greater way to do this than using an example? In particular, I tried to use as an illustration, ...
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Asymptotic decay rate of an oscillator integral
Question:
I want to evaluate the decay estimate of the integral
$I^d(t; v) = \int_0^{\sqrt{d}\pi} dr \, r^{d-2} \int_0^\pi \sin(tr) e^{i\sqrt{d}vtr\cos\theta} \sin^{d-2}\theta \, d\theta $
for ...
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On analytic functions on the complement of a curve without jump across the curve almost everywhere
Question. Suppose $f$ is an analytic function on $\mathbb C\setminus\mathbb R$ and assume that the boundary values of $f$ from above and below the real axis (denoted $f_\pm$ respectively) exist almost ...
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Reference for Mellin inversion; Meijer G-function
We have $$\frac {\Gamma (a)}{2^a}=\int _{(c)}\Gamma (s)\Gamma (a-s)\,ds,$$ see e.g. Exercise C.23 of Montgomery and Vaughan's "Multiplicative Number Theory".
I would like a similar formula ...
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How many roots do $\tan(z)-z^n = 0$, $\sin(z)-z^n=0, \ \cos(z)-z^n=0, $ have?
I asked this question on MSE here.
I am investigating the number of roots of the equations:
$$\tan(z) - z^n = 0$$
$$\sin(z)-z^n=0$$
$$\cos(z)- z^n=0$$
within the vertical strip $|\text{Re}(z)| \leq \...
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Representation of a meromorphic function on a once-punctured complex plane in terms of its zeros and poles
Consider a meromorphic function $f:\mathbb{C}\setminus\{0\}$ such that both $0$ and $\infty$ are its essential singularities with finite order in the sense of value distribution theory (see for ...
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Are real-analytic functions in $\mathbb{R}^2$ holomorphic after suitable change of coordinates?
I'm not sure this is a research-level question, but I couldn't find an answer after a bit of searching, so here goes.
Let $f: \mathbb{R}^2 \to \mathbb{R}^2$ be a real-analytic function. Can we always ...
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Reality of connection or meromorphic function
Let's considering a family of connections: $\nabla^{\lambda}:\mathbb{C}^{*}\rightarrow \Omega^{1}(sl(2,C))$ of trivial rank2 bundle on $\mathbb{P}^{1}-\{ 0,1,\infty \}$ with simple pole. In this case, ...
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SOT and WOT convergence of Toeplitz operators
For the Hardy space $H^2$, every $\phi \in L^\infty (\mathbb T)$ induces a bounded Toeplitz operator $T_\phi$ on the Hardy space and $\lVert T_\phi \rVert = \lVert \phi \rVert _{\infty}$. Consequently,...
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What is the limit of the sequence of iterated cosines?
I asked this question on MSE here.
Define $f_1(z) = \cos(z)$, $f_{n+1}= \cos(f_n (z)) $, The question is: Does $\lim\limits_{n \to \infty}f_n(z)$ exist for certain $z \in \mathbb{C}$? And what is ...
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Functions of bounded boundary rotation on the upper-half plane
It is a fact that there is a one to one correspondence between the space $M(k)$ of finite, signed Borel measures on $\mathbb{S}^1$ with total mass equal to $2$ and total variation equal to some $2 \...
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Reference Request: Continuous extension of conformal maps
currently I am trying to find some references on the continuous extension of conformal maps between two simply connected domains of the Riemann sphere $\hat{\mathbb C}$. Let $\gamma_1,\gamma_2$ be two ...
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1
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A question on Bloch functions
Let $\mathcal{B}(\Delta)$ be the space of Bloch functions in the unit disk $\Delta$. For any $f\in \mathcal{B}(\Delta)$, we define the Bloch norm by
$$
\|f\|_{\mathcal{B}}=\sup_{|z|<1}|f'(z)|(1-|z|^...
- 1,706
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1
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Convergence and meromorphic continuation of a Dirichlet series under RH
Consider a Dirichlet series $F(s) = \sum_{n\geq1} \frac{a_{n}}{n^{s}}$ such that $a_{n} = O(1)$. Suppose the Dirichlet series
$$\zeta(s)F(s)=\sum_{n\geq1} \frac{(a\star1)(n)}{n^{s}}$$
converges ...
- 611
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2
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New series for $\pi$ from string theory
This is a direct followup to the post Possible new series for $\pi$ by Timothy Chow
and its numerous answers and comments.
Using another formula in the same string theory paper by Saha and Sinha one ...
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60
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On an oscillatory property of the Riemann Xi-function
In their paper "The Integral of the Riemann Xi-Function", Lagarias and Montague mention Wintner's 1947 proof that
$$
\Xi^{(-1)}(t) > 0 \quad \text{when} \quad t > 0.
$$
This result ...
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Properties of Poisson Integral and Traces of functions in Sobolev space $W^{1,2} (\mathbb D)$
Let $W^{1,2}(\mathbb D)$ be the complex valued Sobolev space on $\mathbb D$ where $\mathbb D $ is the open unit disk of the complex plane. By definition, $W^{1,2} (\mathbb D)$ is the set of all ...
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2
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Is there a term for a countour integral that disregards direction?
Is there a name for integration of the form $\oint_\gamma f(z) |dz|$?
In other words, the integral that only takes into account the length of the contour and the values of the function but not the ...
- 10.1k
1
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1
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132
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Deriving a specific bound for functions in Hardy Space
Reading some article a while ago I read the following: (here $H^2$ represents the Hardy space)
Let $f\in H^2$ be such that $f(0)=1$, and let $0<\lvert\lambda\rvert<1$, then $$\lVert f(\lambda z)\...
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0
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206
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Partial fraction expansions of meromorphic functions
Sorry if this question (inspired by the recent flurry of activity around a "new" formula for $\pi$) is too naive.
Imitating what one does with Hadamard products, one can try to do the same ...
- 13.1k
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2
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Systematic way to compute $\sum_{n=1}^\infty P(n) / Q(n)$ for polynomials $P$ and $Q$
This may be well known so feel free to downvote.
When $P = 1$ and $Q(x) = x^k$, this is of course the Riemann zeta function. But what about other cases?
For instance is it always possible to express $\...
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