All Questions
3,560 questions
4
votes
1
answer
214
views
Explicit expression for a function in number theory
In their paper "Moyenne de certains fonctions arithmétiques sur les entiers friables", Tenenbaum and Wu proved that for the case of the function $\beta$ which is the indicator function of ...
- 1,257
2
votes
2
answers
363
views
Size of $\zeta'(s)$ at its zeros
How large can the derivative of the Riemann zeta function be at its zeros?
More specifically, let $\rho$ be a zero of the zeta function with $\Im(\rho)\in (0,T]$. What can we say about $|\zeta'(\rho)|...
user536681
2
votes
1
answer
315
views
Where to find or how to establish a general formula for the improper integral $\int_{0}^{\infty}\frac{\sin t}{t}(\ln t)^k\operatorname{d\!} t$?
When I tried to establish the Maclaurin power series expansion of the reciprocal $\frac{1}{\Gamma(z)}$ of the Euler gamma function $\Gamma(z)$, I came across the improper integral $$
I_k=\int_{0}^{\...
- 1,101
0
votes
0
answers
65
views
Rotations and bi-analytic functions
Are the bi-analytic functions $\partial^2_{\overline{z}} f=0$
invariant under rotations?
- 41
9
votes
1
answer
393
views
A hypergeometric series for $\Gamma(1/4)^4/\pi^3$
Sorry if this comes out of the blue. Looking at old notes of mine, I found the identity
$$\dfrac{\Gamma(1/4)^4}{\pi^3}=4+\sum_{n\ge0}\binom{2n+1}{n}^3\dfrac{1}{2^{6n+1}}\;.$$
I cannot remember how I ...
- 13.1k
3
votes
1
answer
116
views
Interpretations of analytic continuations of CDFs to complex probabilities
Are there notable cases where analytic continuations of cumulative distribution functions to complex arguments have a meaningful interpretation or are otherwise useful?
If a one dimensional CDF is ...
- 7,545
2
votes
1
answer
263
views
Complex (i.e., Imaginary) Probability
I’ve been doing some numerical approximation of probability distributions. For continuous $\operatorname{PDF}$s (or $\operatorname{CDF}$s) greater smoothness can be exploited to achieve more ...
4
votes
1
answer
257
views
Asymptotics of an entire function with real zeroes on the real line
Define $ F(x) := A x^m e^{B x} \prod_{k \geqslant 1} (1 - x/\alpha_k)e^{x/\alpha_k} $ where we suppose that $ \alpha_k \in \mathbb{R} $. This function is defined on the whole complex plane and is ...
- 593
10
votes
1
answer
444
views
Analytic continuation gives a covering space (and not just a local homeomorphism)
Let $\mathcal{G}$ be the space of germs of holomorphic functions defined on open subsets of $\mathbb{C}$, topologized in the usual way. There is a natural map $p\colon \mathcal{G} \rightarrow \mathbb{...
- 111
4
votes
0
answers
323
views
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 ...
- 12.6k
0
votes
0
answers
48
views
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 ...
- 41
4
votes
0
answers
148
views
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<...
- 1,879
3
votes
1
answer
229
views
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}.$$ It seems that $g$ has a unique ...
- 37
1
vote
2
answers
310
views
Dirichlet Series that fail to be L-functions
For $\sigma \in \mathbb{R}$, let each $\mathbb{C}_\sigma = \{s \in \mathbb{C} : \Re(s) > \sigma\}$. For a sequence $a_n \in \mathbb{C}$, consider the Dirichlet series $D(s) = \sum_{n\ge 0} a_n n^{-...
- 1,124
0
votes
1
answer
128
views
Characterizing the integral as a function of $n$
Let $\alpha \in [0,3], \beta \geq 1, \lambda \geq 1$ and fix $n \in \mathbb{N}$. Consider the function $f(x;\alpha, \beta, \lambda) = x^{\alpha}\exp(-\lambda x^\beta)$. Let $I(n; \alpha,\beta,\lambda) ...
- 25
8
votes
3
answers
617
views
Uniqueness of Neumann series
Let $f$ be an entire function. Then there exist numbers $a_0,a_1,\ldots$, independent of $z$, such that
$$f(z)=\sum_{n=0}^\infty a_n J_n(z),\quad \forall z\in\mathbb{C}$$
where $J_n$ is the Bessel ...
- 317
4
votes
0
answers
78
views
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 ...
- 353
3
votes
1
answer
127
views
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 ...
0
votes
0
answers
78
views
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 ...
2
votes
1
answer
247
views
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}$, ...
- 5,230
3
votes
1
answer
138
views
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;\...
- 87
13
votes
1
answer
254
views
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_{...
- 50.8k
3
votes
1
answer
179
views
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....
- 1,124
1
vote
0
answers
81
views
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{...
- 11
3
votes
3
answers
312
views
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 ...
1
vote
1
answer
65
views
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
$$
\|f\|_{A^1} ...
4
votes
2
answers
302
views
Can we strengthen this exercise in Forster's book on Riemann surfaces?
Exercise 2.5 in Otto Forster's Lectures on Riemann Surfaces states
Suppose $p_1,\ldots,p_n$ are points on the compact Riemann surface $X$ and $X':=X\setminus\{p_1,\ldots,p_n\}$. Suppose $$f:X'\to\...
- 317
0
votes
0
answers
36
views
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 $$\frac{\partial^{m+n}}{\partial y^m \partial x^n} f(...
- 109
2
votes
1
answer
170
views
Special function in the Hardy space
Let $H^2(\mathbb{D})$ denote the complex Hardy space, this is: analytic functions defined unit disc $\mathbb{D}$ whose coefficients form a sequence in $\ell^2$. Functions in $H^2(\mathbb{D})$ have a ...
- 319
6
votes
0
answers
160
views
Is the map $G^g/G \to \operatorname{Bun}_G X$ locally an isomorphism in good cases?
$\DeclareMathOperator\Bun{Bun}$Suppose you have a closed Riemann surface $X$ constructed by cutting out $2g$ holes into a sphere and sewing pairs of holes together. Given elements $g_1, \dotsc g_{g}$ ...
- 563
10
votes
5
answers
998
views
Integral of $\log|e^{it}-1|$
Does there exist an elegant proof of
$$
\int_0^{2\pi}\log|e^{it}-1|\,dt=0 \ ? \label{1}\tag1
$$
Of course, one can do some $\varepsilon$-$\delta$ stuff and get it, but I look for a nice proof. In the ...
- 1,294
2
votes
0
answers
82
views
Inclusion in Hardy-Smirnov spaces for the analytic continuation of a Cauchy-Type integral with a continuous boundary function
Let $D$ be a bounded simply connected domain in the complex plane $\mathbb{C}=\{z=x+iy\}$ with a Jordan rectifiable boundary $C=\partial D$. Let $P_1$ and $P_2$ be two distinct points on $C$, and let ...
- 31
4
votes
0
answers
168
views
Explicit bounds on gaps between zeros of $\zeta^\prime(s)$
In $\S$9.1 of "Theory of the Riemann Zeta Function", Titchmarsh uses Borel-Carathéodory and Hadamard Three Circles to show that every circle of radius 6 and center $3+iT$ contains a zeros of ...
- 11.1k
2
votes
0
answers
159
views
Conformally mapping between the upper half complex plane, and the plane with a tree on spatial points removed
A stochastic process such as SLE$_{\kappa}$ can be defined by taking the scaling limit of a curve in the upper half complex plane: put simply, one removes a line segment, then another, $n$ times, each ...
- 640
3
votes
0
answers
450
views
What are necessary and/or sufficient conditions for a Dirichlet series to admit analytic continuation?
Let $A = \{a(n)\}_{n \geq 1}$ be a sequence of complex numbers. By normalizing, we may as well assume that $|a(n)| \leq 1$ for all $n \geq 1$. Under this assumption, the Dirichlet series
$\...
- 26.9k
5
votes
2
answers
792
views
How to calculate an integral over the complex unit sphere
We want to calculate the following integral over the complex unit sphere $S^{2n-1}$:
$$\int_{S^{2n-1}} \frac{1 }{|1 - \langle z, \zeta \rangle|^2} \, d\sigma(\zeta),$$
where $ z $ is a fixed point in ...
- 109
23
votes
0
answers
720
views
Which proofs of the fundamental theorem of algebra are "essentially the same" vs. "essentially different"?
The classic MO thread Ways to prove the fundamental theorem of algebra contains $60$ proofs of FTA, and I'm sure there are many more in the literature. It would be nice to have some way to organize ...
- 118k
5
votes
1
answer
226
views
Nontrivial extension of the action of complex hyperbolic group $H$ on $\mathbb{C}$
Inspired by this question about conjugation of reql analytic maps to a holomorphic function and with a group action view point we ask the following question.
The complex Lie group $H=\...
- 366
3
votes
0
answers
116
views
On a functional equation of Mahler?
Recently, I was trying to introduce the concept of natural boundaries to a fellow math student, and what greater way to do this than using an example? In particular, I tried to use as an illustration, ...
- 131
4
votes
1
answer
145
views
Asymptotic decay rate of an oscillator integral
Question:
I want to evaluate the decay estimate of the integral
$I^d(t; v) = \int_0^{\sqrt{d}\pi} dr \, r^{d-2} \int_0^\pi \sin(tr) e^{i\sqrt{d}vtr\cos\theta} \sin^{d-2}\theta \, d\theta $
for ...
- 81
2
votes
0
answers
95
views
On analytic functions on the complement of a curve without jump across the curve almost everywhere
Question. Suppose $f$ is an analytic function on $\mathbb C\setminus\mathbb R$ and assume that the boundary values of $f$ from above and below the real axis (denoted $f_\pm$ respectively) exist almost ...
2
votes
1
answer
125
views
Reference for Mellin inversion; Meijer G-function
We have $$\frac {\Gamma (a)}{2^a}=\int _{(c)}\Gamma (s)\Gamma (a-s)\,ds,$$ see e.g. Exercise C.23 of Montgomery and Vaughan's "Multiplicative Number Theory".
I would like a similar formula ...
- 1,381
6
votes
1
answer
645
views
How many roots do $\tan(z)-z^n = 0$, $\sin(z)-z^n=0, \ \cos(z)-z^n=0, $ have?
I asked this question on MSE here.
I am investigating the number of roots of the equations:
$$\tan(z) - z^n = 0$$
$$\sin(z)-z^n=0$$
$$\cos(z)- z^n=0$$
within the vertical strip $|\text{Re}(z)| \leq \...
- 541
2
votes
1
answer
96
views
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 ...
- 53
9
votes
3
answers
929
views
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 ...
- 5,590
2
votes
0
answers
93
views
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, ...
- 21
1
vote
1
answer
152
views
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,...
- 151
5
votes
2
answers
435
views
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 ...
- 541
1
vote
0
answers
25
views
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 \...
- 111
-1
votes
1
answer
85
views
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 ...
