Questions tagged [ca.classical-analysis-and-odes]

Special functions, orthogonal polynomials, harmonic analysis, ordinary differential equations (ODE's), differential relations, calculus of variations, approximations, expansions, asymptotics.

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Convolution of $\mathscr{F}\{ \log \}(x) * \mu$ with compactly supported measure $\mu$

As I read in this post the Fourier transform of $\psi(\lambda) = \log{|\lambda|}$ must be interpreted in distributional sense and it is given by: $$\mathscr{F}\{\psi\}(x)=-2\pi \gamma \delta(x)-\pi \...
Grandes Jorasses's user avatar
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An inequality for the Darboux integral

As we know, if $f(x)$ are Riemann integrable, we have \begin{gather} \left|\int_a^b f(x)~\mathrm{d}x\right|\leq \int_a^b |f(x)|~\mathrm{d}x. \end{gather} So, for Darboux integrals, such as the upper ...
Daeree's user avatar
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Prove the limit of the integral

Suppose: $f:[0,\frac{\pi}{2}]\to \mathbb{R}$, $f(x)\in C^2[0,\frac{\pi}{2}]$, $\gamma=\lim_{n\to \infty}(1+1/2+\dotsb+1/n-\ln{n})$. Prove:$$ \lim_{s\to +\infty}\left(\int_{0}^{\frac{\pi}{2}} \frac{\...
MathNoob's user avatar
6 votes
3 answers
453 views

Evaluating the infinite product $\prod_{k\geq 2}(1-\frac{1}{k^3})$

Does anyone know how to evaluate the infinite product $$ \prod_{k = 2}^{\infty} \left( 1 - \frac{1}{k^3} \right)? $$ I know that a generalized quadratic version has a nice closed form $$ \frac{\sin(\...
kodlu's user avatar
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Continuous version of ergodic with integral

Let $f\in L^1(\mathbb{T}^n)$ such that $f \geq 0$ and $f$ might have a singularity, but along some curve or line $\xi(t): \mathbb{R}\to \mathbb{T}^n$ the value $f(\xi(t))$ is defined and continuous, ...
Sean's user avatar
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Is it possible to evaluate $ \int_0^1 \tan^{-1}\left[\frac{\tanh^{-1}x-\tan^{-1}x}{1+\tanh^{-1}x-\tan^{-1}x}\right]\frac{dx}{x}? $ [closed]

People suggested me to upload this question on math overflow and many people gave the numerical results by desmos and wolfram alpha Motivation for the problem here: Is it possible to evaluate $$ \...
Sbsty 's user avatar
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Simple convergence of convex compact set implies Hausdorff convergence

I am wondering about the following : In $\mathbb{R}^n$, suppose you are given compact convex bodies $\left\{ C_k : k \geq 1 \right\}$ and $C$, such that for every $x \in \mathbb{R}^n$ $$ \mathbb{1}_{...
Anthony's user avatar
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Show convergence of sets

Consider these sets $$ A\equiv \bigcap_{\delta>0} \liminf_{n\rightarrow \infty} \{x \in X: d(p_n, [\ell(x), u(x)])\leq \delta\} $$ $$ C_n(L_n)\equiv \{x \in X: d(p_n, [\ell(x), u(x)])\leq L_n\} $$ ...
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2 answers
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Existence and uniqueness of solutions to a distributional ordinary differential equation

Suppose that $v$ is a distribution on the real line. Then under what conditions can I solve the differential equation $$ \dot{x}(t) = v(x(t)) $$ which I might interpret as an integral equation $$ -\...
cheshircat's user avatar
2 votes
0 answers
40 views

Expansion of an ODE near irregular singularity

Consider the equation \begin{equation} \frac{df}{dt} = \left(\frac{A}{t^2}+\frac{B}{t} \right)f \end{equation} where $A$ is a diagonal matrix with distinct eigenvalues. According to general theory [1,...
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Reference request for equivalent Lipschitz smoothness conditions

For an open set $Z\subseteq\mathbb{R}^n$, let $f: Z\mapsto \mathbb{R}$ be a continuously differentiable function on $Z$, and let $L>0$ be fixed. Also, suppose that (a) $f$ is nonconvex and (b) $f$ ...
William Kong's user avatar
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2 answers
215 views

Integral involving Legendre polynomial

In a physics problem the following integral shows up $$\int\limits_0^{2\pi}P_m(\cos{(\theta-\alpha)})\,\cos^{m+2}{(n\alpha)}\;d\alpha,$$ where $P_m$ is the Legendre polynomial and $n,m$ are integer ...
Zurab Silagadze's user avatar
5 votes
2 answers
466 views

On the derivative of the Bernstein polynomial

$\newcommand\Z{\Bbb Z}\newcommand\De{\Delta}$For a natural $n$ and a function $g\colon\Z\to\Bbb R$, let $B_n g$ be the corresponding Bernstein polynomial, so that $$(B_n g)(x)=\sum_{k\in\Z} g(k)\binom ...
Iosif Pinelis's user avatar
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Proving convexity of the expected logarithm of binomial distribution

I would like to prove that the following function, for an arbitrary integer $n$: \begin{equation} \begin{split} f(x) & =x\cdot E \ \log(1+\text{Binomial(n,x)}) \\ & = x \cdot \sum_{k=0}^{n} \...
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How to prove these identities for $\log(2)$ based on $_3F_2$ integrals?

In this MO post I have placed 4 Ramanujan-type hypergeometric series found using the LLL algorithm for fast computing of some logarithms. I could prove 3 of them by means of classical methods based on ...
Jorge Zuniga's user avatar
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Gradient flows and particle representations

I was looking into gradient flows and their particle representations, mostly in the context of probability. A simple example of this is the continuity equation. Consider evolving a sample $x \sim \...
CComp's user avatar
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Sufficient initial conditions for "Non-local" PDE

I am studying a problem of the form $$i\, \partial_t \psi(t) = L \psi(t) + \int_0^t U(t-r) \psi(r) \, dr, \qquad \psi(0) = \psi_0,$$ where the evolution occurs in some Hilbert space, $L$ is a self-...
DerGalaxy's user avatar
2 votes
2 answers
171 views

A Inequality in the paper by Kenig, Ponce and Vega

I was trying to read the appendix of the paper by Kenig, Ponce and Vega, "Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle", ...
Sarthak's user avatar
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Conjectured closed form of $\int_0^1\frac{\ln^3(1+x)\,\ln^3x}x\mathrm{d}x$

I posted this question on Math Stack Exchange, but there were no helpful comments or answers https://math.stackexchange.com/q/4874446/1298448 How to integrate $${\displaystyle \int_0^1\frac{\ln^3(1+x)\...
Martin.s's user avatar
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Literature search for results on Bochner-Riesz means for real functions

MathOverflow community! I'm delving into a specific area of Fourier analysis and came across an intriguing lemma (referenced as Lemma 4 of Chen & Chen Paper) stating that for a compact set U in $...
Mohammad A's user avatar
4 votes
2 answers
621 views

Conjectured closed form of $\int\limits_0^1 \frac{\ln y \operatorname{Li}_2 (-y)}{1-y^2} \, dy$

Let's state with $\psi^{(1)}$ the trigamma. Calculate the order: $$\mathcal{S} = \sum_{n=1}^\infty (-1)^{n-1} (\psi^{(1)}(n))^2$$ (Cornel Ioan Valean) I uploaded this question here https://math....
Martin.s's user avatar
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1 answer
135 views

Upper bound of a product of sines

Consider the function $$ f_n(t)= \prod_{1 \leq k \leq n-1,\\ \gcd(k,n)=1} \sin\Big(t-\frac{k \pi}{n}\Big),\quad t \in [0,\pi].$$ I wonder whether it is possible to compute some nontrivial upper ...
AgnostMystic's user avatar
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63 views

Gradient estimates of linear elliptic PDE

Let $\Omega \subset \mathbb{R}^n$ be a bounded smooth domain. Assume that $u(x)$ is the classical solution solving $$a_{ij}(x)\partial_{ij}u(x)+b_i(x)\partial_iu(x)+c(x)u(x)=f(x)$$ $$u(x)\Big|_{\...
mnmn1993's user avatar
3 votes
0 answers
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Perturbation method for time-periodic singular system of ODEs

I am studying a problem arising in physics, and I managed to simplify it to a differential system (initial value problem) of the form: $$ \begin{cases} \dot{x} = \epsilon f_1(x,y,t) + \epsilon^2 f_2(...
squille's user avatar
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1 answer
327 views

Exponential trigonometric integral

I want to compute the normalization constant of some probability density on SO(3). After some simplification, I arrive at the following double integral: $$ \tag{1}\label{eq:1} \int_0^{2 \pi} \int_0^{\...
Peter Johnson's user avatar
2 votes
1 answer
170 views

I am seeking a solution to an Abel equation of the first kind with $f_0=0$, $y’=f_3(x)y^3+f_2(x)y^2+f_1(x)y$

The Abel equation of the first kind with $f_0=0$. $$ y'=f_3(x)y^3+f_2(x)y^2+f_1(x)y \tag{1} $$ where \begin{align} f_3(x)&=(12x^2-2)/x,\\ f_2(x)&=(14x^2-1)/x^2,\\ f_1(x)&=3/x. \end{align} ...
Ali Rabah's user avatar
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1 answer
117 views

Singular integral bounded by Dirichlet form?

We define for some fixed $L$ $$\Omega:=\{(x_1,x_2) \in ([-L,L]^2 \times [-L,L]^2) \setminus \{x_1=x_2\}\},$$ in particular $x_1,x_2 \in \mathbb R^2.$ Let $f \in C_c^{\infty}(\Omega)$, then I am ...
António Borges Santos's user avatar
2 votes
0 answers
133 views

Taylor coefficients of the integral of the ordered exponential

Let $A$ be a continuous $2\times 2$ matrix-valued function on $[0,1]$. Define $X_A$ as the solution of $$ X_A'(t) = A(t) X_A(t), \qquad X(0) = I. $$ In other words $X_A$ is the ordered exponential of $...
Pavel Gubkin's user avatar
1 vote
0 answers
143 views

Laplace transform

\begin{equation} \begin{cases}\mathbb{D}_t^\beta u(x, y, t)=-a(x)\left(u_x(x, y, t)+u_y(x, y, t)\right)+\ell(x, y, t, u(x, y, t)), & x>0, y>0, t>0 \\ u(x, y, 0)=0, & x>0, y>0 \\ ...
TUHOATAI's user avatar
12 votes
2 answers
909 views

Asymptotics of a strange oscillatory function

Consider the function $f:\mathbb{R}\to \mathbb{R}$ defined by $f(x)=\sum_{n\geq 1}\sin(x/n^2)$. It is easy to see that $f(x) = O(\sqrt{x})$ for large real $x$. Is it true that $f(x)>0$ for $x>0$...
Satan's Minion's user avatar
2 votes
0 answers
50 views

Regularization for Newtonian n-body collisions in $\mathbb{R}^3$

In working with binary collisions in the Hamiltonian formulation of the Newtonian $n$-body problem, two common regularization techniques that deal with binary collisions are the Levi-Civita technique, ...
user12994's user avatar
6 votes
0 answers
195 views

Energy of harmonic maps from $\mathbb R^2$ to $S^2$ is quantized

Assume that $U:\mathbb R^2\to S^2=\{y\in\mathbb R^3:|y|=1\}$ is a smooth solution of the equation $\Delta U+|\nabla U|^2U=0$ in $\mathbb R^2$ with $\int_{\mathbb R^2}|\nabla U|^2\,dx<+\infty$. ...
Feng's user avatar
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2 votes
1 answer
280 views

Sets of integers "a little less dense" than the set of prime numbers

Given a set $A \subseteq \mathbb{N}$ of positive integers, put $$ S_A: \mathbb{N} \rightarrow \mathbb{R}, \ \ \ \ N \mapsto \sum_{n \in A \cap \{1, \dots, N\}} \frac{1}{n}. $$ There are obvious ...
Stefan Kohl's user avatar
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Electrostatic potential energy of point-charges at primes up to $x$

Given a positive real (or integral) number $x$ we consider the electrostatic potential energy of equal point charges at all primes up to $x$ given by $$E(x)=\sum_{p_1<p_2\leq x}\frac{1}{p_2-p_1}$$ ...
Roland Bacher's user avatar
7 votes
1 answer
453 views

How to find closed form for $\int_0^1 \frac{x}{x^2+1} \left(\ln(1-x) \right)^{n-1}dx$?

Here in my answer I got the real part for the polylogarithm function at $1+i$ for natural $n$ $$ \Re\left(\text{Li}_n(1+i)\right)=\left(\frac{-1}{4}\right)^{n+1}A_n-B_n $$ where $$ B_n=\sum_{k=0}^{\...
Faoler's user avatar
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0 answers
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The intersection of $ n $ cylinders in $ 3D$ space

I posted the question on here, but received no answer I recently found out about the Steinmetz Solids, obtained as the intersection of two or three cylinders of equal radius at right angles. If we set ...
user967210's user avatar
0 votes
0 answers
208 views

Gauss transformation in fractional Sobolev space

Let $g_{\mu}(x) = \mu^{d/2}\exp(-\pi\mu|x|^2)$ for every $\mu > 0$. Prove that $$ \int_{\mathbb R^{d}}\left|(-\Delta)^{\frac{s}{2}} u\right|^{2} \geq \int_{\mathbb R^{d}}\left|(-\Delta)^{\frac{s}{2}...
Muniain's user avatar
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Characterization of an integral operator with a Bessel kernel

I am considering the following integral operator: $$K(\sigma)(\theta)=\int_0^{2\pi} \sigma(\theta') J_0(|e^{i\theta}-e^{i\theta'}|)\,d\theta',$$ where $J_0$ is the Bessel function of order $0.$ I am ...
Didier Felbacq's user avatar
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0 answers
85 views

Fourier integral operators and parametrix

Consider the classical wave equation in a bounded domain in $\mathbb{R}^n$, assuming vanishing of the function and its normal derivative on the boundary. Question: Is there an expression for the ...
0x11111's user avatar
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1 answer
111 views

An integral involving Riemann xi function

Let $z(t)$ and $w(t)$ be some complex functions of the real variable $t$ with $z(t)$ tending to some complex number as $t \rightarrow 1$ and $w(t)$ tending to $0$ as $t \rightarrow1$ I'm looking at ...
Alexis's user avatar
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1 vote
1 answer
87 views

Bound on $L^1$ norm of solution of two-point boundary value problem

This has to be known, but I have not been able to find it in the literature (probably due to not being too familiar with two-point boundary value problems). I have a function $u:[0,1]\to\mathbb{R}$ ...
gmvh's user avatar
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1 vote
0 answers
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How to derive a lower bound of a MinMax inequality?

Let $x_5,\cdots,x_n\in[0,\alpha]\cup[-\pi,\alpha-\pi]$ where $\alpha$ is a fixed angle $\in(0,\pi/2)$. The goal For a fixed $(A_{ij})_{1\leq i\leq 4,5\leq j\leq n}\in\{-1,+1\}$, verify whether it ...
M.K's user avatar
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3 votes
0 answers
61 views

Admissibility condition of wavelet functions

After a badly formulated question, I decided to make a new post searching for help. The basic problem is the follows: I have a wavelet function $\psi(t)$ (real or complex) and would like to compute (a)...
Luciano Magrini's user avatar
8 votes
2 answers
2k views

Solving 'impossible' integrals with a new (?) trick

The following identities have been suggested based on formulas in a previous question of mine. If complex $\theta_1=\cos^{-1}(p)$ and $\theta_2=\sec^{-1}(p)$, where $p\in(-1, 0) \cup (1, \infty)$, ...
Emmanuel José García's user avatar
1 vote
1 answer
125 views

Functions for which $\lambda f(x)=f(\alpha_\lambda x + \beta_\lambda)$

How many functions $f:\mathbb{R}\to R\subseteq\mathbb{R}$ are there such that there is an $S\subseteq\mathbb{R}$ containing an interval containing $1$ such that for any $\lambda\in S$ there exist $\...
gmvh's user avatar
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2 votes
1 answer
132 views

Behaviour of the solution of a second order ODE

I am currently studying the following second order ODE \begin{cases} \ddot y(x)\left(\ln(x) - 2\ln(y(x))\right) - 2\frac{(\dot y(x))^2}{y(x)} = 0 &\text{in }[0,T]\\ y(0) = 0\\ \dot y(T) = c \end{...
Falcon's user avatar
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2 votes
0 answers
139 views

The Hausdorff measure of intersection of annulus and conformal curve

Recently I came across a problem in my research. Let $g:[0,1]\to\mathbb{C}$ be a restriction of a conformal map that is defined in a simply connected domain $\Omega\subseteq\mathbb{C}$ that include $[...
mathematics is all's user avatar
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0 answers
41 views

Godunov splitting convergence research

The approximation of Godunov splitting on certain differential equations is known to be first order accurate. In 2011, a paper has also shown that it is first order accurate for nonlinear ordinary ...
Redsbefall's user avatar
5 votes
5 answers
983 views

What are the local maxima and minima of $\frac{\sin(nx)}{\sin(x)}$

FYI: I asked this question here couple of days ago but got no answer yet. $n$ is an integer We know the global maximum of the function $\sin(nx)/\sin(x)$ is $n$ (thanks to this question), but what are ...
RajaKrishnappa's user avatar
2 votes
1 answer
145 views

Growth rate of elementary sequences

We consider three sequences $(x_n),(y_n),(z_n)$, where $(x_n) \in \ell^1$ is positive and the other two sequences are merely assumed to be positive, i.e. $y_n,z_n \ge 0$ where $0<z_n<z_{n+1}$ is ...
António Borges Santos's user avatar

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