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.
3,444
questions
0
votes
0
answers
9
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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:
$$\mathscr{F}\{\psi\}(x)=-2\pi \gamma \delta(x)-\pi \...
0
votes
1
answer
67
views
An inequality for the Darboux integral
As we know, if $f(x)$ are Riemann integrable, we have
\begin{gather}
\left|\int_a^b f(x)~\mathrm{d}x\right|\leq \int_a^b |f(x)|~\mathrm{d}x.
\end{gather}
So, for Darboux integrals, such as the upper ...
1
vote
1
answer
181
views
Prove the limit of the integral
Suppose:
$f:[0,\frac{\pi}{2}]\to \mathbb{R}$, $f(x)\in C^2[0,\frac{\pi}{2}]$, $\gamma=\lim_{n\to \infty}(1+1/2+\dotsb+1/n-\ln{n})$.
Prove:$$ \lim_{s\to +\infty}\left(\int_{0}^{\frac{\pi}{2}} \frac{\...
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
$$
\frac{\sin(\...
-1
votes
0
answers
51
views
Continuous version of ergodic with integral
Let $f\in L^1(\mathbb{T}^n)$ such that $f \geq 0$ and $f$ might have a singularity, but along some curve or line $\xi(t): \mathbb{R}\to \mathbb{T}^n$ the value $f(\xi(t))$ is defined and continuous, ...
-1
votes
0
answers
80
views
Is it possible to evaluate $ \int_0^1 \tan^{-1}\left[\frac{\tanh^{-1}x-\tan^{-1}x}{1+\tanh^{-1}x-\tan^{-1}x}\right]\frac{dx}{x}? $ [closed]
People suggested me to upload this question on math overflow and many people gave the numerical results by desmos and wolfram alpha
Motivation for the problem here:
Is it possible to evaluate
$$
\...
1
vote
1
answer
69
views
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 $x \in \mathbb{R}^n$ $$ \mathbb{1}_{...
1
vote
1
answer
256
views
+50
Show convergence of sets
Consider these sets
$$
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\}
$$
...
3
votes
2
answers
274
views
Existence and uniqueness of solutions to a distributional ordinary differential equation
Suppose that $v$ is a distribution on the real line. Then under what conditions can I solve the differential equation
$$
\dot{x}(t) = v(x(t))
$$
which I might interpret as an integral equation
$$
-\...
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,...
0
votes
0
answers
53
views
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) $f$ ...
5
votes
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 ...
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
$$(B_n g)(x)=\sum_{k\in\Z} g(k)\binom ...
2
votes
1
answer
101
views
Proving convexity of the expected logarithm of binomial distribution
I would like to prove that the following function, for an arbitrary integer $n$:
\begin{equation}
\begin{split}
f(x) & =x\cdot E \ \log(1+\text{Binomial(n,x)}) \\
& = x \cdot \sum_{k=0}^{n} \...
5
votes
0
answers
360
views
How to prove these identities for $\log(2)$ based on $_3F_2$ integrals?
In this MO post I have placed 4 Ramanujan-type hypergeometric series found using the LLL algorithm for fast computing of some logarithms. I could prove 3 of them by means of classical methods based on ...
2
votes
1
answer
75
views
Gradient flows and particle representations
I was looking into gradient flows and their particle representations, mostly in the context of probability.
A simple example of this is the continuity equation. Consider evolving a sample $x \sim \...
1
vote
1
answer
79
views
Sufficient initial conditions for "Non-local" PDE
I am studying a problem of the form $$i\, \partial_t \psi(t) = L \psi(t) + \int_0^t U(t-r) \psi(r) \, dr, \qquad \psi(0) = \psi_0,$$ where the evolution occurs in some Hilbert space, $L$ is a self-...
2
votes
2
answers
171
views
A Inequality in the paper by Kenig, Ponce and Vega
I was trying to read the appendix of the paper by Kenig, Ponce and Vega, "Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle",
...
1
vote
0
answers
84
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Conjectured closed form of $\int_0^1\frac{\ln^3(1+x)\,\ln^3x}x\mathrm{d}x$
I posted this question on Math Stack Exchange, but there were no helpful comments or answers
https://math.stackexchange.com/q/4874446/1298448
How to integrate $${\displaystyle \int_0^1\frac{\ln^3(1+x)\...
-1
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0
answers
47
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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 $...
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....
2
votes
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 ...
0
votes
0
answers
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
$$a_{ij}(x)\partial_{ij}u(x)+b_i(x)\partial_iu(x)+c(x)u(x)=f(x)$$
$$u(x)\Big|_{\...
3
votes
0
answers
41
views
Perturbation method for time-periodic singular system of ODEs
I am studying a problem arising in physics, and I managed to simplify it to a differential system (initial value problem) of the form:
$$
\begin{cases}
\dot{x} = \epsilon f_1(x,y,t) + \epsilon^2 f_2(...
7
votes
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:
$$ \tag{1}\label{eq:1}
\int_0^{2 \pi} \int_0^{\...
2
votes
1
answer
170
views
I am seeking a solution to an Abel equation of the first kind with $f_0=0$, $y’=f_3(x)y^3+f_2(x)y^2+f_1(x)y$
The Abel equation of the first kind with $f_0=0$.
$$
y'=f_3(x)y^3+f_2(x)y^2+f_1(x)y \tag{1}
$$
where
\begin{align}
f_3(x)&=(12x^2-2)/x,\\
f_2(x)&=(14x^2-1)/x^2,\\
f_1(x)&=3/x.
\end{align}
...
0
votes
1
answer
117
views
Singular integral bounded by Dirichlet form?
We define for some fixed $L$
$$\Omega:=\{(x_1,x_2) \in ([-L,L]^2 \times [-L,L]^2) \setminus \{x_1=x_2\}\},$$
in particular $x_1,x_2 \in \mathbb R^2.$
Let $f \in C_c^{\infty}(\Omega)$, then I am ...
2
votes
0
answers
133
views
Taylor coefficients of the integral of the ordered exponential
Let $A$ be a continuous $2\times 2$ matrix-valued function on $[0,1]$. Define $X_A$ as the solution of
$$
X_A'(t) = A(t) X_A(t), \qquad X(0) = I.
$$
In other words $X_A$ is the ordered exponential of $...
1
vote
0
answers
143
views
Laplace transform
\begin{equation}
\begin{cases}\mathbb{D}_t^\beta u(x, y, t)=-a(x)\left(u_x(x, y, t)+u_y(x, y, t)\right)+\ell(x, y, t, u(x, y, t)), & x>0, y>0, t>0 \\ u(x, y, 0)=0, & x>0, y>0 \\ ...
12
votes
2
answers
909
views
Asymptotics of a strange oscillatory function
Consider the function $f:\mathbb{R}\to \mathbb{R}$ defined by $f(x)=\sum_{n\geq 1}\sin(x/n^2)$. It is easy to see that $f(x) = O(\sqrt{x})$ for large real $x$. Is it true that
$f(x)>0$ for $x>0$...
2
votes
0
answers
50
views
Regularization for Newtonian n-body collisions in $\mathbb{R}^3$
In working with binary collisions in the Hamiltonian formulation of the Newtonian $n$-body problem, two common regularization techniques that deal with binary collisions are the Levi-Civita technique, ...
6
votes
0
answers
195
views
Energy of harmonic maps from $\mathbb R^2$ to $S^2$ is quantized
Assume that $U:\mathbb R^2\to S^2=\{y\in\mathbb R^3:|y|=1\}$ is a smooth solution of the equation $\Delta U+|\nabla U|^2U=0$ in $\mathbb R^2$ with $\int_{\mathbb R^2}|\nabla U|^2\,dx<+\infty$. ...
2
votes
1
answer
280
views
Sets of integers "a little less dense" than the set of prime numbers
Given a set $A \subseteq \mathbb{N}$ of positive integers, put
$$
S_A: \mathbb{N} \rightarrow \mathbb{R}, \ \ \ \
N \mapsto \sum_{n \in A \cap \{1, \dots, N\}} \frac{1}{n}.
$$
There are obvious ...
2
votes
0
answers
150
views
Electrostatic potential energy of point-charges at primes up to $x$
Given a positive real (or integral) number $x$ we consider the
electrostatic potential energy of equal point charges at all primes up to $x$
given by
$$E(x)=\sum_{p_1<p_2\leq x}\frac{1}{p_2-p_1}$$
...
7
votes
1
answer
453
views
How to find closed form for $\int_0^1 \frac{x}{x^2+1} \left(\ln(1-x) \right)^{n-1}dx$?
Here in my answer I got the real part for the polylogarithm function at $1+i$ for natural $n$
$$ \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}^{\...
1
vote
0
answers
66
views
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 ...
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
$$
\int_{\mathbb R^{d}}\left|(-\Delta)^{\frac{s}{2}} u\right|^{2} \geq \int_{\mathbb R^{d}}\left|(-\Delta)^{\frac{s}{2}...
1
vote
0
answers
54
views
Characterization of an integral operator with a Bessel kernel
I am considering the following integral operator: $$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 ...
0
votes
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 ...
0
votes
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 ...
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 $u:[0,1]\to\mathbb{R}$ ...
1
vote
0
answers
44
views
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 ...
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)...
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 $p\in(-1, 0) \cup (1, \infty)$, ...
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 $\...
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{...
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 $[...
0
votes
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 ...
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 ...
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 ...