# 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.

3,423
questions

2
votes

1
answer

83
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

55
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

32
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

306
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^{\...

1
vote

0
answers

95
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

111
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 ...

1
vote

0
answers

91
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

75
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&...

11
votes

2
answers

845
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

48
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

191
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

276
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

146
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

442
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

63
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

52
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

79
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

108
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

82
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

43
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

58
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)...

7
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

123
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

129
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

135
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

40
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

969
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

144
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 ...

1
vote

1
answer

222
views

### Eigenvalues of a Schrödinger operator

I'm interested in the existence of eigenfunctions and finding eigenvalues of the following operator
$$L(\varphi) = \varphi_{rr} - \frac{1}{r} \varphi_r - [V + \frac{m}{r^2}] \varphi$$
$$\varphi(0) = \...

3
votes

2
answers

428
views

### $\lim_{n \to \infty} \frac{2^n}{n} \left[ 1 - \sum_{k = 1}^{n-1} \frac{(1 - \lambda 2^{-n})^{2^k}}{2^k} \right]$

Find the limit
\begin{equation*}
\lim_{n \to \infty} \frac{2^n}{n} \left[
1 - \sum_{k = 1}^{n-1} \frac{(1 - \lambda 2^{-n})^{2^k}}{2^k}
\right]
\end{equation*}
where $\lambda > 0$.
My guess is that ...

1
vote

1
answer

93
views

### Convergence of ODE with uniform $L^\infty \cap L^1$ bound on nonlinearity

Consider the IVP
$$
\left\{
\begin{aligned}
\frac{d}{dt} \Phi_n(t,x) &= f_n(\Phi_n(t,x)) && \forall t \in \mathbf{R}_+ \\
\Phi_n(0,x) &= x && \forall x \in \mathbf{R}
\end{...

1
vote

1
answer

119
views

### Given a radial symmetric function $f$, the estimate of |$\Delta ^ {m/2}f$| in $R^{2m}$ by induction

This question might be a little strange; my order of Laplacian is related to the dimension of the space.
Actually, I’m reading a result which is obtained by induction; it is the absolute value of ...

3
votes

2
answers

205
views

### Extremum placement for two-variable function

While teaching Calculus 2, one of my students asked me the following
Given 3 points $x_1$, $x_2$, $x_3$. Whether there exists one function $z = f(x,y)$ which has exactly 2 extremum and 1 saddle point:...

4
votes

0
answers

162
views

### Growth rate of a recursively defined sequence

For a side project with a friend (having to do with fractional iterates of the exponential function), we're looking at a sequence $a_n$ defined recursively by equations $a_1 = 1$ and
$$a_n = \sum_{k=1}...

8
votes

1
answer

325
views

### Special Schwartz function on the positive interval

Is there a Schwartz function $\zeta(t)$, defined on $\mathbb{R}$, satisfying the following:
$\int \zeta(t)\: dt=1$,
$\int t^k \zeta(t)\: dt=0$ for all $k\geq 1$,
$\operatorname{supp}(\zeta)\subset (0,...

43
votes

3
answers

2k
views

### Proving $\sum_{i=1}^{n}\sum_{j=1}^{n}\left\{\frac{x_{i}}{x_{j}}\right\}\le \frac{9}{14}n^2$？

For any postive integer $n$ and for any postive real numbers $x_{1},x_{2},\cdots,x_{n}$, show that
$$\sum_{i=1}^{n}\sum_{j=1}^{n}\left\{\dfrac{x_{i}}{x_{j}}\right\}\le \dfrac{9}{14}n^2$$
Let
\begin{...

7
votes

1
answer

451
views

### On a paper by Dimitrie Pompéiu and on one (in two parts) by Edmund Landau

To celebrate the new year and the future of mathematics (or the mathematics of future), I see no better way to ask a question stemming from my researches on power series.
The two papers the title ...

2
votes

1
answer

131
views

### Boundedness of an exit time from a campact set

Let $n\geq 1$ and $v\in\mathcal{C}^1(\mathbb{R}^n,\mathbb{R}^n)$. For $x_0\in\mathcal{O}$, let $\big(x(t)\big)_{t\geq 0}$ be the solution of
\begin{align*}
& x(0)=x_0 \\
& \dot{x}=v(x).
\end{...

8
votes

1
answer

553
views

### What is the minimum of this functional？

Recently I encountered an inequality from mathematical analysis.
Let $f(x)$ be twice continuously differentiable in $[0,1]$ with
$f(0)=f(1)=0$, then for all $x\in(0,1),f(x)\neq 0$, show that:$$\int_{0}...

2
votes

2
answers

130
views

### Upper bound estimation for second-order variable-coefficient ODE

I'm tackling a second-order linear ordinary differential equation with variable coefficients $a(t)$ and seek advice on estimating the upper bound of
$y(t)$ s.t $|y(t)|\le M$. The equation in question ...

18
votes

1
answer

1k
views

### Does summing divergent series using cutoff functions give consistent results?

One way to try to give a value $S$ to a divergent series $\sum_{n=1}^\infty a_n$ is with a smooth cutoff function:
$$
S = \lim_{N\to\infty}\sum_{n=1}^\infty a_n \eta\left(\frac{n}{N}\right)
$$
where $\...

23
votes

5
answers

3k
views

### What phenomena are better modelled by SDE instead of ODE?

Both stochastic differential equations (SDE) and ordinary differential equations (ODE) can be used to model a variety of different phenomena, whether physical or otherwise. Most deterministic ODE ...

3
votes

2
answers

598
views

### Are there any functions $f$ beyond trivial examples where $\int f(x +f(x + f(x +\dotsb)))\,dx$ $= F(x+F(x+F(x +\dotsb))) + C$ for some function $F$?

Basically the title explains most of my question here. Purely out of curiosity (I have no real application), I am wondering if there are any "interesting" functions $f$ where we know of a ...

1
vote

1
answer

149
views

### Derivation of indefinite integral involving hypergeometric function

I am doing a project on projectile motion and I ended up with this integral:
$$\int \frac{m \left(g - \left(\frac{1}{e^t - g^{\frac{m}{c}}}\right)^{\frac{m}{c}}\right)}{c} \, dt$$
where $g, c,$ and $m$...

4
votes

2
answers

1k
views

### Does a function exist which is not Riemann integrable and satisfies the given condition:

I am looking for a function $f:[0,1]\rightarrow \mathbb{R}$ which is not Riemann integrable such that
$$\sum_{k=0}^n |f(x_k)-f(x_{k-1})|^2 <1$$ for every choice of $0=x_0\le x_1 \le \cdots \le x_n =...

0
votes

1
answer

120
views

### Asymptotic behavior of the polylogarithm function and generalisation

So, right now I am writing my master thesis and I need to find a reference for a formula I found in a paper:
$$
\sum_{k=1}^{\infty}k^{-\alpha}(1-\varepsilon)^k\sim b+c\Gamma(1-\alpha)\varepsilon^{\...

0
votes

0
answers

69
views

### Inverse Laplace of the Complex conjugate of the Laplace transform

Let the Laplace transform of f(t) be F(s) and let the inverse Laplace transform of F(s*) be g(t). is there a theorem relating f(t) and g(t)? Basically, looking for a way to calculate g(t) from f(t) ...

4
votes

2
answers

256
views

### Implicit function theorem without uniqueness?

Imagine you are given $f(x,y) := y^2-\sin(x)^2$
and you want to answer the question, if there is a neighbourhood of $x=0$ such that $f(x,y(x))=0$ with $y(0)=0$.
One idea that comes to mind is the ...

2
votes

1
answer

176
views

### Fourier coefficients of the logarithm of a given function

Let $f$ be a $1$-periodic real function that I know is bounded away from zero:
$$
f(x) = \sum_{n = -\infty}^\infty c_n e^{2\pi i n x}
$$
Let me also assume that $f$ is analytic with Fourier ...