All Questions
Tagged with ca.classical-analysis-and-odes reference-request
323 questions
17
votes
1
answer
1k
views
Catalan's constant fast convergent series
NOTE. UPDATE 2 introduces proven series for Catalan's constant that is possibly the fastest currently known.
Working with some conjectured continued fractions that were published here, I have found ...
- 2,836
3
votes
2
answers
808
views
Growth of $L^p$ norms as $p \to \infty$
Let $f$ be a non-negative function defined on the unit interval. It is well known that $N(p) := \left(\int_0^1 f^p(t) dt\right)^{\frac{1}{p}} $ converges to $\operatorname{esssup}_{[0,1]} f$ when $p \...
- 1,211
4
votes
1
answer
195
views
Reference request: Rigorously solving ODEs using divergent asymptotic series
In my research I have come across a divergent asymptotic series $\sum_{n =0}^\infty a_n f_n(x)$ that formally solves a certain fairly simple nonlinear second-order ODE but does not seem to correspond ...
- 775
0
votes
1
answer
191
views
Calculating derivatives of arbitrary-order at an operator's root
Consider roots $f = 0$ of a nicely-behaved real function $f(x, t)$ of two (real) variables.
Namely, points $(x, t)$ on which $f$ vanishes, $f(x, t) = 0$.
Suppose that $x$ can be written as function of ...
- 583
2
votes
1
answer
224
views
The differentiability of the distance function on asymptotically flat manifolds
Let $M = \mathbb{R}^3 \setminus \overline{B_1}$ where $\overline{B_1}$ is the closed unit ball.
Let $g$ be an asymptotically flat metric of the form $g_{ij} = \delta_{ij}+h_{ij}$ in standard ...
- 969
1
vote
1
answer
119
views
Estimating two dimensional theta function
My feeling is that this should be written somewhere but I don't know what to search for.
Let $Q(x,y)$ be a binary quadratic form over $\mathbb{C}$, with $\operatorname{Re}(Q)$ positive definite. Then ...
- 2,178
1
vote
1
answer
149
views
Completeness of asymptotically Euclidean manifolds
Say that you place an asymptotically Euclidean metric on $\mathbb{R}^3,$ e.g. $\mathbb{R}^3$ is endowed Riemannian metric $g$ such that $\text{supp}(g^{ij}-\delta^{ij})\subseteq\{|x|\leq R\}$ for some ...
- 13
2
votes
1
answer
91
views
References for Green's functions right focal boundary-value problem
Could you please give me some references for the computation of a Green's function for a second-order right focal difference equation?
For this problem:
\begin{gather*}
\Delta^2 u(t)=f(t), \; t\in\{0,...
- 27
2
votes
0
answers
2k
views
Stein's book on harmonic analysis
My background :
I am a Math PhD student. I will most probably work in harmonic analysis on Euclidean spaces. I am a fan of Folland's Real analysis and I have thoroughly studied first 8 chapters of ...
- 121
1
vote
1
answer
164
views
Two trigonometric integrals: looking for a transformation
I have two integrals of trigonometric functions and I would like to ask:
QUESTION. Is there a transformation rule (or general principle) to show this equality?
$$\int_0^{\frac{\pi}2}\frac{d\theta}{\...
- 43.2k
1
vote
0
answers
74
views
Calculation of a multi-dimensional Fourier transform
I am interested in the following multi-dimensional Fourier transform:
$$
\int_{\mathbb{R}^{p}} \mathrm{d} \vec{r}_{\parallel}\int_{\mathbb{R}^{q}} \mathrm{d} \vec{r}_\perp \, e^{-\mathrm{i}\, \vec{p}...
- 373
6
votes
2
answers
921
views
Has the "partial Sophomore's Dream function" been studied before?
We can consider the generalized Harmonic numbers $$H_{n,m} := \sum_{k=1}^{n} \frac{1}{k^{m}} $$ as a partial version of the Riemann zeta function, because $$\lim_{n \to \infty} H_{n,m} = \zeta(m). $$
...
- 4,829
1
vote
0
answers
210
views
Questions about iterating the Euler-Maclaurin summation formula
Introduction
The Euler–Maclaurin summation formula is as follows for a positive integer $p$ and a continuous function $f(\cdot)$ that is $p$ times continuously differentiable on the interval $[m,n]$ : ...
- 4,829
0
votes
0
answers
99
views
Does $\sum_{m=0}^{\infty} \left|c_m g(x)^{2m+1}\right|$ converge absolutely to an integrable function?
Consider the integral
\begin{equation}
\int_{0}^{t}J_1(f(t)-f(s))\mathop{ds}=\int_{0}^{t}\sum_{m=0}^{\infty} c_m (f(t)-f(s))^{2m+1}\mathop{ds},
\end{equation}
such that $J_1$ is the Bessel function of ...
- 79
1
vote
0
answers
138
views
An Elementary Inequality [closed]
A friend of mine found in the internet the following exercise:
Let $a, b, c \geq 0$ with $a + b + c = 1$. Show that
$$
\sqrt{a + b^2} + \sqrt{b + c^2} + \sqrt{c + a^2} \geq 2,
$$
where equality is ...
71
votes
3
answers
5k
views
Does iterating the derivative infinitely many times give a smooth function whenever it converges?
I am a graduate student and I've been thinking about this fun but frustrating problem for some time. Let $d = \frac{d}{dx}$, and let $f \in C^{\infty}(\mathbb{R})$ be such that for every real $x$, $$g(...
- 1,763
4
votes
5
answers
2k
views
Reference request: importance of Lipschitz continuity
I see that Lipschitz continuity is a common assumption used in optimisation, statistics, machine learning, etc.
Could you point me in the direction of some literature that discusses why Lipschitz ...
- 161
7
votes
2
answers
2k
views
The source of the Integral
Wolfram alpha calculates the integral
$$\int\limits_0^\infty \frac{x^2\ln{x}}{e^x-1}dx=2\zeta^\prime(3)+3\zeta(3)-2\gamma\zeta(3).$$
However, I need to cite the source of this identity (the table of ...
- 16.5k
4
votes
0
answers
802
views
Reproducing kernel Hilbert space of Matérn kernels
I am trying to read a recent paper titled "Interpolation and learning with scale dependent kernels" by Pagliana, Ruidi, De Vito, and Rosasco. (The paper can be found on ArXiv)
On the top of ...
4
votes
1
answer
461
views
Estimate on $C^1$-norm of solution of the Dirichlet problem for the Laplace equation
Let $\Omega\subset \mathbb{R}^n$ be a bounded domain with $C^\infty$-smooth boundary. Let $\phi\in C^\infty(\partial \Omega)$. Let $u$ be the solution of the Dirichlet problem of the Laplace equation
\...
- 21.8k
6
votes
2
answers
295
views
Which result guarantees convergence of solution of an ODE to a set of non-compact, non-isolated equilibrium?
Consider a continuous ODE,
$$\dot x = f(x), f \in C^1$$
$\dot x = 0$ for all $x \in K \subset \mathbb{R}^n$, where we assume that $K$ is a closed but unbounded set of non-isolated equilibrium. For ...
- 205
1
vote
1
answer
182
views
Proving an estimate for the Neumann problem on $\mathbb{R}^3 \setminus B_1$ in Weighted Sobolev spaces
Let $M := \mathbb{R}^3 \setminus B_1$ where $B_1$ is the unit ball.
It is known that for every $g \in H^{\frac{1}{2}} (\partial M)$, and for an appropriate $\delta$, there exists a unique solution $u$ ...
- 969
2
votes
1
answer
329
views
Is $g(v)=\mathbb{E}[f(v+W)]$ a differentiable function of $v$ when $f$ is continuous and $W$ is multivariate normal?
Suppose $f$ is a continuous function on $\mathbb{R}^n$, and $W$ has a multivariate normal distribution on $\mathbb{R}^n$. If the expectation
$$g(v)=\mathbb{E}[f(v+W)]$$
is defined for all $v \in \...
user44143
2
votes
1
answer
231
views
Solution of Riccati system of ODEs
We have following equation:
$$
w(t,v) = \exp\Bigl(-\phi (t) \frac{v^2}{2}-\psi (t) v -\chi (t)\Bigr),\quad (t,v)\in [0,T]\times \mathbb{R},
$$
where $(\phi, \psi ,\chi)$ are solutions of the Riccati ...
- 43
3
votes
1
answer
613
views
Searching for the proof of a certain claim in Arnold's ODE book from 1992
I was reading today the book of Stephen Wiggins called "Global Bifurcations and Chaos" (the 1988 edition).
On pages 12-13 he writes the following:
Consider the following ordinary ...
- 1,594
2
votes
1
answer
163
views
Approximation of $C^1$-smooth equivariant maps by infinitely smooth ones
Let $M,N$ be smooth closed manifolds acted by a finite group $G$. Let $f\colon M\to N$ be a $C^1$-smooth $G$-equivariant map.
Is it true that for any $\varepsilon>0$ there exists a $C^\infty$-...
- 21.8k
2
votes
1
answer
473
views
An identity for the elliptic theta function
For real $s>0$, let
$$S(s):=\sum_{n=-\infty}^\infty e^{-n^2/(2s^2)}
=\vartheta _3\left(0,e^{-1/(2 s^2)}\right),$$
where $\vartheta$ is the elliptic theta function.
Plotting suggests that the ...
- 128k
6
votes
2
answers
513
views
Need a reference for a trigonometric inequality
In my old high school notebook (20 years ago), the following inequality appears with its proof:
$$1+\cos x + \frac{1}{2}\cos 2x + \cdots + \frac{1}{n}\cos nx \geq 0$$
for any real $x$ and positive ...
- 335
7
votes
1
answer
352
views
Tight upper bounds on trigonometric polynomials
According to D. Hajela's chapter in Open Problems in Communications and Computation the following question was open as of the late 1980s. I have been unable to find any references so any results or ...
- 10.4k
4
votes
1
answer
591
views
The constant $e$ represented by an infinite series
In this Wikipedia article the constant $\pi$ is represented by the following infinite series: $$\pi=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}-\...
- 2,661
3
votes
2
answers
552
views
Second order differential equation with oscillating behavior
I consider a differential equation $y^{\prime \prime} (x) + V(x) y(x) = 0$ in the interval $[0,\infty)$, where $C_1 \leq V(x) \leq C_2$ for all $x \in [0,\infty)$ for some constants $C_2 > C_1 >...
- 5,562
13
votes
2
answers
968
views
Have any proposals been advanced for the analytic continuation of the divisor function?
While I was working on the evaluation of a certain series, the following limit came up:
\begin{align} \lim_{n \to 1} \frac{d(n)-1}{n(n-1)} &= \lim_{n \to 1} \frac{d'(n)}{2n-1} \\
&= d'(1) .\...
- 4,829
1
vote
2
answers
106
views
Green function of symmetric stable process in dimension 1 and 2
Are the results in this paper on the Green function of a symmetric stable process available also in space dimension $d =1$ and $d=2$? The main theorems here are stated only for $d \ge 3$.
- 99
3
votes
1
answer
183
views
On integral representation of Whittaker $W$ functions
According to NIST, the integral representation of Whittaker $W$ functions
$$
W_{\kappa,\mu}\left(z\right)=\frac{z^{\mu+\frac{1}{2}}2^{-2\mu}}{\Gamma\left(%
\frac{1}{2}+\mu-\kappa\right)}\int_{1}^{\...
- 373
1
vote
1
answer
122
views
Existence and uniqueness for the equation $u_t + \nabla |u| = 0$
How does one prove the existence, uniqueness, and regularity for the equation
$$u_t + \nabla_x |u| = 0 $$
with initial data $u(0,x) = u_0(x)$ and where the unknown function is $u:\mathbb (0,\infty)\...
- 41
9
votes
2
answers
1k
views
A tricky integral to evaluate
I came across this integral in some work. So, I would like to ask:
QUESTION. Can you evaluate this integral with proofs?
$$\int_0^1\frac{\log x\cdot\log(x+2)}{x+1}\,dx.$$
- 43.2k
1
vote
1
answer
195
views
Existence and regularity for fractional elliptic problem with gradient term: $ (-\Delta)^s u + v\cdot \nabla u = 0$ with $v \in \dot H^s$
Let us consider the problem
$$ (-\Delta)^s u + v\cdot \nabla u = 0 \quad \text{ in } \mathbb R^n, $$
where $s \in (0,1)$, $(-\Delta)^s$ is the fractional Laplace operator and
$v:\mathbb R^n \to \...
- 99
3
votes
2
answers
312
views
Complex Hermite polynomial orthogonality on weighted space
Consider the "probabilist's" Hermite polynomials given by $$H_n(x)=(-1)^ne^{\frac{x^2}{2}}\partial_x^ne^{-\frac{x^2}{2}}.$$
These polynomials trivially extend to functions of $w\in\mathbb{C}$...
- 854
1
vote
1
answer
196
views
Taylor expansion of Modified Mathieu functions
Do we know the Taylor expansion at $0$ of the radial Mathieu functions $(\mathsf{Mc}_n^{(j)}(\,\cdot\,, \sqrt{q}))_{n \ge 0}$ and $(\mathsf{Ms}_n^{(j)}(\,\cdot\,, \sqrt{q}))_{n \ge 1}$, for $q \in \...
- 310
2
votes
0
answers
108
views
Do you know of orthogonal-polynomial families with complex measure on the square? I'm just looking for family names to read up on
I'm looking for the name(s) of a family or families of polynomials whose normalization and orthogonality are defined by integrals (inner product) over the complex square $\{u+iv\, |\, u, v \in [-1,1]\}...
- 49
2
votes
0
answers
169
views
Functions whose Fourier coefficients satisfy $ \sum_{k=1}^\infty |c_k| < 1 $?
Let $f:(0,1) \to \mathbb R$ be a function that can be written as $$f(x) = \sum_{k=1}^\infty c_k \phi_k(x),$$ where $\phi_k(x) = \cos(\pi k x)$. What is the minimal assumption required on $f$ to ...
- 839
5
votes
1
answer
297
views
A scaled fractional ''Sobolev inequality''
Does a fractional interpolation inequality similar to $$
\int_{B_R(0)} |u| dx \le C R^{2} \sqrt{\log(2R)} \Big( \int_{\mathbb R^2}\int_{\mathbb R^2} \frac{|u(x)-u(y)|^2}{|x-y|^{2+2s}} dxdy + \int_{B_1(...
- 109
1
vote
2
answers
624
views
Prove Liouville theorem without using mean value property
How can I prove the following Liouville theorem without using the mean value property?
If $u$ is harmonic on $\mathbb{R}^n$ and $\int_{\mathbb{R}^n}|\nabla u|^2 dx \leq C$ for some $C > 0$, then $...
- 217
2
votes
0
answers
86
views
Eigenvalues of the operator $A = -v'' + B(x) v$
How can I prove that for the eigenvalues of the operator $$A := -v'' + B(x) v$$ on $(0,L)$ with zero Dirichlet boundary condition it holds that
$$
\left| \lambda_n - \frac{\pi^2n^2}{L^2}\right| \le ||...
- 217
11
votes
1
answer
725
views
Chebyshev's other inequality
It is a simple fact, the granddaddy of correlation inequalities that if $f,g$ are monotone functions on $[0,1]$ then $$\int_0^1 f(x)g(x) dx \ge \int_0^1 f(x) dx \int_0^1 g(x) dx.$$ In their famous ...
- 827
10
votes
0
answers
345
views
Is this elliptic integral identity known?
Thinking about some physical problem, I came across the following identity:
$$\phi^2\Pi\left(-\phi,\frac{1}{\sqrt{2}}\right)+\phi^{-2}\Pi\left(\phi^{-1},\frac{1}{\sqrt{2}}\right)=\frac{\pi}{\sqrt{2}}+...
- 16.5k
1
vote
0
answers
690
views
What is the Jacobi-Anger expansion of $\sin^{[k]} (\theta) $?
Cross-post from MSE.
The Jacobi-Anger expansion gives expressions for functional iterates of trigonometric functions in terms of Bessel functions. For instance, we have $$\sin(z \sin(\theta)) = 2 \...
- 4,829
3
votes
0
answers
93
views
reciprocal spherical harmonic expansion
Let $f$ be a spherical function in $L^2(S^2)$ with spherical harmonic expansion $$f = \sum_{lm} f_{lm} \, Y_{lm} \;.$$
The function $g = 1/f$ is a spherical function. Assuming $g \in L^2(S^2)$, what ...
user114668
2
votes
0
answers
55
views
References for generalized confluent hypergeometric differential equation
According to Wolfram, a generalization of the confluent hypergeometric differential equation is given by:
$$y''+\left(\frac{2R}{x}+2F'+p\frac{H'}{H}-H'-\frac{H''}{H'}\right) y'+\left[\left(p\frac{H'}{...
user161091
0
votes
1
answer
86
views
Kolmogorov entropy of a subset of $L^1$
How can we estimate the Kolmogorov $\epsilon$-entropy
$$H_\epsilon (A,L^1(\mathbb R))$$
where
$
A = \{f:\mathbb R \to [0,K] \text{ s.t. $f \in L^1$ and has total variation $TV(f) \le M$}\}
$?
- 839