All Questions
Tagged with ca.classical-analysis-and-odes reference-request
323 questions
4
votes
1
answer
162
views
Definite integral of power of sine ratio
I stumbled on the following rather appealing trigonometric definite integral,
\begin{equation}
\int_0^y \left(\frac{\sin x}{\sin (y-x)}\right)^a \mathrm{d}x = \pi \frac{\sin(ya)}{\sin(\pi a)}
\end{...
- 3,927
6
votes
1
answer
584
views
Integral representation of $\frac{355}{113}-\pi$? [duplicate]
It is well known that
$$
\frac{22}{7} - \pi = \int_0^1 (x-x^2)^4 \frac{dx}{1+x^2},
$$
Given that $\frac{355}{113}$ is an excellent approximation of $\pi$, is there any known integral representation of ...
- 1,608
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: $$\pi=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}-\...
- 2,661
3
votes
1
answer
268
views
References for $\int_{-\infty}^\infty\binom{1}{t}^3\,\mathrm dt=\frac{3}{2}+\frac{6}{\pi^2}$ and related integrals?
In user dxdydz's answer to the question "Unexpected appearances of $\pi^{2}/6$", the identity $$\int_{-\infty}^\infty\binom{1}{t}^3\,\mathrm dt=\frac{3}{2}+\frac{6}{\pi^2}$$ is mentioned.
...
- 4,829
9
votes
2
answers
1k
views
A tricky integral to evaluate
I came across this integral in some work. So, I would like to ask:
QUESTION. Can you evaluate this integral with proofs?
$$\int_0^1\frac{\log x\cdot\log(x+2)}{x+1}\,dx.$$
- 43.2k
2
votes
1
answer
317
views
Recommendation for books on boundary-value problems that include perturbed boundaries and many solved problems
I am looking for a book or resource that contains applied math analytical methods and a lot of solved problems in Boundary-Value Problems for second-order PDEs, and if it could be related to wave-...
- 183
27
votes
2
answers
1k
views
Rademacher theorem
If $f:\mathbb{R}^n\to\mathbb{R}^m$ is of class $C^1$ and $\operatorname{rank} Df(x_o)=k$, then clearly $\operatorname{rank} Df\geq k$ in a neighborhood of $x_o$. It is not particularly difficult to ...
- 28k
3
votes
1
answer
488
views
Strict inequality in decoupling inequality
I am working on the decoupling inequality developed by Bourgain and Demeter: https://arxiv.org/abs/1604.06032.
Is there an example where we have strict inequality in Theorem 1.1, say in the case $n=2$ ...
- 751
59
votes
7
answers
29k
views
Learning roadmap for harmonic analysis
In short, I am interested to know of the various approaches one could take to learn modern harmonic analysis in depth. However, the question deserves additional details. Currently, I am reading Loukas ...
3
votes
0
answers
141
views
Direct analytic proof of positive definiteness of stable characteristic functions
Is there a direct analytic proof that the function
$$
f ( t ) =
\exp\left(-|t|^\alpha \big[ \lambda + i \theta \operatorname{sign} ( t ) \big]\right),
\qquad
\lambda > 0, \quad
|\theta| < \...
- 620
18
votes
11
answers
5k
views
Applications of measure, integration and Banach spaces to combinatorics
I'm going to be teaching a Master's level analysis course (measure theory, Lebesgue integration, Banach and Hilbert spaces, and if there's time, some spectral or PDE stuff) in the fall. My problem is ...
- 1,665
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
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
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
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
13
votes
5
answers
3k
views
Reference request: Oldest calculus, real analysis books with exercises?
Per the title, what are some of the oldest calculus, real analysis books out there with exercises? Maybe there are some hidden gems from before the 20th century out there.
Edit. Unsolved exercises ...
3
votes
0
answers
101
views
A special type of differential equations
Working in optimal control of PDEs, I came across a type of evolution problem that has instead of an initial condition a link between the initial state and the final state.
Here is a simplified ...
- 1,759
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
11
votes
1
answer
2k
views
Reference request: proof of Ramanujan's Cos/Cosh Identity
The Ramanujan Cos/Cosh Identity, as stated here, is
$$\left[1+2\sum_{n=1}^{\infty}\frac{\cos n\theta}{\cosh n\pi}\right]^{-2}+
\left[1+2\sum_{n=1}^{\infty}\frac{\cosh n\theta}{\cosh n\pi}\right]^{-2}=...
2
votes
1
answer
473
views
An identity for the elliptic theta function
For real $s>0$, let
$$S(s):=\sum_{n=-\infty}^\infty e^{-n^2/(2s^2)}
=\vartheta _3\left(0,e^{-1/(2 s^2)}\right),$$
where $\vartheta$ is the elliptic theta function.
Plotting suggests that the ...
- 128k
3
votes
1
answer
613
views
Searching for the proof of a certain claim in Arnold's ODE book from 1992
I was reading today the book of Stephen Wiggins called "Global Bifurcations and Chaos" (the 1988 edition).
On pages 12-13 he writes the following:
Consider the following ordinary ...
- 1,594
1
vote
0
answers
100
views
N-wave solution of conservation law $u_t + (u - u^2)_x = 0$
How can we compute the "N-wave" source-solution of the conservation law
$$u_t + (u - u^2)_x = 0, $$
that is, the entropy solution of this conservation law with the initial data $u(0,\cdot) = ...
- 839
4
votes
1
answer
195
views
Reference request: Rigorously solving ODEs using divergent asymptotic series
In my research I have come across a divergent asymptotic series $\sum_{n =0}^\infty a_n f_n(x)$ that formally solves a certain fairly simple nonlinear second-order ODE but does not seem to correspond ...
- 775
3
votes
2
answers
552
views
Second order differential equation with oscillating behavior
I consider a differential equation $y^{\prime \prime} (x) + V(x) y(x) = 0$ in the interval $[0,\infty)$, where $C_1 \leq V(x) \leq C_2$ for all $x \in [0,\infty)$ for some constants $C_2 > C_1 >...
- 5,562
3
votes
1
answer
175
views
Is there a classical textbook/reference on numerical discretization schemes?
I found that it is relatively easy to find a book that discusses Euler discretization or Runge-Kutta discretization, but I am not aware of one that is well-known and/or common knowledge (i.e., field-...
- 479
5
votes
0
answers
109
views
Asymptotics in the Chebyshev-type optimization problem
Let $g(x)\colon [-2,2]\to \mathbb{R}$ be a continuous function. Let $f_n(x)$ be a polynomial of degree $n$ such that $\log |f_n(x)|\leqslant ng(x)$ for all $x\in [-2,2]$. Then the maximal possible ...
- 109k
2
votes
2
answers
109
views
Regular Lagrangian flow for explicit ODE with discontinuous right-hand side
Consider the problem
$$(\star) \quad \begin{cases} \frac{d}{dt} X(t,x) = \begin{cases} - 1 & \text{ if } X(t,x) >0, \\
1 & \text{ if } X(t,x) < 0 \end{cases}, &t \in [0,T],\\
X(0,x) ...
- 839
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
15
votes
5
answers
12k
views
Beginners text on calculus of variations
I want to begin learning Calculus of Variations. What texts would MathOverflow recommend? Amazon shows up quite a few options.
I work on Machine Learning, and that where I intend to apply this.
...
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 ...
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
2
votes
0
answers
85
views
Multipole expansion
In Simon's book Harmonic Analysis, example 3.5.12 shows:
Fix $x \in \mathbb{R}^{\nu}$ (with $\left.\nu \geq 3\right), x \neq 0$ and define $f$ on $\{y:| y|<| x |\}$ by
$$
f(y)=|x-y|^{-(\nu-2)}.
$$
...
- 21
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
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?
$$\int_0^{\frac{\pi}2}\frac{d\theta}{\...
- 43.2k
7
votes
5
answers
1k
views
Generalizations of the Euler–Maclaurin Summation Formula
I'm using the Euler–Maclaurin formula in a research project I'm working on. While brilliant, the elementary proof found in Apostol - An Elementary View of Euler's Summation Formula does not give me ...
- 3,574
1
vote
2
answers
106
views
Green function of symmetric stable process in dimension 1 and 2
Are the results in this paper on the Green function of a symmetric stable process available also in space dimension $d =1$ and $d=2$? The main theorems here are stated only for $d \ge 3$.
- 99
2
votes
1
answer
329
views
Is $g(v)=\mathbb{E}[f(v+W)]$ a differentiable function of $v$ when $f$ is continuous and $W$ is multivariate normal?
Suppose $f$ is a continuous function on $\mathbb{R}^n$, and $W$ has a multivariate normal distribution on $\mathbb{R}^n$. If the expectation
$$g(v)=\mathbb{E}[f(v+W)]$$
is defined for all $v \in \...
user44143
11
votes
1
answer
566
views
Integral representation of product of two Whittaker functions
Does anyone know anything about the following formula involving special functions:
$$\begin{multline*}
W_{\kappa,\mu}(z)W_{\lambda,\mu}(w)=\frac{e^{-(z+w)/2}(zw)^{\mu+1/2}}{\Gamma(1-\kappa-\lambda)}\...
- 373
26
votes
9
answers
14k
views
Graduate ODE textbook
Suppose that a hypothetical math grad student was pretty comfortable with first-year real variables and algebra, and had even studied some other things (algebraic geometry, Riemannian geometry, ...
2
votes
0
answers
64
views
Scaling limit of ODE with double-well potential
Let us consider the ODE
$$
\frac{d}{dt}x_\epsilon(t) = -\frac{1}{\epsilon} W'(x_\epsilon(t))
$$
where $W$ is a double-well potential -- for example, $W(x)=\frac{1}{4}(x^2-1)^2$ so that the ODE reads
$$...
- 839
2
votes
1
answer
163
views
Approximation of $C^1$-smooth equivariant maps by infinitely smooth ones
Let $M,N$ be smooth closed manifolds acted by a finite group $G$. Let $f\colon M\to N$ be a $C^1$-smooth $G$-equivariant map.
Is it true that for any $\varepsilon>0$ there exists a $C^\infty$-...
- 21.8k
1
vote
0
answers
138
views
An Elementary Inequality [closed]
A friend of mine found in the internet the following exercise:
Let $a, b, c \geq 0$ with $a + b + c = 1$. Show that
$$
\sqrt{a + b^2} + \sqrt{b + c^2} + \sqrt{c + a^2} \geq 2,
$$
where equality is ...
1
vote
1
answer
119
views
Estimating two dimensional theta function
My feeling is that this should be written somewhere but I don't know what to search for.
Let $Q(x,y)$ be a binary quadratic form over $\mathbb{C}$, with $\operatorname{Re}(Q)$ positive definite. Then ...
- 2,178
25
votes
3
answers
7k
views
Analysis from a categorical perspective
I have not studied category theory in extreme depth, so perhaps this question is a little naive, but I have always wondered if analysis could be taught naturally using categories. I ask this because ...
- 5,759
12
votes
4
answers
2k
views
History of ODE and PDE reference request
Is there any reference (book or articles) which made the history (up to the modern times) and the conceptual development of Ordinary Differential Equations and Partial Differential Equations? It will ...
- 1,759
1
vote
1
answer
149
views
Completeness of asymptotically Euclidean manifolds
Say that you place an asymptotically Euclidean metric on $\mathbb{R}^3,$ e.g. $\mathbb{R}^3$ is endowed Riemannian metric $g$ such that $\text{supp}(g^{ij}-\delta^{ij})\subseteq\{|x|\leq R\}$ for some ...
- 13
2
votes
1
answer
231
views
Solution of Riccati system of ODEs
We have following equation:
$$
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
4
votes
1
answer
2k
views
How to mathematically characterize a feedback loop in ODEs?
I have a biological system that exhibits a feedback type of behavior. The diagram is a schematic of the system of ODEs. In this system, the total amount of $x_1, x_2, x_3$ is conserved; however, there ...
- 513
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
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