All Questions
Tagged with ca.classical-analysis-and-odes reference-request
323 questions
10
votes
1
answer
580
views
About certain infinite products with the property $f(a)=f(1/a)$
In the paper Transformations of infinite series, Bryden Cais gives the following transformations of infinite products
Theorem 4. If
$$
f(t) = \frac{\cosh(\pi t)-1}{\sinh(\pi t)}\frac{\cosh(2\pi t)+1}{...
- 5,624
10
votes
0
answers
345
views
Is this elliptic integral identity known?
Thinking about some physical problem, I came across the following identity:
$$\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
10
votes
0
answers
245
views
A slightly generalized existence and uniqueness theorem for integral equations (reference request)
I want to use the following statement without including the proof, which is completely straightforward but rather tedious:
Let $G_0:\mathbb R\times\mathbb R^m\to\mathbb R^m$, $\Theta_0:\mathbb R\...
- 61.9k
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
9
votes
2
answers
2k
views
Interpolation of Sobolev spaces
I know quite a bit about the abstract theory of Interpolation of Banach spaces. Today I had a colleague from Environmental sciences (who used to be in our Applied Maths department) come and ask me ...
- 18.7k
9
votes
1
answer
749
views
property of convex functions
I am able to give a proof to the following inequality for convex functions. Most likely this is well known, but I am unable to find a reference. I would appreciate if someone more knowledgeable in the ...
- 1,211
9
votes
2
answers
1k
views
Hilbert transforms of measures
Given a finite measure $\mu$ on the real line $\mathbb R$, one definition of its Hilbert transform is $(H\mu)(y) =\frac{1}{\pi}(PV)\int \frac{d\mu(x)}{x-y}$ which is known to exist almost everywhere ...
- 91
9
votes
1
answer
630
views
$C^{k,\alpha}$ diffeomorphisms and vector fields
This feels like something I should know, but I have a hard time finding a definite reference.
Let $M$ be a compact (Riemannian) manifold, $k\ge 1$ be an integer and $\alpha\in(0,1)$. When v is a $C^k$...
- 14.4k
8
votes
4
answers
4k
views
Approximation by exponential polynomials
Let $u(t) = \Sigma_{k=1}^n c_k e^{\lambda_k t} (c_k \in \mathbb C, \lambda_k \in \mathbb C) $ be an exponential polynomial of order $n$.
Define $E_n$ to be the collection of all exponential ...
- 1,795
8
votes
1
answer
5k
views
How to "globalize" the inverse function theorem?
Let $F: V \times W\rightarrow Z$, where $V,W,Z$ are finite-dimensional smooth (or analytic) manifolds and $F$ is smooth (or analytic). Assume that $\dim W=\dim Z$ and the usual inverse function ...
- 81
8
votes
1
answer
437
views
Function $\phi$ such that $f(\phi(x,y)) = f(x) + f(y)$
I have a continuous function $f:\mathbb{R}^n\to\mathbb{R}$, and I am looking for a continuous (or at least measurable) function $\phi:\mathbb{R}^{2n}\to\mathbb{R}^n$ such that $f(\phi(x,y))=f(x)+f(y)$....
- 3,065
8
votes
2
answers
2k
views
Divergent series expansion in Apéry's proof of the irrationality of $\zeta(2)$ and $\zeta(3)$
UPDATE. I am now making this a CW in the hope someone can improve the content of this question and/or correct the text.
This is a concise version of this math.SE question of mine. I've got an answer ...
8
votes
1
answer
338
views
A representation of the Bernoulli numbers
Let
\begin{equation}
\ell_{m,p}:=\sum_{j=1}^m\gamma_{m,j}\sigma_{p,j},
\end{equation}
where
\begin{equation}
\gamma_{m,j}:=\frac{2 (-1)^{j-1} }{j }\binom{2 m}{m+j}\Big/\binom{2 m}{m},\quad
\...
- 128k
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
7
votes
4
answers
3k
views
How does curvature change under perturbations of a Riemannian metric?
Let $M$ be a compact subset of $\mathbb R^2$ with smooth boundary, and let $g$ be a Riemannian metric on $M$. If $g'$ is another Riemannian metric which is "close" to $g$, then they should have ...
- 8,512
7
votes
2
answers
661
views
For a manual evaluation of a definite integral
I note that Mathematica could yield the identity
$$\int_0^1\frac{\log(1+x^2(x-1)/2)}{x^2(x-1)}dx=\frac{\pi(\pi-4)-12\log^22+24\log2}{16}.\tag{1}\label{1}$$
But I don't know how Mathematica got this.
...
- 15.6k
7
votes
2
answers
2k
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Cantor Sets Inside Cantor Sets
(Or: "I heard you liked Cantor Sets...")
I'm working on a student project, and the following construction came up very naturally: If $C$ is the usual Cantor Set, build a countable union of copies of ...
- 4,586
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
7
votes
4
answers
4k
views
Estimating the probability that one Poisson RV is larger than another
Let $X$ and $Y$ be Poisson random variables with means $\lambda$ and $1$, respectively. The difference of $X$ and $Y$ is a Skellam random variable, with probability density function
$$\mathbb P(X - Y ...
- 8,512
7
votes
1
answer
771
views
Famous but unavailable paper of Jan Boman
The following paper is well known, but hard to find:
J. Boman, $L^p$-estimates for very strongly elliptic systems, Report 29, Department of Mathematics, University of Stockholm, 1982.
In this paper ...
- 28k
7
votes
2
answers
880
views
Survey of the history of calculus?
Boyer 1939 is a nice readable survey of the history of the calculus, but it's showing its age. Discussing the notion of instantaneous velocity, he has:
Mathematics knows no minimum interval of ...
user21349
7
votes
2
answers
521
views
How large (small) can be the measure of a set where a polynomial takes small values ?
A $n$-th degree polynomial has precisely $n$ roots. So it is natural to ask the question
how large ( and small) can be the measure of a set where a polynomial takes small values ?
This, and other ...
- 1,795
7
votes
1
answer
488
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 ...
- 6,367
7
votes
1
answer
494
views
Carleson's lectures at UCLA
It seems that Professor Lennart Carleson gave a series of Lectures at UCLA in 1985. For example, one could find several mentions about these lectures in the book by Garnett & Marshal (see for ...
- 747
7
votes
1
answer
488
views
Ref. request: Additive probability measure on $\mathcal P({\bf N})$ supplies subset of $\mathbf R$ without Baire property
ZFC proves, among the other things, the existence of a (finitely) additive probability measure $\theta: \mathcal P(\mathbf N) \to \mathbf R$ on the power set of $\mathbf N$ such that $\theta(X) = 0$ ...
- 10.5k
7
votes
2
answers
697
views
Local positivity of solutions to linear differential inequalities (Chaplygin's theorem)
According to the entry "Differential inequality" of the Encyclopedia of Mathematics
http://www.encyclopediaofmath.org/index.php/Differential_inequality
the following result is due to Chaplygin (1919)...
- 1,371
7
votes
2
answers
1k
views
For what spaces is the Hardy-Littlewood maximal operator of strong type $(p,p)$ if and only if $p > p_0 > 1$?
(This is essentially a continuation of my previous question, here.)
Let $(X,d,\mu)$ be a metric measure space, i.e. $\mu$ is a Borel measure on the metric space $(X,d)$. Further assume (though you ...
user14166
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
7
votes
3
answers
2k
views
Collections of examples and counterexamples in (real, complex, functional) analysis, ODEs and PDEs
What books collect examples and counterexamples (or also "solved exercises", for some suitable definition of "exercise") in
real analysis,
complex analysis,
functional analysis,
ODEs,
PDEs?
The ...
user60665
7
votes
2
answers
948
views
Uniform variant of Stirling's approximation
Stirling's formula is usually stated in the form $\log \Gamma(s) = (s-\frac12) \log{s} - s + \log\sqrt{2\pi} + E(s)$, where
$E(s) = c_1/s + c_2/s^2 + \dots + O(s^{-K})$ for certain absolute ...
- 4,671
7
votes
2
answers
593
views
Dependence of a solution of a linear ODE on parameter
Is the following theorem known, or can be easily derived from known results?
Consider the differential equation
$$w''-kz^{-1}w'=(\lambda+\phi(z))w,$$
where $k>0$ is fixed, $\lambda$ is a large (...
- 91.8k
7
votes
0
answers
356
views
Is this proof of Basel identity known?
Today, to divert myself, I tried to find a new proof of Basel identity $\boxed{\sum_{j=1}^\infty\frac{1}{j^2}=\frac{\pi^2}{6}}$. I came up with the following, which essentially interprets the identity ...
- 3,146
6
votes
1
answer
796
views
A Poincaré-like inequality
Is it true that for some real $K>0$ and all real $u\in C_0^\infty((0,1))$ we have
$$\int_0^1 (u'(x)^2+u(x)^2)\,dx\,\int_0^1 u(x)^2\,dx
\le K\Big(\int_0^1 x\,u'(x)^2\,dx\Big)^2\text{ ?}$$
- 128k
6
votes
2
answers
516
views
Texts about Dwork's work
I want to ask about references to papers, that probably exist, which explain the articles of Bernard Dwork starting from "The rationality of the zeta function of an algebraic variety" to "On the ...
- 69
6
votes
2
answers
513
views
Need a reference for a trigonometric inequality
In my old high school notebook (20 years ago), the following inequality appears with its proof:
$$1+\cos x + \frac{1}{2}\cos 2x + \cdots + \frac{1}{n}\cos nx \geq 0$$
for any real $x$ and positive ...
- 335
6
votes
2
answers
921
views
Has the "partial Sophomore's Dream function" been studied before?
We can consider the generalized Harmonic numbers $$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). $$
...
- 4,829
6
votes
2
answers
295
views
Which result guarantees convergence of solution of an ODE to a set of non-compact, non-isolated equilibrium?
Consider a continuous ODE,
$$\dot x = f(x), f \in C^1$$
$\dot x = 0$ for all $x \in K \subset \mathbb{R}^n$, where we assume that $K$ is a closed but unbounded set of non-isolated equilibrium. For ...
- 205
6
votes
2
answers
390
views
Solution uniqueness for ODE
I have a vectorial, non-linear second order ordinary differential equation
$$Z''=f(Z)$$
for which I have a solution $Z^0$ on $[0,1]$ with $Z^0_i(0)=0$ and $Z^0_i(1)=1$. I would like to know under ...
- 14.4k
6
votes
3
answers
423
views
Infinite electrical networks and possible connections with LERW
I've been exposed to various problems involving infinite circuits but never seen an extensive treatment on the subject. The main problem I am referring to is
Given a lattice L, we turn it into a ...
- 85.6k
6
votes
1
answer
584
views
Integral representation of $\frac{355}{113}-\pi$? [duplicate]
It is well known that
$$
\frac{22}{7} - \pi = \int_0^1 (x-x^2)^4 \frac{dx}{1+x^2},
$$
Given that $\frac{355}{113}$ is an excellent approximation of $\pi$, is there any known integral representation of ...
- 1,608
6
votes
1
answer
309
views
Survey paper on isoperimetry
I am searching for a comprehensive survey article (or more different articles) on the subject of isoperimetric problems from ancient Greece to modern mathematical physics. Could you point out some ...
user60665
6
votes
2
answers
1k
views
Existence of a measure-preserving bijection
Let $f, g \, \colon \mathbb{R}^n \rightarrow \mathbb{R}$ be two Borel-measurable functions such that $f$ is non negative and
g is radially symmetric,
the function $ (0, \infty )\ni t \mapsto g (tx)$ ...
- 161
6
votes
1
answer
2k
views
Approximation by analytic functions
Dear all.
Let
$$
f(x) = \sum_{k \in \mathbb{Z}} \hat{f}(k) \exp(2\pi \mathrm{i} kx)
$$
be a function given by usual fourier series.
Since my original question hasn't got any answer yet, and I ...
- 3,343
6
votes
1
answer
134
views
Multi-parameter stationary phase asymptotic expansion
I am looking for an asymptotic expansion of the oscillatory integral of the form
$$\int_{\mathbb{R}^n}f(x)\exp(i(\lambda_1\phi_1(x)+\dots+\lambda_k\phi_k(x))dx,$$
as $\lambda_i\to \infty$ ...
- 1,692
6
votes
0
answers
227
views
Origins of the generalized shift operator exp(t*g(z)d/dz)
Charles Graves in the 1850s investigated iterated operators of the form $g(x) \frac {d}{dx}$ (see page 13 in The Theory of Linear Operators ... (Principia Press, 1936) by Harold T. Davis). Graves ...
- 10.5k
6
votes
0
answers
200
views
Spectral theory for Dirac Laplacian on a funnel
I would like to study the spectral theory of the Dirac Laplacian on a non-compact quotient of the hyperbolic plane by a discrete group (I am particularly interested in the simple case where the ...
- 313
6
votes
0
answers
373
views
Paper by Y. Amice
Can anybody help me find the following:
Y. Amice, Duals, Proceedings of a Conference on p-adic Analysis, Nijmegan (1978), 1-15, Math. Institut Katholische Univ. , 1978
- 61
5
votes
2
answers
1k
views
real analyticity, Fourier coefficients [duplicate]
Question. Suppose $f$ is periodic in $[0,2\pi]$. What conditions on the Fourier coefficients of $f$ would guarantee real analyticity of $f$? Please provide me with a reference.
- 43.2k
5
votes
2
answers
952
views
Good references for analytic solutions to nonlinear ordinary differential equations?
I am faced with a non-autonomous initial value problem for a function $x:[0,\infty) \to \mathbb{R}^2$ of the form
$$ x'(t) = f(t,x(t)) $$
for $f: [0,\infty) \times \mathbb{R}^2 \to \mathbb{R}^2$ with ...
- 33.3k
5
votes
2
answers
644
views
On the derivative of the Bernstein polynomial
$\newcommand\Z{\Bbb Z}\newcommand\De{\Delta}$For a natural $n$ and a function $g\colon\Z\to\Bbb R$, let $B_n g$ be the corresponding Bernstein polynomial, so that
$$(B_n g)(x)=\sum_{k\in\Z} g(k)\binom ...
- 128k