All Questions
Tagged with ca.classical-analysis-and-odes reference-request
323 questions
14
votes
3
answers
664
views
(Sharp) inequality for Beta function
I am trying to prove the following inequality concerning the Beta Function:
$$
\alpha x^\alpha B(\alpha, x\alpha) \geq 1 \quad \forall 0 < \alpha \leq 1, \ x > 0,
$$
where as usual $B(a,b) = \...
- 315
1
vote
2
answers
859
views
Linear independence of exponential functions: a reference
Is there a publication containing this obvious fact: For any real $T>0$, any natural $n$, any complex $c_1,\dots,c_n$, and any distinct complex $z_1,\dots,z_n$ such that $\sum_1^n c_k e^{tz_k}=0$ ...
- 128k
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
0
answers
74
views
Calculation of a multi-dimensional Fourier transform
I am interested in the following multi-dimensional Fourier transform:
$$
\int_{\mathbb{R}^{p}} \mathrm{d} \vec{r}_{\parallel}\int_{\mathbb{R}^{q}} \mathrm{d} \vec{r}_\perp \, e^{-\mathrm{i}\, \vec{p}...
- 373
24
votes
2
answers
2k
views
Reference for exponential Vandermonde determinant identity
I recently stumbled upon the following identity, valid for any real numbers $\alpha_1,\dots,\alpha_n$ and $\lambda_{n1} \leq \dots \leq \lambda_{nn}$:
$$ \mathrm{det}( e^{\alpha_i \lambda_{nj}} )_{1 \...
- 114k
3
votes
1
answer
202
views
Stability of nonsmooth, Lipschitz continuous, autonomous system of differential equations
Consider the following autonomous system of differential equations:
$$\frac{\mathrm d\mathbf x}{\mathrm dt} = \mathbf v(\mathbf x)$$
where $\mathbf x, \mathbf v \in \mathbb R^n$. Assume that $\...
- 884
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 $$\sin(z \sin(\theta)) = 2 \...
- 4,829
7
votes
3
answers
2k
views
Collections of examples and counterexamples in (real, complex, functional) analysis, ODEs and PDEs
What books collect examples and counterexamples (or also "solved exercises", for some suitable definition of "exercise") in
real analysis,
complex analysis,
functional analysis,
ODEs,
PDEs?
The ...
user60665
5
votes
1
answer
297
views
A scaled fractional ''Sobolev inequality''
Does a fractional interpolation inequality similar to $$
\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
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
$$
f(t) = \frac{\cosh(\pi t)-1}{\sinh(\pi t)}\frac{\cosh(2\pi t)+1}{...
- 5,624
11
votes
2
answers
1k
views
Two divergent series conspiring?
Consider the sequence $a_n=2^{2n}\binom{2n}n^{-1}$. Stirling's approximation shows that $a_n\sim \sqrt{\pi n}$, thus
$$\sum_{n\geq0}\frac{\pi}{2a_n}\qquad \text{and} \qquad
\sum_{n\geq0}\frac{a_n}{2n+...
- 43.2k
5
votes
1
answer
1k
views
Continuous dependence on initial parameters of an ODE for non-Lipschitz functions?
For ODEs, the standard theorem of continuous dependence of initial parameters deals only with functions that are Lipschitz. Do there exist more general results holding for non-Lipschitz functions? If ...
- 215
1
vote
0
answers
47
views
Scaling limit of transport equation with double-well potential
Let us consider the transport PDE
$$
u^\epsilon_t + u^\epsilon_x= -\frac{1}{\epsilon} W'(u^\epsilon)
$$
where $W$ is a double-well potential -- for example, $W(x)=\frac{1}{4}(x^2-1)^2$ so that the PDE ...
- 839
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
809
views
Square-integrable unbounded function
In R.D. Richtmyer, Principles of Advanced Mathematical Physics, p.85 an example is given of a continuous and square-integrable on $\bf{R}$ function, which is not bounded at infinity:
$$f(x)=x^2\exp{(−...
- 16.5k
3
votes
1
answer
183
views
On integral representation of Whittaker $W$ functions
According to NIST, the integral representation of Whittaker $W$ functions
$$
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
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
10
votes
0
answers
345
views
Is this elliptic integral identity known?
Thinking about some physical problem, I came across the following identity:
$$\phi^2\Pi\left(-\phi,\frac{1}{\sqrt{2}}\right)+\phi^{-2}\Pi\left(\phi^{-1},\frac{1}{\sqrt{2}}\right)=\frac{\pi}{\sqrt{2}}+...
- 16.5k
1
vote
1
answer
426
views
$L^p$ compactness for a sequence of functions from compactness of cut-off
Fix $p \in [1,\infty)$. Let $f_n:[a,b] \to \mathbb R$, $n \in \mathbb N$, be a sequence of $C^1$ functions. For every fixed $m\in \mathbb N^*$, suppose that the sequence of functions $$\{f_{n}\psi_m(...
- 161
5
votes
1
answer
4k
views
Rigorous multivariate differentiation of integral with moving boundaries (Leibniz integral rule)
The Leibniz integral rule, in its multivariate form, deals with differentiation of the following sort:
$$ \frac{\partial}{\partial t} \int_{D(t)} F({\bf x}, t) \, d{\bf x} \, , \qquad D(t)\in \mathbb{...
- 3,574
7
votes
1
answer
771
views
Famous but unavailable paper of Jan Boman
The following paper is well known, but hard to find:
J. Boman, $L^p$-estimates for very strongly elliptic systems, Report 29, Department of Mathematics, University of Stockholm, 1982.
In this paper ...
- 28k
1
vote
1
answer
285
views
Why are the two ODE solutions linearly independent?
I notice that some second-order ODEs can be related to the triconfluent Heun's equation
$$u''(z)-(3z^2+\gamma)u'(z)+(\alpha-(3-\beta) z)u(z)=0.$$
And people usually say the general solution of the ...
- 155
2
votes
0
answers
169
views
Functions whose Fourier coefficients satisfy $ \sum_{k=1}^\infty |c_k| < 1 $?
Let $f:(0,1) \to \mathbb R$ be a function that can be written as $$f(x) = \sum_{k=1}^\infty c_k \phi_k(x),$$ where $\phi_k(x) = \cos(\pi k x)$. What is the minimal assumption required on $f$ to ...
- 839
6
votes
1
answer
134
views
Multi-parameter stationary phase asymptotic expansion
I am looking for an asymptotic expansion of the oscillatory integral of the form
$$\int_{\mathbb{R}^n}f(x)\exp(i(\lambda_1\phi_1(x)+\dots+\lambda_k\phi_k(x))dx,$$
as $\lambda_i\to \infty$ ...
- 1,692
1
vote
1
answer
196
views
Taylor expansion of Modified Mathieu functions
Do we know the Taylor expansion at $0$ of the radial Mathieu functions $(\mathsf{Mc}_n^{(j)}(\,\cdot\,, \sqrt{q}))_{n \ge 0}$ and $(\mathsf{Ms}_n^{(j)}(\,\cdot\,, \sqrt{q}))_{n \ge 1}$, for $q \in \...
- 310
1
vote
1
answer
387
views
$L^p$ compactness for a sequence of functions from compactness of product with cut-off
Fix $p \in [1,\infty)$. Let $f_n:[a,b] \to \mathbb R$, $n \in \mathbb N$, be a sequence of $C^1$ functions. For every fixed $m\in \mathbb N^*$, suppose that the sequence of functions $$\{f_{n}\psi_m(...
- 161
3
votes
2
answers
322
views
Hausdorff dimension of the graph of the sum of two continuous functions
How can one prove the following result on the Hausdorff dimension of the graph of the sum of two continuous functions:
Let $f,g:[0,1] \to \mathbb R$ be two continuous functions. Suppose that $$\...
user139844
19
votes
2
answers
2k
views
Integral representation of higher order derivatives
I'm quite curious about the following phenomena, that still puzzle me although I have a proof, and I'd be really glad if someone may shred some light, showing an interpretation or a generalization. I ...
- 60.5k
12
votes
1
answer
1k
views
Proof of Green's formula for rectifiable Jordan curves
$\newcommand{\Ga}{\Gamma}$
I am trying to find a proof of Green's formula for rectifiable Jordan curves $\Ga$ (and the corresponding interior regions $R$). There is a proof by Ridder, followed by ...
- 128k
18
votes
2
answers
5k
views
Nonvanishing of Jacobians implies global injectivity?
I am interested in obtaining injectivity of a $C^1$ map from the nonvanishing minors of its Jacobian matrix. Here is a brief history of the topic.
In 1953, Samuelson asked the following:
If the ...
- 981
10
votes
1
answer
833
views
This is not a dyadic cosine-product
The double-angle formula, $\sin2x=2\sin x\cos x$, turns the scary-looking integral
$$\int_0^{\infty}dz\prod_{k=1}^{\infty}\cos\frac{z}{2^k}$$
into fun once you realize $\prod_k\cos\frac{z}{2^k}=\frac{\...
- 43.2k
1
vote
2
answers
424
views
Regular Lagrangian flow for the problem $\frac{d}{dt} X(t,x) = \chi_{\{x>0\}}(X(t,x))$
Consider the problem
$$(\star) \quad \begin{cases} \frac{d}{dt} X(t,x) = \chi_{\{x>0\}}(X(t,x)), &t \in [0,T],\\
X(0,x) = x, &x \in \mathbb R
\end{cases}
$$
where $\chi$ denotes the ...
- 839
1
vote
0
answers
151
views
Fourier transforms exhibiting symmetries about their critical points
Upon looking at the graphs of various Fourier sine and cosine transforms (ones without Dirac deltas in their domain) I've noticed a pattern that is probably already known, but that I thought would be ...
- 113
0
votes
0
answers
230
views
Ramanujan's infinite sum for pi
Ramanujan's famous pi formula states that
\begin{equation}
\frac{1}{\pi}=\frac{2\sqrt{2}}{99^2}\sum_{k=0}^{\infty}\frac{(4k)!}{k!^4}\frac{26390k+1103}{396^{4k}}
\end{equation}
How can one prove this?...
- 45
2
votes
0
answers
86
views
Eigenvalues of the operator $A = -v'' + B(x) v$
How can I prove that for the eigenvalues of the operator $$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
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
3
votes
1
answer
301
views
Reference request: Oldest books on series with unsolved exercises?
Per the title, what are some of the oldest books on series out there with unsolved exercises? Maybe there are some hidden gems from before the 20th century out there.
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
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
3
votes
1
answer
172
views
Translation to English of Brillouin's analysis of Airy's integral
I am trying to read the following paper by Leon Brillouin (the part on page 16 onwards):
Léon Brillouin, Sur une méthode de calcul approchée de certaines intégrales dite méthode du col, Annales ...
- 1,594
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
4
votes
0
answers
84
views
Elementary functions (growing faster than exponential ones) with elementary Legendre–Fenchel transforms
Let $F$ be the set of all convex functions $f\colon[0,\infty)\to[0,\infty)$ with $f(0)=0=f'_+(0)$ and $f_+(\infty-)=\infty$, where $f'_+$ is the right derivative of $f$. For any function $f\in F$, its ...
- 128k
3
votes
1
answer
224
views
Flow of ODE with monotone source
Let $\Phi$ be the flow (defined as in page 6 of this paper) of the ODE
$$\begin{cases}
\frac{d}{dt}\Phi(x,t) = f(\Phi(x,t),t) \quad t >0 \\
\Phi(x,0) = x \quad x \in \mathbb{R}.
\end{cases}$$
Is ...
user124345
3
votes
1
answer
142
views
Non-trivial examples of regular Lagrangian flow in BV case
What is a concrete example of BV vector field $v$ with $\mathrm{div}\, v = 0$ that makes Ambrosio's theory of regular Lagrangian flow relevant?
With concrete I mean that we can compute the flow ...
- 839
5
votes
0
answers
163
views
Minimizing total variation
Let $f:\mathbb{R}\to \mathbb{R}$ be a function of bounded variation. Define $\overline{f}(x)$ by
$$\overline{f}(x) = \limsup_{\mu(I)\to 0} \frac{1}{\mu(I)} \int_I f(x) dx,$$ where $I$ ranges over ...
- 20.2k
2
votes
2
answers
317
views
Concrete example of BV function $u:\mathbb{R}^2 \to \mathbb{R}$ with singular derivative
What are examples of two BV functions $u:\mathbb{R}^2 \to \mathbb{R}$ with singular derivative?
More precisely, I'd like to see an example (and a plot using Mathematica or Matlab) of
a function
$$...
- 839
13
votes
1
answer
661
views
Poincaré on analytic dependence on parameters of solutions of linear differential equations
There is the following important General Principle: if a parameter enters
in a linear differential equation additively, for example
$$\frac{d^2w}{dx^2}+(q(x)+\lambda)w=0,$$
where the parameter is $\...
- 91.8k
1
vote
0
answers
218
views
Asymptotic inverses and de Bruijn conjugates (etc.) for complex-valued functions
I recently got my hands on a copy of Regular Variation by Bingham, Goldie, and Teugels ("BGT"), and it's been an absolute revelation for my research. The thing is, my current work centers ...
- 1,284
3
votes
0
answers
93
views
reciprocal spherical harmonic expansion
Let $f$ be a spherical function in $L^2(S^2)$ with spherical harmonic expansion $$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