30 votes

Intuitive explanation why "shadow operator" $\frac D{e^D-1}$ connects logarithms with trigonometric functions?

This is basically a lightly transformed version of Euler's cotangent partial fraction expansion $$ \pi \cot(\pi z) = \frac{1}{z} + \sum_{n=1}^\infty \frac{1}{z-n} + \frac{1}{z+n}$$ (the log derivative ...
Terry Tao's user avatar
  • 108k
15 votes

Do distance functionals separate probability measures?

No. Suppose that $\Omega$ consists of four points arranged in a square, where adjacent points have distance 1 between them and opposite points have distance 2. Specifically, if the points are labeled ...
George Lowther's user avatar
12 votes
Accepted

Representation of the Dirac delta function

As usual in such examples, there is no need to integrate against a test function. One can simply use the fact that if a sequence (or net) of distributions converges in the distributional sense, then ...
klempner's user avatar
  • 206
11 votes

Intuitive explanation why "shadow operator" $\frac D{e^D-1}$ connects logarithms with trigonometric functions?

The op $$T_x = \frac{D_x}{e^{D_x}-1} = e^{b.D_x},$$ where $(b.)^n = b_n$ are the Bernoulli numbers, is (mod signs) often referred to as the Todd operator (maybe originally given that name by ...
Tom Copeland's user avatar
  • 9,937
11 votes
Accepted

On an asymptotic integral decay

I'll change $[-1,0]$ to $[0,1]$ (so $t\to -t$), which seems easier on the brain. $f(t)=e^{-1/t}$ is a counterexample: For $0<t<1/\sqrt{\lambda}$, we have $f\le e^{-\sqrt{\lambda}}$, so this part ...
Christian Remling's user avatar
10 votes

volume over a hypercube, over simplex: twist by Euler numbers

This is only a partial answer. The Beukers-Kolk-Calabi change of variables $$x_1=\frac{\sin{u_1}}{\cos{u_2}},\;\;x_2=\frac{\sin{u_2}}{\cos{u_3}},\ldots, \;x_{n-1}=\frac{\sin{u_{n-1}}}{\cos{u_n}},\;\;...
Zurab Silagadze's user avatar
7 votes
Accepted

Asymptotic behavior of integral with gamma functions

If I just insert the large-$z$ asymptotics of $\Gamma(z)\rightarrow \sqrt{2 \pi } e^{-z} z^{z-\frac{1}{2}}$, and take $z>1/2$ real for simplicity, I find $$5^z\,F(z)\rightarrow \int_0^\infty \left(...
Carlo Beenakker's user avatar
7 votes

Do distance functionals separate probability measures?

On the positive side, the answer is affirmative if $\Omega$ is the unit interval $[0,1]$ with its standard distance. In this case $\phi_\mu$ is a convex $1$-Lipschitz function (in fact, it is also ...
Pietro Majer's user avatar
  • 56.5k
6 votes

Why Mellin transform is omitting infinite terms?

This question has not been well received, but I am intrigued by the delta function Mellin transform and would like to respond. I found this 2004 paper by Norbert Südland and Gerd Baumann instructive. ...
Carlo Beenakker's user avatar
6 votes
Accepted

Injectivity of a class of integral operators

If the support of $\mu$ is contained in $[0, b]$ for some $b \in (0,1)$, then $T_\mu f = 0$ for any $f$ that is $0$ on $[0,b]$, so it is not injective. EDIT: Another example, where the support of $\...
Robert Israel's user avatar
6 votes

Integral with 4 Bessel functions and an exponential

Let's consider the second integral, which can be written in the following form: $$ I(p, q, i, j, k, l; a, b, c, d) := \int_0^\infty dt\, \exp(-p t^2) t^q j_i(a t) j_j(b t) j_k(c t) j_l(d t) $$ where ...
JCGoran's user avatar
  • 159
6 votes

Representation of the Dirac delta function

$\newcommand\ep\epsilon\newcommand\de\delta\newcommand\R{\mathbb R}$Consider any family of nonnegative measurable kernels $K_\ep\colon(0,\infty)\to\mathbb R$ for $\ep\in(0,\infty)$ such that $\int_0^a ...
Iosif Pinelis's user avatar
5 votes

Relationship between the Radon transform and Twistor spaces

The correspondence between the Radon transform (from a space of real-valued functions on $\mathbb{R}^2$ to the space of functions on the manifold of straight lines in $\mathbb{R}^2$) and the Penrose ...
Carlo Beenakker's user avatar
5 votes
Accepted

An integral identity relate to the Gamma function or the Beta function

To evaluate $\int_{-\infty}^\infty {dx\over (1+ix)^a\,(1-ix)^b}$, view this as $\int_{-\infty}^\infty \widehat{f_a}(x)\,\overline{\widehat{f_{\overline{b}}}(x)}\,dx$ where $f_c$ are functions whose ...
paul garrett's user avatar
  • 22.5k
5 votes
Accepted

Infinite dimensional version of a simple fact on certain singular matrices

For the first question, the answer is not necessarily. Very rough idea: The rank-nullity theorem doesn't always hold on infinite dimensional spaces. Rough idea: Let the operator $A$ be defined on $...
Willie Wong's user avatar
  • 37.4k
5 votes

2D Fourier transform of log function

If $v(r)=\log r$, then $\Delta v=\delta$ and $\hat {(\Delta v)}=1$, that is $-(\xi^2+\eta^2)\hat {v}(\xi, \eta)=1$. However this yields $-\hat{v}=1/(\xi^2+\eta^2)$ which has no sense since this ...
Giorgio Metafune's user avatar
5 votes
Accepted

Fast computation of convolution integral of a gaussian function

Convolution with a Gaussian kernel of an $n$-point function has $n^2$ complexity, while Fourier transformation (FFT), multiplication, and inverse Fourier transformation is only of complexity $n\log n$....
Carlo Beenakker's user avatar
5 votes

Solution to $\int_{0}^{y} x^{-a} \exp \left[- \frac{(b - cx^{-d})^2}{2} \right] dx$

Maple does not know a symbolic answer for this. The very special case $a=2,b=0,c=1,d=2$ is evaluated by Maple in terms of the Whittaker M function or a $\;{}_1F_1$ hypergeometric function: $$ \int_{0}...
Gerald Edgar's user avatar
  • 40.2k
5 votes
Accepted

Is there a solution to $\int_{\theta-1}^{x} \left(\frac{y+1-\theta}{\theta} \right)^{-\frac{1}{\epsilon+3}} y^{-\frac{\epsilon+4}{\epsilon+3}}dy$?

With some effort (the lower integration limit requires care) I found this answer for the definite integral: $$I(x)=\int\limits_{\delta}^{x} \left(y-\delta\right)^{-\frac{1}{ \epsilon+3}} y^{-\frac{\...
Carlo Beenakker's user avatar
5 votes

How to solve the following $0= \int_{-\infty}^\infty e^{-\frac{(bt+\omega)^2}{2}} f(t+\omega) \frac{1}{i t} dt, \forall \omega \in \mathbb{R}$

It may be helpful to rewrite this in a way that avoids the principal value: $$0=\int_{-\infty}^\infty e^{-(bt+\omega)^2/2} f(t+\omega) \frac{2}{i t} dt=\int_{-\infty}^\infty dt \int_{-\infty}^{\infty}...
Carlo Beenakker's user avatar
5 votes
Accepted

Definite integral of Bessel function of the first kind times $x^{3/2}$

The integral requires $\nu>-5/2$ for convergence, and then becomes a hypergeometric function: $$\int_0^a x^{3/2} J_\nu (bx) dx=\frac{2^{1-\nu} a^{\nu+\frac{5}{2}} b^{\nu}}{(2 \nu+5) \Gamma (\nu+1)}\...
Carlo Beenakker's user avatar
5 votes

A solution satisfying an integral inequality is bounded

Apparently, the question should be interpreted as follows: Let $y\colon[0,\infty)\to[0,\infty)$ be a measurable function satisfying the inequality \begin{equation} y(t)+\int_0^t y(s)\,ds\le c_1\int_0^...
Iosif Pinelis's user avatar
5 votes
Accepted

Integral of a function changing sign

In fact, this conjecture fails to hold for $x$ in a right neighborhood of $0$. Indeed, let $g(x,s)$ denote the integrand. Note that $g(x,s)$ is continuous in $x\in[0,1)$ for each real $s>0$, and $$...
Iosif Pinelis's user avatar
5 votes

A density claim

This is not an answer to the question, but a proof of something much weaker: there is a sequence $c_m\to0$ such that if $\lVert g_m\rVert\leq c_m$ for all $m$, then the result from the question is ...
Saúl RM's user avatar
  • 7,916
5 votes

A density claim

Following Jochen Wengenroth suggestion, I sketch the proof of the result from a comment. The claim follows from the following Lemma. If $\mu$ is a measure supported on $[0,X]$, where $X>1$ such ...
Oleg Eroshkin's user avatar
4 votes

Does the inverse Laplace transform of the square root exist?

According to Wolfram Alpha, it is $\frac{1}{\Gamma(-1/2)x^{3/2}}$. This, of course, involves regularization, so to get the full answer, we have to subtract the divergent part so that its Laplace ...
Anixx's user avatar
  • 9,306
4 votes

how to reduce the integral into hypergeometric function?

Letting $\beta=\frac{2\Pi}{1+\Pi^2}$, write the integrand as $$\sqrt{1+\Pi^2}\left(\sqrt{1+\beta \sqrt{1-y^{2}}}-\sqrt{1-\beta \sqrt{1-y^{2}}}\right).\tag1$$ Use the binomial expansion to express (1), ...
T. Amdeberhan's user avatar
4 votes
Accepted

Fourier sine and cosine transforms and Laguerre polynomials

Here is a derivation starting from the Fourier transforms given in Orthogonal polynomials on the unit circle associated with the Laguerre polynomials. The Fourier cosine and sine transforms of $\...
Carlo Beenakker's user avatar
4 votes

Infinite dimensional version of a simple fact on certain singular matrices

Willie Wong answered the general case, and I'd like to give a counterexample for the symplectic case: Let $M=S^2$, the standard 2-sphere embedded in $\mathbb{R}^3$ as the unit sphere, with the ...
kosta's user avatar
  • 375
4 votes

Relationship between the Radon transform and Twistor spaces

To understand the relationship between the Penrose and Radon transforms, it's hard to do better than the outline given by Atiyah in [1]. See chapter VI, section 5 (pages 78--81). (It even looks like ...
Oliver Nash's user avatar
  • 1,404

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