All Questions
Tagged with integral or integration
1,506 questions
-1
votes
1
answer
128
views
Riesz energy for open sets in dimension $1$
This is a continuation of the question Calculation of Riesz energy for balls . As there are three questions,;I am posting a new question here. Riesz energy for a ball $B(x_0,r)$ is given by
$$I_s(B(...
- 87
22
votes
1
answer
2k
views
A difficult integral for the Chern number
Cross post from Maths stack exchange
The integral
$$
I(m)=\frac{1}{4\pi}\int_{-\pi}^{\pi}\mathrm{d}x\int_{-\pi}^\pi\mathrm{d}y\phantom{,} \frac{m\cos(x)\cos(y)-\cos x-\cos y}{\left( \sin^2x+\sin^2y +...
- 333
2
votes
1
answer
112
views
Uniqueness of the zero of $f-f*G_\sigma$ with $f$ convex/concave
I am struggling with the following problem. Let $f$ be a real smooth function:
strictly convex on $(-\infty,0)$,
strictly concave on $(0,\infty)$,
strictly increasing.
For $\sigma>0$, how can one ...
- 393
3
votes
1
answer
163
views
Asymptotic behavior of the integral that contains $\delta$ function
The integral I want to calculate is defined as
$$
P(s)=\int_{-\infty}^{\infty}{\rm d}x\int_{-\infty}^{\infty}{\rm d}y\ \delta\left(\frac{(x+y)^2+4x^2y^2}{(x+y)^2+(x+y)^4}-s\right)e^{-\left(x^2+y^2\...
- 375
3
votes
1
answer
263
views
Surface integration w.r.t Hausdorff measure
I am reading at Evans' book Measure Theory and Fine Properties of Functions, Revised Edition, p. 165 and I can't see how one gets the transition from the first dotted (🔴) integral to the second one --...
- 255
4
votes
1
answer
205
views
Show that $\frac{1}{2 \pi i} \oint_{\mathbb{S}^1} \frac{1-\hat{f}(\xi)}{1-\xi}\cdot \frac{\mathrm{d} \xi}{\xi^{n+1}} \to 0$ as $n \to \infty$
Let $f = (f_0,f_1,\ldots,f_n,\ldots) \in \mathcal{P}(\mathbb N)$ be a probability distribution on $\mathbb N$ and denote by $$\hat{f}(z) = \sum_{n\geq 0} z^n f_n$$ for its probability generating ...
- 730
2
votes
0
answers
63
views
Evaluation of a certain area
I asked a version of this question on Math Stack Exchange 6 days ago, but without any responses: The area of a certain region
I am interested in evaluating the area of the region defined by
$$A_{L_1, ...
- 26.9k
1
vote
1
answer
341
views
Convolution of a Hermite polynomial with Gaussian kernel
Let $H_n$ be the $n$th probabilistic Hermite polynomial of degree n and $\eta = \exp(-x^2/2)/\sqrt(2 \pi)$ be the standard Gaussian density.
I would like to compute the integral $f_n(x) = \int H_n(x - ...
6
votes
0
answers
131
views
Complex beta function $\int_{\mathbb{R}^2} (x^2+y^2)^{\alpha-1}((1-x)^2+y^2)^{\beta-1} \,dx\,dy$
I am interested in showing that the integral
\begin{align}
& \int_{\mathbb{C}} |z|^{2\alpha-2}|1-z|^{2\beta - 2} \,dA(z) \\[8pt]
= {} & \int_{\mathbb{R}^2} (x^2+y^2)^{\alpha-1}((1-x)^2+y^2)^{\...
- 177
2
votes
1
answer
228
views
Distance between root of $f$ and its Gaussian convolution
Let $f$ be a :
$f\in\mathcal{C}^\infty(\mathbb{R},\mathbb{R})$,
for all $x> 0,~f(x)>0$,
for all $x< 0,~f(x)<0$,
I am struggling to find a bound for the distance between the root of $f$ ...
- 393
2
votes
0
answers
90
views
Computing a complex integral with many poles
For an integer $k\geq 1$, let $f:\mathbb{C}^k\to\mathbb{C}$ be such that $f$ is analytic in the region $\text{Re}(u_i) > -1$ (say) for each $1\leq i \leq k$, and decays rapidly on vertical lines (i....
- 1,990
3
votes
2
answers
352
views
Calculation of Riesz energy for balls
I was reading stuffs about Riesz energy which is defined for an open subset $U\in\mathbb{R}^d$ by $I_s(U)=\int_U\int_U|x-y|^{-s}\ dx\ dy$ where $dx$ and $dy$ are Lebesgue measure in $U$. Now if I take ...
- 87
7
votes
2
answers
518
views
Integral of the $\delta$ function
I want to calculate the integral defined as
$$
P(s)=\iint \mathrm dx \, \mathrm dy\ \ \delta\left(\frac{(x+y)^2+4x^2y^2}{(x+y)^2+(x+y)^4}-s \right).
$$
The integration is taken within the rectangle $-...
- 375
3
votes
0
answers
231
views
Could the patterns in the roots of the Riemann-Liouville differintegral for $(s-1)\,\zeta(s)$ be explained?
The well-known integral expression for the entire function:
$$(s-1)\,\zeta(s) = \frac{-i\,\pi}{2}\int_{1/2-i\infty}^{1/2+i\infty} \frac{\csc(\pi\,u)^2}{u^{s-1}} \, du \qquad s \in \mathbb{C} \tag{0}$$
...
- 4,169
4
votes
0
answers
180
views
Do the difficulties in generalising Henstock-Kurzweil still exist if every subset of $\mathbb R^n$ is Lebesgue measurable?
There are apparently some difficulties generalising the Henstock-Kurzweil integral from functions of signature $\mathbb R\to\mathbb R$ to functions of signature $\mathbb R^n \to \mathbb R$. One ...
- 4,943
8
votes
2
answers
1k
views
A curious family of integrals that give $\pi$
I have noticed experimentally that:
$$\int_0^1 \frac{\color{red}{x}}{x^4+2x^3+2x^2-2x+1} \,dx=\color{blue}{\frac{\pi}{8}},\tag{1}$$
$$\int_0^1 \frac{\color{red}{1-x^2}}{x^4+2x^3+2x^2-2x+1} \, dx=\...
3
votes
1
answer
136
views
The function $H(x) =(4\pi t)^{-n/2} \int_{\mathbb{R}^d} e^{\frac{-|x-y|^2}{4t}}(1+|y|)^m\,dy$ attains its maximum in $x=0.$
In this paper, Lemma 6, Pinsky proves that $H(x) =(4\pi t)^{-n/2} \int_{\mathbb{R}^d} e^{\frac{-|x-y|^2}{4t}}(1+|y|)^m \, dy$ attains its maximum in $x=0.$ I didn't understand the proof very well, but ...
- 677
6
votes
0
answers
357
views
Is there a uniform version of Lebesgue's differentiation theorem?
Let $\mu$ be a finite measure on $\mathbb R$ and $f,g : \mathbb R \to \mathbb R_{\geq 0}$ two measurable maps such that $\int_{x\in\mathbb R} f(x)\ \mu(dx) \leq 1$ and that $g(x) \leq 1$ for all $x$. ...
4
votes
1
answer
470
views
Iterated Duhamel's formula for solutions of Boltzmann equation
My question comes from a computation in the paper Central limit theorem for Maxwellian molecules and truncation of Wild expansion. Specially, consider the following Boltzmann equation
$$\frac{\partial ...
- 730
2
votes
1
answer
179
views
Is this integral solvable analytically?
I have this integral that comes from my research with some Fourier Transforms of spectrum functions:
$$ G(\tau) = \int_{0}^{\infty} e^{-\Lambda x} x^n e^{i \tau ( c_1 - c_2 e^{-c_3 x} ) } dx $$
where $...
- 159
1
vote
0
answers
80
views
Biot-Savart-like integral for a toroidal helix
The following problem originates from Physics, so I apologize if I will not use a rigorous mathematical jargon.
Let us consider a toroidal helix parametrized as follows:
$$
x=(R+r\cos(n\phi))\cos(\phi)...
- 255
7
votes
2
answers
660
views
For a manual evaluation of a definite integral
I note that Mathematica could yield the identity
$$\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}$$
But I don't know how Mathematica got this.
...
- 15.6k
9
votes
0
answers
223
views
On the conditions of convergence in the generalized Riemann-Lebesgue lemma
I am reposting the following question that I asked in the MSE site here.
As I mentioned there, The following generalizations of the Riemann-Lebesgue lemma are rather well known (Kahane, C. S., ...
- 271
0
votes
0
answers
77
views
Completeness of a normed space
We consider the set $\mathcal{PC}([-r,0],X)$
$$\mathcal{PC}([-r,0],X):=\{\varphi:[-r, 0] \rightarrow X: \varphi \text{ is continuous everywhere except
for a finite number
of points } t_* \text{ ...
- 39
3
votes
1
answer
205
views
Bound on an integral representing a difference of two relative entropies
Let $ f : [0,1] \to \mathbb{R} $ be a function satisfying: 1.) $ |f(x)| \leqslant a $ for some $ a < 1 $, and 2.) $ \int_0^1 f(x) {\mathrm d}x = 0 $. I would like to know whether the following ...
- 503
1
vote
1
answer
249
views
Approximation for a Bessel function integral
I'm trying to calculate hit probabilities on a dart board if the dart thrower has some Gaussian angle distribution function with width $\Delta$ and some systematic angle offsets $\phi_0, \theta_0$. ...
- 19
0
votes
1
answer
79
views
Convergence in sequential Lebesgue spaces
Consider a strictly increasing sequence $1\leq q_0<q_n<q_{n+1}<q$ such that $q_n\to q$ as $n\to \infty$. Let $B\subset \Bbb R^d$ be a ball, so that $L^{q}(B)\subset L^{q_{n+1}}(B)\subset L^{...
- 1,101
0
votes
1
answer
131
views
Solution or approximation to $\int x^{-a} \Gamma\left( b, c x^{-d} + e \right) dx$
I'm looking for a solution or approximation to the following indefinite integral $$\int x^{-a} \Gamma\left( b, c x^{-d} + e \right) dx.$$
I've tried Mathematica, but it does not converge to a solution....
2
votes
1
answer
290
views
Numerical methods for integral eigenvalue equation
I have an integral equation which is not exactly an eigenvalue type equation, but similar:
$$\int d^3 y\, K(x,y,\lambda) f(y) = f(x)$$
Here $\lambda$ can be thought of as an eigenvalue, so it is ...
- 1,150
1
vote
2
answers
163
views
Integral with linear function, Normal PDF, Normal CDF
I am trying to calculate the following integral:
$$\int_a^\infty x \Phi(cx+d) \phi\left(\frac{x-\mu}{\sigma}\right) dx,$$
where $\Phi$, $\phi$ denote the CDF and PDF of the standard Normal $N(0,1)$.
I ...
- 49
1
vote
1
answer
118
views
Volterra Processes (integration wrt Brownian motion): reference request
I need some references about Volterra processes $Y=(Y_t)_{t\geq0}$ defined as
$$ Y_t:=\int_{0}^{t} g(t,s)dB_s, \ \ t\geq 0,$$
where $B=\left(B_t\right)_{t\geq0}$ is a brownian motion and $g$ satisfies
...
- 95
7
votes
1
answer
381
views
Consistency of a strong Fubini type theorem for measure zero sets
Is the following statement (†) consistent with ZFC?
If $E \subseteq [0,1]^2$ is such that $E_x := \{y\in[0,1] : (x,y)\in E\}$ has measure zero for almost all $x$, then $E^y := \{x\in[0,1] : (x,y)\in ...
- 32.5k
16
votes
1
answer
1k
views
Integral inequality: an elementary proof?
I have a very indirect proof of the following property involving a parametrized integral. If $a,a_1,\ldots,a_n\in\mathbb R^n$ (here $n\ge2$), let me denote $V(a,a_1,\ldots,a_n)$ the volume of the ...
- 52.3k
0
votes
1
answer
135
views
When integrating by part produces a singularity
I'm currently interesting in the following operator:
$$O[f](x):=f(x)-2xe^{x^2}\int_x^{+\infty}dt \, e^{-t^2} f(t)$$ for all $x\in\mathbb{R}$ and $f$ smooth and decaying at infinity fast enough with ...
- 41
4
votes
1
answer
424
views
An exercise on log-concave random variable on the real line
Let $X$ be a real random variable with log-concave density $f$. Assume that $E(X) =0$ and $E(X^2)=1$.
Show that there is a universal (independent of $X$) constant $c>0$ such that:
$$P(X\in[-1/2;0])\...
- 245
2
votes
1
answer
159
views
Integral in the Lamb shift calculation – fourth order
I was trying to solve some integrals that appear in quantum electrodynamics but I was not able to do it on my own.
$$1/6\int_0^1 \int_0^1 { u^3 z^2(1-z^2/3) \over [u^2(1-z^2)+4(1-u)]}dudz $$
I know ...
1
vote
0
answers
55
views
Convergence of Farey series integral of a "density" function as the order tends to infinity
Let $F_n$ denote the $n$-th Farey sequence, and let $q$ be a rational number such that $0 \leq q \leq 1$. I am studying the convergence of a specific integral related to Farey series, defined as ...
- 375
6
votes
1
answer
408
views
On an asymptotic integral decay
Let $f\in C^{\infty}([-1,0])$ be real-valued and suppose that
$$ \left| \int_{-1}^{0} f(t)\,e^{\lambda t} \,{\rm d} t \right| \leq e^{-\sqrt{\lambda}},$$
for all $\lambda > 0$. Does it follow that $...
- 4,135
1
vote
1
answer
223
views
Analytic expression for $\int_0^\infty \mathrm{d}p\frac{e^{-p \sin (\phi )} \sin (p \cos (\phi ))}{p \left(e^{c p}+1\right)}$
I am looking for ways to do this integration analytically
\begin{equation}
\int_0^\infty \mathrm{d}p\frac{e^{-p \sin (\phi )} \sin (p \cos (\phi ))}{p \left(e^{c p}+1\right)}
\end{equation}
For ...
- 199
2
votes
1
answer
72
views
Uniqueness of a solution to an equation
Let $t \in [0,1]$ or $t \in (0,1)$ be distributed according to $F(t)$.
Now consider the following equation:
\begin{equation}
\frac{\int_{\underline{t}}^{\overline{t}}(\gamma-t(2\gamma-1))dF(t)}{\int_{...
- 23
0
votes
1
answer
124
views
Double integral of two Gaussians and few complex poles
Recently encountered an integral:
$$ \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \dfrac{ e^{-i(x_1+x_2)k} \exp\left(-\frac{(x_1-x_0)^2}{2\sigma^2} -\frac{(x_2-x_0)^2}{2\sigma^2}\right) }{(x_1+x_2-\...
- 101
12
votes
1
answer
437
views
Slick proofs using the Henstock–Kurzweil integral?
I enjoyed Iosif Pinelis's slick answer to another MO problem using the Henstock–Kurzweil integral. Are there other examples of problems whose statement does not explicitly involve the Henstock–...
7
votes
1
answer
337
views
If $f(x) = \sum_{n=0}^\infty a_n x^n$, then $\int_{-\infty}^\infty f(x^2) dx = \pi i a_{-\frac{1}{2}}$
I've noticed a curious relationship between the coefficient $a_n$ for a power series and the integral of the real line. For instance, take $f(x) = e^{-x} = \sum_{n=0}^\infty \frac{(-1)^n}{n!} x^n$. ...
- 1,730
2
votes
2
answers
382
views
Asymptotics of an integral requested
Given an integer $n\geq2$, consider the following integral
$$I_n:=\int_0^1nx^{n-1}\sqrt{\left\vert \frac{\log(1-x)}{\log n}\right\vert} \, dx.$$
QUESTION. Is this true? It appears to be so.
$$\lim_{n\...
- 43.2k
3
votes
1
answer
162
views
Proof of $\Re\int_0^{\infty}\exp(-ik-k/\sqrt{1-4ik})dk=\Im\int_0^{1/4}\exp(-k+ik/\sqrt{1-4k})dk$
As mentioned in the title, I want to show that
$$
\Re\int_0^{\infty}\exp\left(-ik-\frac{k}{\sqrt{1-4ik}}\right)dk=\Im\int_0^{1/4}\exp\left(-k+\frac{ik}{\sqrt{1-4k}}\right)dk.
$$
I try to proof this ...
- 375
0
votes
1
answer
204
views
How to prove approximation for fresnel integral converges
I was looking at the fresnel integral $S(x)=\int^x_0\sin(s^2)ds$. From reading I learned that this integral approaches $\frac{1}{2} \sqrt{\frac{\pi}{2}}$ as $x \rightarrow \infty$. Through messing ...
4
votes
2
answers
305
views
Generalization of van der Corput's estimate on oscillatory integrals
Question: Given exponents $0<\alpha<\beta$ and an interval
$[a,b]\subset(0,\infty)$ are there constants $C,d>0$ such that for any
$\lambda_1,\lambda_2\in\mathbb{R}$,
$$\left|\int_a^be(\...
- 1,701
1
vote
2
answers
135
views
Lemma about the weighted interpolation inequality
In this article Interpolation inequalities with weights
Chang Shou Lin the following lemma is stated and proved.
Lemma: Suppose $\dfrac{1}{p}+\dfrac{\alpha}{n} > 0$, then there exists a constant $C$...
- 677
6
votes
1
answer
230
views
Integration along fibres of continuous map on compact Hausdorff spaces
Let $p:Z\to X$ be a continuous surjective map between compact Hausdorff spaces.
Does there exist a family $m=(m_x)_{x\in X}$ of Radon probability measures on $Z$, such that
the support of $m_x$ is ...
user473423
1
vote
1
answer
152
views
The monotonicity of the bivariate normal with non-isotropic covariance
Let $Y = (Y_1, Y_2) \sim N(0, 11^T + I)$, be a bivariate normal random variable with non-isotropic covariance.
Define $y = (y_1, y_2)$ and let
\begin{align}
F_{\delta}(y) = \Pr[Y_1 > y_1 - \delta, ...
- 113