All Questions
Tagged with integral or integration
1,507 questions
1
vote
2
answers
74
views
Opial type inequalities
Let $x(t)\in C^1[0,h]$ be such that $x(0)=x(h)=0$ and $x(t)$ in (0,h) ,then the following inequality
$ \int^h_0 |x(t)x^{'}(t)|dt \leq \frac{h}{4}\int^h_0(x^{'}(t))^2dt$
my question: I would like ...
- 57
0
votes
0
answers
115
views
Bayesian Bandits - What's the probability that choice K is the best?
I have $K$ very unfair coins. I don't know how unfair they are, but they all seem to have different probabilities of landing heads. I'd like to figure out which one is best as quickly as possible.
...
- 9
0
votes
1
answer
212
views
All roots of a polynomial are inside the unit circle [closed]
How to compute the following integral?
$\int_0^1 P_j(x)dx$, where $P_j(x) = \prod\limits_{i=0,i\ne j}^n(x+i)$?
I met this problem when doing estimation on the roots of the following polynomial.
All ...
- 31
7
votes
1
answer
806
views
Can I cover a compact set by balls {B} such that {2B} has bounded overlap?
Suppose I have a compact set $K \subset B_1(0) \subset \mathbb{R}^n$. Can I always find a family of open balls $\{B_{r_j}(x_j)\}$ such that
$x_j \in K$ and $B_{r_j}(x_j) \subset B_1(0)$ for each $j$;
...
- 1,179
1
vote
1
answer
258
views
If $h$ is a decreasing function then $\psi$ is an increasing function
Let $h : [0,1] \to [0,1]$ be a $\mathcal{C}^1$ function such that $h'(x)<0$ for all $x \in (0,1)$. Consider the function
$$
\psi(x) = \frac{\int_0^1yh(|x-y|) dy}{\int_0^1 h(|x-y|)dy}
$$
I am trying ...
- 31
1
vote
0
answers
145
views
An integral involving many exponential terms with quadratic exponents in the denominator
Given $k$ points $\{p_1,\cdots, p_k\}$ in $\mathbb{R}^n$ and positive constants $r_1, ..., r_k$ and another positive constant $\alpha>0$. Is there a way to compute/approximate the following (...
- 461
1
vote
1
answer
344
views
Is this relation between divergent intergals justifiable?
Graf's book on hyperfunction theory says (page $36$) that
$$\frac1{(x-i0)^n}=\frac{(-1)^{n-1}\pi i}{(n-1)!}\delta^{(n-1)}(x)+\operatorname{fp}\frac1{x^n},$$
while the table of Fourier transforms ...
- 10.1k
3
votes
0
answers
238
views
Dominated convergence Theorem
I am struggling to understand the proof in the paper, Learning Temporal Evolution of Spatial Dependence with
Generalized Spatiotemporal Gaussian Process Models.
Theorem 2.1 in the page 33 uses ...
- 31
5
votes
0
answers
254
views
Is there a practical application of natural integral or differintegral?
The following formulas give natural differintegral (that is one with naturally fixed integration constant):
$$f^{(s)}(x)=\sum_{m=0}^{\infty} \binom {s}m \sum_{k=0}^m\binom mk(-1)^{m-k}f^{(k)}(x)$$
$$f^...
- 10.1k
2
votes
1
answer
81
views
Perform certain constrained integrations over an ordered subsection of a 3-simplex, yielding "absolute separability" probabilities
Let us order the four points $\lambda_1 \geq \lambda_2 \geq \lambda_3 \geq \lambda_4 \geq 0$ of a 3-simplex, $\lambda_1+\lambda_2+\lambda_3+\lambda_4=1$, giving us a subsection $L$.
Integration over $...
- 723
0
votes
1
answer
167
views
Taylor expension of a simple integral [closed]
I'm trying to derive some weights expression for a boosting algorithm on a L2-ISE loss function, and i have trouble with the taylor expension.
Suppose that $f$ and $g$ are two densities from $\...
- 686
1
vote
0
answers
107
views
Change variable in integration with symmetry
Not sure if I can ask such fundamental problem here.
Let
$G$ be a finite group, $\sigma \in G$. Consider linear group actions fo $G$ on $\mathbb{R}^n$.
$\sigma(K)=\{x\in \mathbb{R}^n: \sigma^{-1}(...
- 345
2
votes
1
answer
139
views
Example of evaluation of $\int_0^1\left(\sum_{k=0}^n (f(x))^k\right)^{\alpha}dx$, for some choice of $f(x)$ satisfying certain requirements
Let $0<\alpha\leq\frac{1}{2}$ a fixed real number. I wondered if it is possible to evaluate the sequence of definite integrals $$\int_0^1\left(\sum_{k=0}^n (f(x))^k\right)^{\alpha}dx\tag{1}$$
for ...
3
votes
3
answers
383
views
Density of a functional space
Let $D$ be a bounded domain of $\mathbb{R}^n$ with smooth boundary $\partial D$. Is the following subspace dense in space $L^2(D)\times L^2(\partial D)$:
$$\{(f,f\rvert_{\partial D}) : f\in C^\infty(\...
user135093
3
votes
1
answer
963
views
Risch's algorithm for symbolic integration and its variations
I want to explore symbolic integration, but for this I initially need to imagine what are the algorithmic achievements in this area today, so I have some questions about Risch's algorithm and all its ...
- 101
2
votes
1
answer
636
views
Sufficient condition for function of conditional probability density to be increasing
Let $Y$ and $W$ be two jointly distributed random variables; $Y$ takes values on $(y_1,y_2)$ and $W$ takes values on $(w_1,w_2)$. The conditional probability density of $W$ given $Y$ is given by $f_{W|...
- 143
4
votes
2
answers
303
views
Express $\int_0^{\pi/2}\{ \operatorname{gd}^{-1}(x)\}dx$ as series of special functions, with $\operatorname{gd}^{-1}(z)$ the inverse Gudermannian
I know that in the literature there are a lot of integrals involving the fractional part (and other floor and ceiling functions). For some of these integrals is provide its evaluation as a series or ...
1
vote
1
answer
112
views
Inequalities for the Gudermannian function of the type $\operatorname{gd}(x)\operatorname{gd}\left(\frac{1}{x}\right)<\text{upper bound}$, where $x>0$
After I've read the solution of Problem 4327 (see [1]) I wondered if an inequality for a similar upper bound in the RHS is feasible for the known as Gudermannian function
$$\operatorname{gd}(x)\...
2
votes
1
answer
498
views
Uniform sampling on a Riemannian manifold via tangent space and exponential map
Given a Riemannian manifold $(\mathcal{M}, \{g_x\}_{x \in \mathcal{M}})$ and a fixed point $x \in \mathcal{M}$, does the following procedure yield uniform samples from $\{y \in \mathcal{M} : d_\...
- 281
1
vote
1
answer
299
views
Examples of Steffensen's inequality at undergraduated level studies
I've known few days ago the known as Steffensen's inequality, see the article Steffensen's inequality from Wolfram MathWorld and the cited bibliography. It seems that there are applications (I don't ...
16
votes
2
answers
2k
views
Challenge: Non-Gaussian quartic integral and path integral in Quantum field theory
(1) It is well-known that we can get a Gaussian integral of this type, where $x$ is in $\mathbb{R}$:
$$
\int_{-\infty}^{\infty} dx e^{-ax^2}=\sqrt{(2\pi)/a}. \tag{i}
$$
We can generalize this ...
- 2,147
1
vote
1
answer
151
views
A marginal space splitting $\{ \psi \}^{\perp}$
Let $\psi \in L^2(\mathbb R^2,\mathbb C)$. Is there a continuous projection from $\{ \psi \}^{\perp}$ onto
$$
\left\{ \varphi \in L^2(\mathbb R^2) \:\:\Big| \int \overline{\psi}(x,y) \varphi(x,y)\...
- 31
0
votes
1
answer
266
views
On a type of sequence of integrals inspired in the Borwein integral and an integral due to Furdui
This post is inspired in the Borwein integral, and in a problem proposed by Ovidiu Furdui in Crux Mathematicorum, that is the Problem 3707, in page 151.
I've considered integrals of the form $$\int_0^...
5
votes
1
answer
2k
views
A rather curious equality: is this true?
I came across (coincidentally) two integral evaluations, which seem to agree according to numerical tests. It did not seem easy to convert one into the other.
QUESTION. Is this true?
$$\int_0^1\...
- 43.2k
1
vote
0
answers
101
views
Reparametrization of a closed curve that balances sum of first derivatives
(Question in the yellow box below.)
A few weeks ago I was wondering about the existence of a scalar function $f(s): S^1 \rightarrow \mathbb{R}$ and a turning angle $\phi(s):S^1 \rightarrow \mathbb{R}/...
- 405
1
vote
1
answer
117
views
Limiting value of definite integral [closed]
$$I = \int_{-4}^4 e^{in\pi x/4}\frac{\sinh(b\pi/4)}{\sin^2\left(\frac{a-x}{8/\pi}\right)+\sinh^2(b\pi/8)}\,dx$$
I am unable to integrate the above equation when when $b$ tends to $0$, because of a ...
- 11
1
vote
2
answers
378
views
Prove that a certain integration yields the value $\frac{7}{9}$
Numerical methods surely indicate that $\int_0^{\frac{1}{3}} 2 \sqrt{9 x+1} \sqrt{21 x-4 \sqrt{3} \sqrt{x (9 x+1)}+1} \left(4 \sqrt{3} \sqrt{x (9
x+1)}+1\right) \, dx= \frac{7}{9}$.
Can this be ...
- 723
6
votes
1
answer
433
views
Triangle inequality for Ito integral?
For Lebesgue integrals one has the triangle inequality saying that for continuous functions let's say
$$\left\vert\int_0^t f(s) \ ds\right\vert \le \Vert f \Vert_{\infty} \int_0^t \ ds$$
Now if ...
- 536
1
vote
0
answers
511
views
Weak derivative under the integral sign
Let $\Omega$ be a bounded and regular open subset $\Omega$ of $\mathbb{R}^N$ and $u:[0,\infty)\times \Omega\to \mathbb{R}$ be a smooth function (for example a smooth solution to a PDE). Thus the ...
- 161
4
votes
1
answer
170
views
Functions orthogonal to powers of $1/{\left(1+x^2\right)}$
Let $f,g:\mathbb{R}\to\mathbb{R}$ be continuous functions with the following properties:
$f(x)$ and ${g(x)}/x$ are bounded;
${g(x)}/{\left(1+x^2\right)}\in L^1\left(\mathbb{R}\right)$;
$\lim_{x\to0}f(...
- 537
0
votes
1
answer
126
views
Proof of $\sum\limits_{k\in\mathbb{Z}}\int_{\mathbb{R}}|f(x+k)f'(x)|dx<\infty$ for Schwartz function $f$
For a function $f$ from the Schwartz space $\mathcal{S}(\mathbb{R})$ do we have that
$$\sum\limits_{k\in\mathbb{Z}}\int_{\mathbb{R}}|f(x+k)f'(x)|dx$$
converges?
- 123
2
votes
1
answer
230
views
Does exist a variant of Frullani's theorem valid for $f(x)=\pi(x)/x$ or $f(x)=\psi(x)/x$, where the numerators are prime-counting functions?
Frullani's theorem is a deep theorem in real analysis with applications, see the Wikipedia Frullani integral and other uses and contexts (see [2]). I wrote two imaginative examples of what can be ...
0
votes
0
answers
136
views
A complex integration formula
I'm trying to integrate a formula but I can’t figure it out. Its calculation involves an error function. Here's the formula:
$f(a, b, c)=\int_{0}^{+\pi} d \theta \exp (a \cos \theta) \operatorname{...
1
vote
2
answers
73
views
Expectation of $\left| \frac{(\textbf{x}+\textbf{y})^{H} \textbf{x} }{\| \textbf{x} + \textbf{y} \|^2} \right|^2$, with complex Gaussians?
Given that following two random variables $\textbf{x} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{x}^{2}\textbf{I}_{M})$ and $\textbf{y} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{y}^{2}\textbf{I}_{M})$ ...
11
votes
2
answers
603
views
Function orthogonal to powers of $1/\left(1+x^2\right)$
Does there exist any continuous function $f:\mathbb{R}\to\mathbb{R}$, $f(x)/(1+x^2)\in L^1(\mathbb R)$, such that $f(0)=1$ and
$$\int_{-\infty}^{\infty}\frac{f(x)}{\left(1+x^2\right)^p}dx=0$$ for ...
- 537
1
vote
2
answers
889
views
Simplify Wasserstein distance between Gaussians with binary cost function
Let $\mu_1$ and $\mu_2$ be 1D gaussian distributions with means $m_1$ and $m_2$ respectively and common variance $\sigma$. Let $\Omega$ be a closed subset of $\mathbb R^2$, and consider the cost ...
- 6,853
2
votes
1
answer
543
views
On $\zeta(7)$ as the integration of the product of an indefinite integral due to Lobachevskii by a power of the inverse Gudermannian function
In this post I invoke certain function from a post of this site MathOverflow it is [1] (please see further references from the post, authors from the Springer link of the cited literature and answers ...
4
votes
3
answers
2k
views
Legendre Polynomial Integral over half space
I need to compute the following integral
$$
I_{n,m} := \int_0^1 P_n(x) P_m(x) \; \mathrm{d}x
$$
where $P_n$ is the Legendre polynomial.
For an even sum $n+m=2l$ it is easy to show that
$$
I_{n,m} = \...
- 213
2
votes
0
answers
249
views
Calculating $\int_1^{\infty}\frac{\operatorname{ali}(x)}{x^3}dx$, where $\operatorname{ali}(x)$ is the inverse function of the logarithmic integral
It is well-known that we can compute the closed-form of the integrals $$\int_1^{\infty}\frac{\log x}{x^2}dx$$
and $$\int_1^{\infty}\frac{\operatorname{li} (x)}{x^3}dx,$$
where $\operatorname{li} (x)$ ...
0
votes
0
answers
185
views
Express $\zeta(n)$, in particular the Apéry's constant, in terms of series involving Gregory coefficients or the Schröder's integral
The idea and motivation of this post is to know if it is possible to express integer arguments of the Riemann zeta function $\zeta(n)$, in particular $\zeta(3)$, in terms of series involving the so-...
7
votes
1
answer
681
views
Change of variables for $p$-adic integral
Say $p$ is an odd prime. Suppose I have a measure $\mu$ on $\mathbf{Z}_p$. As in II.4.3 in Colmez - Fonctions d'une variable $p$-adique, I can restrict $\mu$ to $1+p\mathbf Z_p$, and there is a ...
- 503
8
votes
1
answer
805
views
Integration with values in a topological vector space
Is there a general theory of integration of functions with values in a topological vector space (not necessarily locally convex)?
Browsing through mathoverflow posts, I came across a discussion ...
- 151
3
votes
1
answer
495
views
Help in proving, that $\int_{0}^{\infty} \frac{1}{\Gamma(x)} d x=e+\int_{0}^{\infty} \frac{e^{-x}}{\pi^{2}+(\ln x)^{2}} d x$ using real methods only
I hope this does not seem like a too easy question for Overflow. I would like to find an easier method than mine to prove the above statement for the Fransén-Robinson Constant. My first method was to ...
- 169
1
vote
1
answer
246
views
Limits of a family of integrals
Assume $\lambda_1+\lambda_2=1$ and both $\lambda_1$ and $\lambda_2$ are positive reals.
QUESTION. What is the value of this limit? It seems to exist.
$$\lim_{n\rightarrow\infty}\int_0^1\frac{(\...
- 43.2k
2
votes
1
answer
935
views
Exterior derivative independence from coordinate systems
In the book Mathematical Methods of Classical Mechanics by V.I. Arnold, the author introduces (p.189) the concept of exterior derivative as "the principal linear part of the increment" of the function ...
- 141
5
votes
3
answers
475
views
Closed form $\int_{0}^{\frac{r}{2}} {\binom{n}{p} \binom{n-p}{r-2p} 2^{r-2p}}{\binom{2n}{r}^{-1}} \ \text{d}p$
Note: This is exact copy of my Math.SE question, which I am reposting here, as despite bounty it did not receive any answers.
Let there be $n$ pairs of shoes in a box.
The the probability that from ...
- 67
5
votes
0
answers
202
views
Invariant measure on coset space and integrable functions
Let $G$ be a locally compact abelian group, and $H$ a closed subgroup. Let $C_c(G)$ be the space of continuous, compactly supported complex valued functions on $G$. General theory of Haar measure ...
- 6,180
2
votes
0
answers
248
views
Intrinsic volume - is there a simplified formula?
I'm struggling to understand how can I compute the instrinsic volumes (or Minkowski functionals) of a submanifold $\Omega$ of $\mathbb{R}^N$. I found a formula, but I really can't understand it...
It ...
- 1,759
4
votes
1
answer
351
views
Asymptotic behaviour of function using Fox $H$-function representation
In equation (9) of this paper, it is claimed that the limiting behaviour
$$
\int_0^\infty \frac{1-\cos(kx_0)}{s+Dk^\alpha}dk
\sim
\frac{\Gamma(2-\alpha)\sin(\pi(2-\alpha)/2)x_0^{\alpha-1}}{(\alpha-1)D}...
- 85
3
votes
0
answers
487
views
Homeomorphism between $L^p$-spaces on metric spaces and $L^p$-spaces on Euclidean space
Setup:
Fix $p \in [1,\infty)$.
Let $(X,d_X,x_0)$ and $(Y,d_Y,y_0)$ be complete pointed metric spaces and $\mu$ be Borel. Let $E^n,E^D$ be Euclidean spaces of respetive dimensions $n$ and $D$ and ...
- 5,405