Questions tagged [integration]
Questions related to various forms of integration including the Riemann integral, Lebesgue integral, Riemann–Stieltjes integral, double integrals, line integrals, contour integrals, surface integrals, integrals of differential forms, ...
1,442
questions
6
votes
0
answers
111
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Area of generalized ellipse
An ellipse $E$ can be defined by two foci, $p,q\in\mathbb{R}^2$, and a length parameter $\ell$ as follows:
$$ E = \{x\in\mathbb{R}^2 : ||p-x||+||q-x||\le\ell
\}.$$
The area of $E$ is uniquely ...
4
votes
1
answer
175
views
Does a nontrivial notion of integral under logarithmic deformations of $\mathbb{R}_+$ exist?
Background
The upper and lower Maslov dequantizations are respectively the limits $h \downarrow 0$ and $h \uparrow 0$ of deformations of the semifield $(\mathbb{R}_+,+,\cdot)$ defined for $0 \ne h \...
1
vote
1
answer
267
views
Interpolation between $L^1$ and $L^2$ spaces
I was wondering whether the following interpolation between $L^1$ and $L^2$ spaces is true:
Let $f \in \mathbb{R}^n$ be such that
$$ \alpha_1:= \int_{\mathbb{R}} \left\lVert f(x_1,\cdot,....\cdot) \...
23
votes
1
answer
1k
views
Number theoretic interpretation of the integral $\int_{-\infty}^\infty\frac{dx}{\left(e^x+e^{-x}+e^{ix\sqrt{3}}\right)^2}=\frac{1}{3}$?
Is there any explanation based on algebraic number theory that the integral
$$
\int_{-\infty}^\infty\frac{dx}{\left(e^x+e^{-x}+e^{ix\sqrt{3}}\right)^2}=\frac{1}{3}\tag{1}
$$
has a closed form? ...
7
votes
2
answers
569
views
"sinc-ing" integral
Let $a_1,\dots,a_n, b$ be positive real numbers.
*Question.** Is this true?
$$\int_{-\infty}^{\infty}\frac{\sin(bx+a_1x+\cdots+a_nx)}{x}\prod_{j=1}^n\frac{\sin(a_jx)}{a_jx}\,\,dx=\pi.$$
My ...
3
votes
0
answers
102
views
Extension of the Gagliardo Inequality
The Gagliardo Inequality generalizes Fubini's Theorem: let $f_j$ be $d-1$ non-negative measurable functions over ${\mathbb R}^{d-1}$. Let us form the function
$$f(x)=\prod_{j=1}^df_j(\widehat{x_j}),$$
...
-6
votes
2
answers
2k
views
Is there a transformation or a proof for these integrals?
Here are certain weighted Gaussian integrals I have encountered for which numerical computation reassures equality.
Question. Is this true? If so, is there an underlying transformation or just a ...
9
votes
1
answer
663
views
Conductor as volume of the integers ring
I am working on Tate's thesis, and I have some problems with computations, yet the result seems to be a good natural motivation for introducing the arithmetic conductor of a character.
Let $F$ be a ...
3
votes
0
answers
327
views
Norm of a singular integral operator
Let $H$ be the characteristic function of $(0,+\infty)$ and let us define for $(x,y)\in \mathbb R^2$, $x\not=y$
$$
k(x,y)=\frac{H(x+y)}{iπ(x-y)}.
$$
For $u\in C^1_c(\mathbb R)$, we define for $x\in \...
3
votes
2
answers
396
views
Can you work out the following integral
I want to integrate
$$
\int_0^{\infty}dx\,e^{-ax}\frac{1-(2x)^b}{1-2x}
$$
where $a,b>0$. My The naive approach was to consider $b$ to be an integer, in which case you get a truncated geometric ...
2
votes
0
answers
128
views
An estimation for holomorphic functions in the unit disc
I have proved using Green's theorem the following equality:$$\frac{1}{2\pi i}\left(\frac{1}{\zeta}-\bar{\zeta}\right)\int_{\partial \mathbb{D}}f(z+\zeta)\log\left(\frac{z}{z-\zeta}\right)dz=f(\zeta)+\...
1
vote
0
answers
59
views
When do quasiperiodic functions on $\mathbb{R}^2$ have an average over the plane?
For example, consider the following
\begin{equation}
\lim_{T\rightarrow\infty}\frac{1}{(2T)^2}\int^T_{-T}\int^T_{-T}\big[\cos(v/a)-\cos(u/b)\big]\cos(\sqrt{u^2+v^2})\ du\ dv,
\end{equation}
where $...
4
votes
0
answers
244
views
English language and Mathematics
I have a question maybe more relevant to an English language section of StackExchange, but I doubt that anybody but a Mathematician could properly answer my question.
Let $\mathcal M$ be a smooth ...
1
vote
0
answers
66
views
Generalising a one-dimensional integral identity involving Bessel functions to higher dimensions
This was originally posted on Math Stack Exchange, but no responses were received. I recently came across the following remarkable identity, due to Hardy:
$$\displaystyle \int_{-\infty}^{\infty} \...
3
votes
1
answer
83
views
Does $p$ integrability in n-1 dimensions give higher integrability in $n$ dimensions?
Restrict everything to a ball in $n$ dimensions, let $x$ represent the first $n-1$ variables, and $t$ the $n-$th variable. It is obvious by Holder's Inequality that
$$
\int\limits_t\left(\int\limits_x|...
-2
votes
1
answer
430
views
Expectation of random integral of deterministic function
Suppose I have some random variable $W$ along with its expectation $\mathbb{E}[W]$. My goal it to compute the integral
\begin{equation}
\mathbb{E}\left[\int_{0}^{W}f(t)dt\right] = \int_{0}^{\mathbb{E}...
2
votes
2
answers
838
views
Monotone convergence theorem for stochastic integrals
I was wondering if there exists an equivalent of a monotone convergence theorem for stochastic integrals. I looked into plenty of books and papers, but I haven't found anything useful. I would expect ...
1
vote
0
answers
263
views
Does a growing manifold fixed at a point converge to its tangent plane?
Let $M$ be a smooth compact $(n-1)$ dimensional submanifold in $\mathbb{R}^n$. Let $H$ be the $(n-1)$ dimensional Hausdorff measure. Let $f(x,y,t)$ is a function for $x\in\mathbb{R}^n$, $y\in\mathbb{R}...
14
votes
2
answers
1k
views
Expected number of lines meeting four given lines or "what is 1.72..."
Given four random lines in $\mathbb{R}P^3$, how many lines intersect all of those lines?
In the recent paper Probabilistic Schubert Calculus, Peter Bürgisser and Antonio Lerario
discuss this question ...
1
vote
0
answers
185
views
A question concerning positive definite matrix functions
Let $C(e^{i\theta})$ be an $m\times n$ ($m\ge n$) matrix-valued continuous function of $\theta\in[-\pi,\pi]$. Let $A_1(e^{i\theta})$ and $A_2(e^{i\theta})$ be two $n\times n$ positive definite matrix-...
3
votes
1
answer
146
views
volume of region between two manifolds
The question is motivated by a simple example: the area of a ring is $\pi(R^2-r^2)$, where $R$ and $r$ are the radii of the outer and inner circles respectively. Let $C$ be the 'middle circle' with ...
2
votes
0
answers
545
views
Integrating a product of integrals involving Bessel functions
I have asked similar questions on Math Stack Exchange, but not been able to receive many helpful responses. Therefore, I am posting this problem here, and any input would be extremely valuable.
I ...
4
votes
0
answers
657
views
Proofs of the second fundamental theorem of calculus
I am referring to the following version of the theorem, in the setting of the Lebesgue integral.
Theorem Let $f: [a,b] \rightarrow \bf R$ be an everywhere differentiable function whose derivative is ...
5
votes
1
answer
725
views
Gaussian integral over a ball
How to compute the following integral?
$$\int_{\|x\|^2\leq R} \exp(-x^\ast G x+2\mathcal{Re}(x^\ast a)) \,dx,$$
where $x$ is an $M \times 1$ vector ($M\gg 1$), $G$ is a positive definite matrix, and $...
3
votes
2
answers
2k
views
cumulative binomial equals integral
Can somebody explain why the cumulative binomial equals an integral expression?
Thanks!${}$
$$
\sum_{j=0}^{k-1}\binom{n}{j}\theta^j(1-\theta)^{n-j} = 1- \binom{n}{k}k\int_0^\theta t^{k-1}(1-t)^{n-k}...
10
votes
1
answer
823
views
This is not a dyadic cosine-product
The double-angle formula, $\sin2x=2\sin x\cos x$, turns the scary-looking integral
$$\int_0^{\infty}dz\prod_{k=1}^{\infty}\cos\frac{z}{2^k}$$
into fun once you realize $\prod_k\cos\frac{z}{2^k}=\frac{\...
5
votes
1
answer
336
views
rigorous derivation of isoperimetric inequality from ideal gas equation
I'm an undergraduate math student that learned about classical ideal gases and the associated maxwell-boltzmann distribution for particle velocities in a statistical physics course. Now, starting from ...
1
vote
1
answer
94
views
Implications of a Regularity Condition for Functions
$
\newcommand{\essSup}{\mathop{\rm sup_{ess}}\nolimits}
$
What can be concluded from the fact, that $f: X\ni x\mapsto f(x)\in [a,b]\subset\mathbb{R}\setminus\lbrace{-\infty,+\infty\rbrace} $ ...
6
votes
1
answer
421
views
Definite integral involving inverse regularized incomplete beta functions
In my research I encountered the following integral
$$J = \int_0^1 I_t^{-1}(a_1,b_1) \: I_t^{-1}(a_2,b_2) \: \mathrm{d}t$$
which I would like to evaluate as a closed form expression, that is, as a ...
4
votes
0
answers
238
views
Generalize $\frac1{48^{1/4}\,K(k_3)}\,\int_0^1 \frac{dx}{\sqrt{1-x}\,\sqrt[3]{x^2+27x^3}}=\,_2F_1\big(\tfrac13,\tfrac13;\tfrac56;-27\big)=\frac47\,$?
These involve separate cases $a=\tfrac13,\,a=\tfrac14,\,a=\tfrac16$ of,
$${_2F_1\left(a ,a ;a +\tfrac12;-u\right)}=2^{a}\frac{\Gamma\big(a+\tfrac12\big)}{\sqrt\pi\,\Gamma(a)}\int_0^\infty\frac{dx}{(1+...
5
votes
1
answer
368
views
sum, integral of certain functions
While working on some research, I have encountered an infinite series and its improper integral analogue:
\begin{align}\sum_{m=1}^{\infty}\frac1{\sqrt{m(m+1)(m+2)+\sqrt{m^3(m+2)^3}}}&=\frac12+\...
1
vote
3
answers
277
views
Evaluation of Gaussian density integral
Is there any closed form, asymptotics, and/or approximations for the following integral:
$$f(c) := \frac{1}{\sqrt{2\sigma^2\pi}}\int e^{-x^2/2\sigma^2}\frac{x^2}{1-cx^2}dx,$$ where $\sigma^2$ is real ...
1
vote
0
answers
706
views
The derivative of an integral function with indicator and max function as integrand
I encounter the following type of problem:
\begin{equation}
F(x) = \int_a^b \mathbf{1}_{\{v+x-h(v)\geq 0\}}\max\{h(v)-y-x,0\}dv
\end{equation}
where $\mathbf{1}_{\{z\geq 0\}}=1$ if event $z\geq 0$ ...
9
votes
2
answers
1k
views
On the definition of "almost-everywhere" for non-complete measure spaces
If $(X,\mathcal{B},\mu)$ is a (non-necessarily complete) measure space, we can give two different notions of a property $P(x)$ that is true almost-everywhere :
(D1) There is a measurable set $A$ ...
7
votes
0
answers
292
views
An integral for the tribonacci constant and the general case
When I asked for integrals involving the tribonacci constant $T$, user @nospoon gave the nice answer,
$$\int_0^{\infty} \eta( i \, x)\,\eta(i \,11 x) \,dx = \frac{ \ln T}{\sqrt{11}} $$
However, the ...
2
votes
1
answer
215
views
Is there a way to get the closed form approximate result of $\int_ 0^a\frac {e^{-x - \frac {1} {x}}} {x}\, dx$ [closed]
It is known that
$$\int_ 0^{\infty}\frac {e^{-x - \frac {1} {x}}} {x} dx=2 K_0(2),$$ but now I want to get the closed form approximate result of
$$\int_ 0^a\frac {e^{-x - \frac {1} {x}}} {x} dx.$$
I ...
2
votes
0
answers
447
views
Does equality almost everywhere on a product imply equality almost everywhere on sections [closed]
(This question was on MSE, with no answers)
Consider two $\sigma$-finite measure spaces $X_1$ and $X_2$, and $X=X_1\times X_2$ the product measure space (a priori non-completed).
Take two functions ...
-1
votes
1
answer
403
views
Topological properties of complex valued Riemann sum limit curve and a particular integral inequality
I am studying under what conditions the following integral inequality would hold ($a$ real, $a>0$):
$$ \int_{-\infty} ^{\infty} \frac{f(ix)}{a\pm ix}dx\ = 0 \ \ \ \ \Rightarrow \ \ \ \int_{-\...
2
votes
1
answer
259
views
An Stokes type theorem for some operations other than integral
Let $M$ be a compact $m$ dimensional manifold with boundary $\partial M$.
Assume that $I_{1}, I_{2}$ are two linear functionals on $\Omega^{m}(M), \Omega^{m-1}(\partial M)$, respectively.
Assume ...
14
votes
2
answers
786
views
Integral of power of binomials equal to sum of power of binomials?
Inspired by this MO question about integrating binomial coefficients and the answers, I was wondering whether integrating powers of binomial coefficients also relates to the respective sums. And ...
1
vote
0
answers
294
views
Is there a general solution to a multivariate gaussian integral under constraints?
I'm interested in an integral of the form
$$\int d\vec{\theta} e^{-\beta (\vec{\theta}-\vec{\theta}_0) \cdot K \cdot (\vec{\theta}-\vec{\theta}_0)} \delta(f(\vec{\theta}))$$
where $\vec{\theta}$ ...
4
votes
1
answer
302
views
Fourier transform in terms of special function?
I have a Fourier integral
$$\int\limits_{-\infty}^{\infty}\mathrm{d}t\,\frac{1}{t^2}\exp\left({\mathrm{i}\frac{t^3}{3}+\mathrm{i}Yt+\frac{\mathrm{i}\lambda^2}{4t}}\right),$$
where $Y$ and $\lambda$ ...
-3
votes
3
answers
335
views
Can we decide whenever a function is the derivate of another function in this Language?
Our EXP functions are made in the following way:
Any constant $ \in \Bbb R$ is a EXP
$X \in \Bbb R$ is a EXP
$sin( g(x))$, $cos( g(x))$ are in EXP if $g(x)$ is a EXP
$tan( g(x))$ is a EXP if $g(x)$...
6
votes
1
answer
698
views
Аrе thеsе integrals known?
While studying some dark matter related stuff, I came across to the following interesting identities:
$$\int\limits_0^\infty\sqrt{\frac{y}{xp}}\,e^{-y}\left(K(p)-E(p)\right)dy=
\frac{\pi x}{4} \left[...
38
votes
4
answers
3k
views
Binomial again, and again
Let $\lceil a\rceil=$ the smallest integer $\geq a$, otherwise known as the ceiling function. When the arguments are real, interpret $\binom{a}b$ using the Euler's gamma function, $\Gamma$.
Recently, ...
1
vote
0
answers
295
views
Estimating an integral involving Bessel functions
I would like to preface this question by saying that I have asked a series of questions on this topic on Math Stack Exchange, but have almost never received any fruitful responses, with the exception ...
5
votes
0
answers
472
views
integrating with respect to parameters in beta function
I would like to evaluate an integral:
$$\int_t^1\frac{1}{B(1+s\phi,1+\phi(1-s))}p^{s\phi}(1-p)^{\phi(1-s)}ds,$$
where $B(a,b)$ is a beta function and $p\in(0,1)$ and $\phi>0$ are some parameters. ...
105
votes
5
answers
9k
views
integral of a "sin-omial" coefficients=binomial
I find the following averaged-integral amusing and intriguing, to say the least. Is there any proof?
For any pair of integers $n\geq k\geq0$, we have
$$\frac1{\pi}\int_0^{\pi}\frac{\sin^n(x)}{\...
3
votes
1
answer
871
views
Stokes theorem for manifolds with boundary as disjoint union of submanifolds
Looking at the generalizations of Stokes theorem, I did find a version for manifold with corners, but I was surprised this generalization doesn't contain a simple example such as the cone. So my ...
10
votes
2
answers
996
views
Can the integration of integrable sections of a measurable function of two variables ever result in a non-measurable function?
I spent some time searching MathOverflow for a problem that would resemble the one given below, but it turned out to be a rather futile endeavor. I was led to this problem in my attempts to construct ...