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, ...

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35
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Are we better in computing integrals than mathematicians of 19th century?

When I started to learn mathematics, I was fascinating by legendary «Демидович»: problems in mathematical analysis. Fifteen years later, when I open chapters about integrals, I see a long list of ...
22
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522 views

A multiple integral

Let us consider the multiple integral $$I_{n}=\int_{-\infty }^{\infty }ds_{1}\int_{-\infty}^{s_{1}}ds_{2}\cdots \int_{-\infty }^{s_{2n-1}}ds_{2n}\;\cos {(s_{1}^{2}-s_{2}^{2})}\;\cdots \cos {(s_{2n-1}...
16
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405 views

Reference request for Grothendieck's work on “Integration with values in a topological group”

Disclaimer. This question was already asked in Mathematics Stack Exchange (see the link here). I wanted the question to be migrated here but I was told by a moderator that a question that old is ...
13
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454 views

Is it possible that the following integral is $0$?

Given any integer $n\geqslant1$, let $E,F$ be two subsets of $\{\{i,j\}:1\leqslant i<j\leqslant n\}$ such that every two sets in $F$ are disjoint. It is not difficult to see that $$\int_{1<|z|&...
13
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400 views

Reference for a proof of the fiberwise Stokes theorem

The fiberwise Stokes theorem says that given a differential form on a smooth fiber bundle whose fibers have boundary, the difference between the fiberwise integral of the differential and the ...
11
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343 views

Integrability property of polynomials in several variables

This might be very trivial, or not. Let $p\colon\mathbb{R}^n\to \mathbb{R}$ be a polynomial of even degree, at most $n-2$. Assume that $p(x)\leq 0$ for any $x\in\mathbb{R}^n$. Assume that there ...
11
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126 views

Assymptotics of a Selberg type integral

Does any one know some references/ ideas on how to study the assymptotics as $N$ goes to $\infty$ of the following Selberg type integral $$\int _{\mathbb R^N} e^{-|x|^2}\ \prod_{1\le i<j\le N} \...
9
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327 views

Symmetry of function defined by integral

(Originally posed in Math.SE in Jan 2013. Received no complete answers as of yet.) Define a function $f(\alpha, \beta)$, $\alpha \in (-1,1)$, $\beta \in (-1,1)$ as $$ f(\alpha, \beta) = \int_0^{\...
9
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0answers
572 views

Reference request : Grothendieck's topological space valued integral

As I am learning the different kind of Banach space valued integrals (Pettis, Bochner), I know that Grothendieck made a "mémoire" in his youth about this topic, but I don't know if it is available ...
7
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147 views

A function with unexpectedly simple Legendre transformation

Let $I(x) = \frac{1}{2\pi} \int_{-2}^2 \sqrt{4-y^2}\ln|x-y|dy$. Then $I(x)$ is a concave function and \begin{equation} I(x)= \begin{cases} \frac{1}{4}x^2-\frac{1}{2}, &\text{if } |x|\leq2 \\ \...
7
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128 views

Hilbert series for invariant ring

I would like to compute the Hilbert series of the ring of invariants of certain irreducible representations of some groups (namely $SO(5)$ to begin with). To put it in some broader context, let $G$ ...
7
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214 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 ...
7
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299 views

An inequality which involves a sum of integrals

Please help me to prove $$ \sum\limits_{j=2}^n \frac{1}{j^\alpha (j-1)^\alpha} \int\limits_{j-1}^j \frac{dx}{x^{1-\alpha}(n-x)^\alpha} \leq \int\limits_0^1 \frac{dx}{x^{1-\alpha}(n-x)^\alpha},\quad \...
6
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84 views

Conceptual meaning of a non-linear relation connecting $6$ Mordell integrals?

Define Mordell integral by $$ \phi_\alpha(\theta)=\int\limits_0^\infty\frac{\cos\pi \theta x}{\cosh \pi x}\,e^{-\pi \alpha x^2}dx.\tag{1} $$ There are a lot of linear relations connecting integrals ...
6
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114 views

A reference for an integrability property?

In a recent paper of mine (Compensated integrability), I established a functional inequality which has nice consequences. For instance, it contains the isoperimetric inequality, and it gives a new ...
6
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194 views

Evaluating $\iint_{\mathbb{R}\times \mathbb{R}^{+}}e^{-|w-c_{1}|^2-|u-c_{2}|^2}\frac{1}{w_{1}+iw_{2}-u_{1}-iu_{2}}dw_{1}dw_{2}du_{1}du_{2}$

For $c_{1},c_{2}\in \mathbb{H}:=\{Im(z)>0\}$ I want to compute the following integral or prove it doesn't exist: $$\int_{\mathbb{R}\times \mathbb{R}^{+}}\int_{\mathbb{R}\times \mathbb{R}^{+}}e^{-|...
6
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91 views

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 ...
6
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189 views

Integral-like concepts

I am looking for interesting concepts (I guess you could say functionals from the function space $[a;b]\to\mathbb R$) that are like integrals in some respect. The background is that I have proven a ...
6
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987 views

Interchange of integral and infimum

Can anyone please suggest how to justify widely used formula for interchange of integral and infimum: $ \inf_{u(t)\in U}\int_{t_0}^{t_1}g(t,u(t))dt=\int_{t_0}^{t_1}\inf_{u\in U}g(t,u)dt, $ where $ U\...
6
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0answers
255 views

Integral identity related with cubic analogue of arithmetic-geometric mean

This is re-post from MSE as I did not get an answer there. Let $a,b$ be positive real numbers and we define two sequences $\{a_{n}\},\{b_{n}\}$ as follows: $$a_{0}=a,b_{0}=b,a_{n+1}=\frac{a_{n}+2b_{n}...
6
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224 views

How to take this Grassmann integral?

I'm trying to reconstruct and understand what is explained in a paragraph of this paper. I am trying to check if the method they describe actually gives us the Laughlin state. The integral I'm facing ...
6
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0answers
1k views

The Perturbation of Non Hamiltonian algebraic Vector fields

In this question we are interested in the number of limit cycles which appear in the following perturbational system: \begin{equation}\cases{ x'=y -x^{2}+\epsilon P(x,y) \\ y'=-x+\epsilon Q(x,y) } \...
6
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605 views

Minkowski's Inequality for Integrals in Orlicz spaces

EDIT: I have changed the question to have less parameters, fitting it into the context of Orlicz spaces. Suppose $f:[0,\infty)\to[0,\infty)$ is convex and increasing, $f^{-1}:[0,\infty)\to[0,\infty)$...
5
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61 views

Extending gauge integral to higher dimensions/spaces and analogue of Riemann rearrangement theorem for it

The gauge/Henstock-Kurzweil integral allows for the integration of a very large set of functions in $\Bbb R$, at the cost of many of the nice properties of Lebesgue integration, of which it is a ...
5
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0answers
174 views

More or less universal formula for regularization of divergent integrals?

Is there a simple formula that would produce the regularized value for the most common divergent integrals? I know, there is a formula for Cesaro integration, but it is applicable only to Cesaro-...
5
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0answers
182 views

Riesz Representation Theorem for $L^2(\mathbb{R}) \oplus L^2(\mathbb{T})$?

The spaces $L^2(\mathbb{R})$ (square-integrable functions) and $L^2(\mathbb{T})$ (1-periodic square-integrable functions, considered over the real line $\mathbb{R}$) are two subspaces of the space of ...
5
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0answers
147 views

Modified energy method for transformed Fokker-Planck equation (tricky integration by parts…)

I came across Villani's paper titled "Hypocoercive diffusion operators" and couldn't figure out a computation that is skipped in that paper. Specifically, consider the following transformed ...
5
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178 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): $$f^{(s)}(x)=\sum_{m=0}^{\infty} \binom {s}m \sum_{k=0}^m\binom mk(-1)^{m-k}f^{(k)}(x)$$ $$...
5
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0answers
286 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
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0answers
99 views

Remainder term in an integral linked to the Riemann zeta function

Sorry if this is not research level, but the following problem occurs in my own research: it is trivial to show that for $k\ge2$ integral we have $\zeta(k)=(1/(k-1)!)\int_0^\infty t^{k-1}/(e^t-1)\,dt$ ...
5
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0answers
533 views

why do we mainly integrate with respect to martingales?

Although my resarch focuses on PDEs (optimal transport, these days), I am currently trying to learn stochastic calculus and integration. I am just beginning in this topics, but I was wondering: why do ...
5
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0answers
357 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. ...
5
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0answers
193 views

The integral of $\exp(-|x-a|)$ over an even dimensional sphere

I'm after a reference for an integral. For $m$ a positive integer and $R>0$ let $S^{2m}_R\subset \mathbb{R}^{2m+1}$ denote the radius $R$ sphere of dimension $2m$. Suppose that $a$ lies inside ...
5
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0answers
209 views

Contour integral $\int_{\gamma}e^{-t\mu}(-\mu)^{-m}\log\sqrt{-\mu}d\mu$

Motivation: In my research I try to find the asymptotics of the heat trace $\text{tr}e^{-t\Delta}$ as $t\to0$, where $\Delta$ is Laplace operator on a manifold with singularities. First I find the ...
5
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0answers
355 views

From Selberg integral to Dyson integral

My question is about the derivation from Selberg integral to Dyson integral in this paper: Selberg integral : $$ S_n(\alpha,\beta,\gamma) := \int_0 ^1 \cdots \int_0 ^1 \prod_{j=1}^n t_j^{\alpha-1}(...
5
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0answers
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Hubbard-Stratonovich Transformation

Hello, The Hubbard-Stratonovich transformation $\exp(x^2) = \frac{1}{\sqrt{4 \pi}} \int_{-\infty}^{+\infty} du \exp(-\frac{u^2}{4} - xu)$ allows one to wirte the exponential of a the square of a ...
5
votes
1answer
550 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 $...
5
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1answer
2k views

Fourier transforms via Kurzweil-Henstock integral on locally compact commutative groups

Is it possible to define Fourier transforms on locally compact commutative groups using the Kurzweil-Henstock integral instead of the Lebesgue integral?
4
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0answers
114 views

Length of the arc of a Fourier series

I'm working modeling the behavior of periodic variable stars and I have a question about reducing the expression of a parameter involved in this analysis. Let $f(t)$ be a Fourier series define as: $$f(...
4
votes
0answers
140 views

The Poincaré Lemma

Let me consider an $L^1(\mathbb R^N)$ function $f$ such that $$ \int_{\mathbb R^N} f(x) dx =0. $$ Then I claim that the $N$-form $f(x) dx_1\wedge\dots\wedge dx_N$ is closed, i.e. there exists a vector ...
4
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0answers
76 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 ...
4
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185 views

Can this integral be made nonpositive?

Let $M^2 \subset \mathbb{S}^3$ be a closed and orientable embedded (and minimal, if important) surface. Choose a unit normal vector field $\eta: M \to \mathbb{S}^3$ along $M$ and a point $p_0 \in \...
4
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0answers
90 views

Integration on a family of differential forms

Let $X$ be a smooth manifold, and denote by $\Omega^*(X)$ the set of all smooth differential forms on $X$. Assume we have a family of differential forms $\omega_t \in \Omega^*(X)$, $t\in E$, ...
4
votes
0answers
165 views

Computing the volume of intersection between a ball and a box

$C$ is the set of vectors which are coordinate-wise less than $\overline{c}\in [-1,1]^d$ and greater than $\underline{c}\in [-1,1]^d.$ Is there a procedure not exponentially complex in $d$ that ...
4
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0answers
256 views

Is the following integral positive or not?

Let $n$ be a given even positive integer. We have the following integral \begin{eqnarray} &&\int_0^1\cdots\int_0^1\prod\limits_{i=1}^n\prod\limits_{j=1}^n(x_i-y_j)dx_1\cdots dx_ndy_1\cdots ...
4
votes
0answers
311 views

Fractional integral inequality (Hardy-Littlewood-Sobolev)

I am investigating the following integral \begin{equation} I^*(x) = \int_{\mathbb{R}} \frac{f(y) \ln |y-x| }{|y - x|^{\mu}} \, dy \end{equation} where $f \in L_p(\mathbb{R})$, $ 1 < p < q <...
4
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0answers
232 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 ...
4
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0answers
225 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+...
4
votes
0answers
254 views

Verifying a source that lacks a citation

In this German Mathematics Wikibook page, formula $0.5$ lists the following equation $$\int_0^1 \sin(\pi x) x^x (1-x)^{1-x} \ \mathrm{d}x = \frac{\pi e}{24}$$ as supposedly attributed to Ramanujan (...
4
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167 views

Divergence theorem on stratified spaces

It is very common in physics and engineering to apply the divergence theorem to compact spaces whose boundary is not smooth. For example, in the wikipedia link I just gave, the picture illustrating ...

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