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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11
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0answers
257 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
votes
0answers
325 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}...
10
votes
0answers
213 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 ...
9
votes
0answers
75 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} \...
7
votes
0answers
412 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, $f^{-1}:[0,\infty)\to[0,\infty)$...
6
votes
0answers
127 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^{\...
6
votes
0answers
220 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
votes
0answers
111 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
votes
0answers
110 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 ...
5
votes
0answers
117 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 ...
5
votes
0answers
305 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 ...
4
votes
0answers
208 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
votes
0answers
135 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 ...
4
votes
0answers
86 views

Numerical integration error bounds on the unit sphere

A sequence of points $x_1,x_2,\dots$ on the unit sphere $S^{D-1}$ is said to be uniformly distributed if \begin{align} \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{j=1}^N f(x_j) = \int_{x \in S^{D-...
4
votes
0answers
180 views

Is there a coordinate free proof of the Morrey--Kohn--Hormander identity?

The Morrey--Kohn--Hormander identity is the key to proving vanishing/existence results on bounded pseudoconvex domains in $\mathbb{C}^n$, or more generally, Stein domains. See, for instance, the ...
4
votes
0answers
155 views

The Haar integral on uniform spaces

Does anybody know what the current research status is on the topic of Haar measures on uniform spaces? I'm specifically interested in compact uniform spaces and its corresponding Haar probability. As ...
4
votes
0answers
266 views

Inverse of matrix-valued function

Given $c>0$. Let $\gamma_c:{\cal M}_{k \times k}^+\mapsto {\cal M}_{k \times k}^+$ is a function defined by \begin{equation} \gamma_c(\Omega)=\frac1{\sqrt{(2\pi)^{k}|\Omega|}}\int_{\mathbb{R}^k}\{(-...
4
votes
0answers
152 views

About arithmetic-geometric mean

It's well known that if we set $a_0=x \geq 0, \ g_0=y \geq 0$, and $$ a_{n+1}=\dfrac{1}{2}(a_n+g_n), \ g_{n+1}=\sqrt{a_n g_n} ,$$ then both $\{a_n\}$ and $\{g_n\}$ will converge to $AGM(x,y)$. ...
4
votes
0answers
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?
3
votes
0answers
76 views

interpretation of a singular integral

There is a post on MSE about a principal value integral in this paper. It has not received much attention even with a bounty, and since it concerns a published paper, I believe this is a better forum ...
3
votes
0answers
80 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 ...
3
votes
0answers
80 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 ...
3
votes
0answers
90 views

Using and understanding the Atiyah-Bott localization theorem/integration formula

I posted this on r/math, but was told I might have better success here given the level of the question. Basically, I need to learn how to use the localization theorem to compute integrals on ...
3
votes
0answers
85 views

Numerical inversion involved confluent hypergeometric (1F1) (or Kummer function)

Edit: The question is solved !! The code is actually correct. There is not error in the codes. I miss-used it. Thank you for your attention : ) This problem arises when I tried to compute the valua ...
3
votes
0answers
56 views

Dual cone of 'positive' Bochner integrable functions

If we consider the space of integrable functions $L^1([0,1];\mathbb{R})$, it can be ordered by the convex cone of positive integrable functions $L^1([0,1];\mathbb{R}_+)$. It is known that the ...
3
votes
0answers
204 views

A difficult integral which the Risch algorithm shows is not elementary

For reasons which aren't conceptually related to the problem a few of my colleagues and I are in need of finding an expression for the following integral in terms of $a$ and $\delta$: $$\int_{\delta}^...
3
votes
0answers
172 views

An upper bound for a average of a function in $L_{p}([0,1))$

Suppose that $ f $ is $ 1 $-periodic and that $ f \in {L^{p}}([0,1) $, where $ p > 1 $. Let $$ (D_{n})_{n \in \mathbb{N}_{0}} = \left( \left\{ I^{n}_{j},~ 1\leq j \leq 2^{n} \} \right\} \right)_{n ...
3
votes
0answers
111 views

motivic integration and jacobian ideal

When we consider the change of variables in motivic integration, we have a birational map $f:Y\rightarrow X$ with Y smooth and we have to consider two invariants the order of the Jacobian ideal of $X$ ...
3
votes
0answers
107 views

What are the criteria for an elementary function to be infinitely integrable in elementary functions?

What features of elementary functions define a class of functions whose consecutive indefinite integration also gives an elementary function? Is there a way to check whether a given elementary ...
3
votes
0answers
239 views

When does this method for integrals of fractional/integer parts work?

In a question Agno suggested an interesting way to compute $\{x\}$ and $\zeta(s)$. Define $$ F(x) = \{x\} = x - \lfloor x \rfloor = \frac{i \, \log\left(-e^{\left(-2 i \, \pi x\right)}\right)}{2 \, \...
3
votes
0answers
1k views

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 ...
3
votes
0answers
164 views

An isoperimetric inequality for “order” polytopes

I am looking for an isoperimetric inequality for order-like polytopes. An order polytope $K\in \mathbb{R}^n$ is defined by a set of linear inequaities: $$ \forall i \; 0\leq x_i \leq 1 $$ and $ ...
2
votes
0answers
50 views

Do any of these integrals have closed forms in terms of special functions?

I've been looking at nonelementary integrals of the form $\frac{1}{f(x) + g(x)}$, where $f$ and $g$ are simple but different enough to be interesting. Mathematica can't evaluate any of these integrals,...
2
votes
0answers
167 views

A question about multidimensional integral

Consider the function $$\Omega(N,E)=\int dE_1 \int dE_2 \cdots \int dE_N \Omega_1(E_1)\Omega_2(E_2) \cdots \Omega_N(E_N)\delta(E-E_1-E_2\cdots -E_N)$$ Is there a necessary condition on the $\Omega_i$'...
2
votes
0answers
247 views

Is there a Bayesian theory of deterministic signal? Prequel and motivation for my previous question

This is a prequel to my question: What's the probability distribution of a deterministic signal? (functional integrals in probability theory) Clearly my question looks at the same time fairly ...
2
votes
0answers
731 views

What's the probability distribution of a deterministic signal or how to marginalize dynamical systems? (functional integrals in probability theory)

In many signal processing calculations, the (prior) probability distribution of the theoretical signal (not the signal + noise) is required. In random signal theory, this distribution is typically a ...
2
votes
0answers
170 views

How to analytically evaluate this n-dimensional iterated integral?

I would very much appreciate any suggestions and/or pointers to references relevant for the analytic evaluation of the following n-dimensional iterated integral $$\int_{-\infty}^{+\infty}dx_1\int_{-\...
2
votes
0answers
81 views

Summation of an integral involving Laguerre polynomial and Bessel function

In an engineering setting, I reduced my problem to calculating the following sum: $$\sum_{n=0}^\infty \frac{n!}{(k+n)!}\left[\int_0^a \left(\frac{x}{u}\right)^kL_n^{(k)}\left(\frac{x^2}{u^2}\right)\...
2
votes
0answers
53 views

Integral of a parametrized commutator

I am trying to solve the following integral $$ \int_{-1}^{1}\;db\;||[t_{b}(A),J]||_{F}^{2} $$ where $t_{b}$ is the entrywise threshold of the matrix A ($0$ if $a_{ij}<b$, $a_{ij}$ if $a_{ij}>b$, ...
2
votes
0answers
123 views

Closed form for Gaussian-like integral

Let $X$ be a tall $M\times N$ matrix with complex elements, i.e. $M >> N$, and $h$ an $N\times 1$ complex vector. Furthermore, $c$ is an $M\times 1$ vector, $\Sigma_h$ an $N\times N$ diagonal ...
2
votes
0answers
97 views

Approximating a divergent integral with modified Bessel functions of the first and second kinds

I am a physicist who needs to evaluate the following (divergent at the origin) integral involving the modified Bessel functions of the first and second kinds $$I = \int_0^{\infty} \frac{\cos(ax)}{x} ...
2
votes
0answers
91 views

Swapping sums and integration for a kernel in Fourier space (the non absolutely convergent case)

Under what conditions on $c_{r}^{m}$ does $$\int_0^{2\pi} k(p,q)\exp(-inq)dq=\sum_{r=0}^{\infty}c_{r}^{m}\exp\left(-imp\right) \text{ in } L^2_{per}$$ hold for $$k(p,q)=\sum_{r=0}^{\infty}\sum_{l=...
2
votes
0answers
67 views

expectation involving normal pdf and Rayleigh distribution

I need to calculate following definite integral \begin{equation*} \frac{1}{2\pi }\int_0^\infty \frac{x^2 e^{-x^2/\sigma^2 } }{\sigma} \frac{e^{-\frac{\lambda}{{ax^2+b}}}}{\sqrt{ax^2+b}} ~~dx. \end{...
2
votes
0answers
64 views

Trigonometric multiple integral identity

How this alleged multiple integral identity can be proved? $$\int\limits_{-\infty }^{\infty }ds_{1}\int\limits_{-\infty}^{s_{1}}ds_{2}\cdots \int\limits_{-\infty }^{s_{2n-1}}ds_{2n}\;\cos { (s_{1}^{2}-...
2
votes
0answers
225 views

Inequality with CDF of order statistics

here is a problem I have been struggling with for a while now. This is for a paper I am working on. Any help would be appreciated! Here we go: Each bidder's valuation $\theta _{i},$ $i=1,...,N$, is ...
2
votes
0answers
228 views

A integral equation with Discrete to result by inverse problem

Problem I asked a question at Math.SE last year and later offered a bounty for it, but it remains unsolved even in the simplest case. So I finally decided to repost this case here, (I know the ...
2
votes
0answers
235 views

Integral involving a Meijer-G function

I am having trouble with calculating the following integral: $$ \int_{0}^{\infty} \ln{(1 + \alpha x)\, G^{k,0}_{k,k}\left[e^{-x}\left|^{(a_k)}_{(b_k)} \right. \right]} \, dx, $$ where $\alpha > ...
2
votes
0answers
108 views

Mean and variance of a general multivariate skew normal distribution

I have a problem about a general multivariate skew normal distribution. There is a $p\times 1$ vector, $\mathbf{y}=(\mathbf{y}_1',\mathbf{y}_2',\ldots,\mathbf{y}_n')',p>n$, which has the density as ...
2
votes
0answers
133 views

An integral problem related to matrix determinant

I am stuck in an integral problem: $$\int_{\mathbb{R}^d}(\det(\mathbf{A}+k(\mathbf{x}-\mathbf{y}_0)(\mathbf{x}-\mathbf{y}_0)^T))^{-r_1}\prod_{i=1}^n(\det(\mathbf{B}+(\mathbf{x}-\mathbf{y}_i)(\mathbf{...
2
votes
0answers
110 views

How to show the identity $\int_0^T \int_{\Gamma(t)}f(s,t)\;dsdt = \int_S f(\sigma)(1+(\mathbf w \cdot \mathbf n)^2)^{-\frac{1}{2}}\;d\sigma$?

I am reading this paper. Let $\Gamma(t)$ be a smooth closed connected oriented hypersurface for each $t \in [0,T]$. Define the set $$S = \bigcup_{t \in (0,T)}\Gamma(t) \times \{t\}.$$ On page 5 of ...