**11**

votes

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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 ...