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**11**

votes

**0**answers

229 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 ...

**10**

votes

**0**answers

288 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 ...

**9**

votes

**0**answers

64 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

388 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, ...

**6**

votes

**0**answers

91 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:
...

**6**

votes

**0**answers

100 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 ...

**4**

votes

**0**answers

75 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 ...

**4**

votes

**0**answers

287 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

163 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

132 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

247 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}
...

**4**

votes

**0**answers

146 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

182 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 $\delta$:
...

**3**

votes

**0**answers

127 views

### Computing local volumes : the case of Hecke p-adic subgroups

I am quite interested in knowing how to compute some volumes of groups defined on local fields $K$, mainly in order to evaluate the identity term in trace formulas. It is something well done in the ...

**3**

votes

**0**answers

161 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

102 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

90 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

233 views

### How is the deconvolution of a fat gaussian from a polynomial derived?

We have a 2D order-2 polynomial, a Gaussian and a 'box' indicator function. Let:
$\begin{eqnarray}
p(x,y) &=& c_0+c_1x+c_2y+c_3xy+c_4x^2+c_5y^2+c_6xy^2 \\
G(x,y) &=& ...

**3**

votes

**0**answers

230 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

905 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

148 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

30 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 ...

**2**

votes

**0**answers

50 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

87 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

87 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
...

**2**

votes

**0**answers

46 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 ...

**2**

votes

**0**answers

54 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.
...

**2**

votes

**0**answers

52 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 {
...

**2**

votes

**0**answers

204 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

226 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

176 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

128 views

### Regularity of measures in the theorem of Riesz

There are two concurrent theories of measure/integration on a locally compact topological spaces: either as positive linear forms on the space of continuous functions with compact support, or as Borel ...

**2**

votes

**0**answers

83 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

123 views

**2**

votes

**0**answers

109 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 ...

**2**

votes

**0**answers

348 views

### How to perform this matrix integral?

Edit: some backgrouds added.
In quiver matrix model which is reviewed DV or CKR, the path integral reduce to the matrix integral
$$Z \sim \int \prod_{i=1}^r d\Phi_i \prod_{<a,b>} dQ_{ab} ...

**2**

votes

**0**answers

222 views

### An integral inequality

Let $g:\mathbb{R}\rightarrow\mathbb{R}$ be bounded with derivative $g'$. I have shown that the following inequality holds for all $w\in\mathbb{R}$,
...

**2**

votes

**0**answers

255 views

### Definite integral probably equal to zeta with known (but unusable) closed form for the indefinite integral

Related to this and
this questions.
Basically got definite integral that experimentally equals
$\zeta(s)$ both numerically and symbolically.
Closed form for the indefinite integral is known, but I ...

**2**

votes

**0**answers

310 views

### Expectation involving the ratio of normal pdf to normal cdf?

i need to calculate some expectations which involving the ratio of normal pdf to normal cdf.
Specifically, they are $E\{\phi(x)/\Phi(x)\}$ and $E\{x\phi(x)/\Phi(x)\}$ where $x\sim N(0,1)$.
Written ...

**2**

votes

**0**answers

200 views

### Marginalizing multivariate normal over defined interval

Hello everyone,
I am trying to obtain an analytic expression for the following Gaussian integral
$$\frac{1}{\sqrt{(2 \pi)^n |\Sigma|}} \int \kern-0.2em \cdots \kern-0.2em \int d\mathbf{x}_{\sim i} ...

**2**

votes

**0**answers

224 views

### Coutour Integral of Gamma Functions

How do I solve the Integral
$$ \frac{1}{2\pi j} \oint
\frac{b^{ - s} \Gamma[2 + i - s] \Gamma[s] \Gamma[-1 - i + s]}{
(2 + i - s) \Gamma[3 + i - s]} \:\mathrm{d}s$$
This integral is an inverse ...

**1**

vote

**0**answers

51 views

### Generalized “elliptic integrals”

I am interested in evaluating the following type of integrals. Here we a polynomial $q(x)$ of degree $d \geq 2$ with no non-negative roots. Then is there a name for integrals of the shape
...

**1**

vote

**0**answers

52 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} ...

**1**

vote

**0**answers

50 views

### A slight generalization of Mehta's integral.

I am trying to find the value of following integral
$$\int_{-\infty}^{\infty}\dots\int_{-\infty}^{\infty}\prod_{i=1}^ne^{-\frac{t_i^2}{2}+\alpha_i t_i}\prod_{1\le i<j\le ...

**1**

vote

**0**answers

113 views

### Is the implication ($f$ is Riemann integrable over $D_1$ and $D_2$) $\Rightarrow $ ($f$ is Riemann integrable over $D=D_1\cup D_2$) true?

Let $D_1,D_2$ be a bounded subset of $\mathbb{R}^n$ and $\partial D_1,\partial D_2$ are both of Lebesgue measure zero (that is to say: $D_1,D_2$ are Jordan measurable).
Also, let $f:D_1\cup ...

**1**

vote

**0**answers

250 views

### convolution integral involving modified Bessel functions of the first kind

I'm stuck with this convolution integral ($z \geq 0$)...
\begin{equation}
f_{Z}(z)=\int^{\infty}_{-\infty}f_{1}(x)f_{2}(z-x)dx = \mbox{ } ???
\end{equation}
which represents the pdf of the sum $Z = ...

**1**

vote

**0**answers

81 views

### Convergence of solutions of the volterra integral equation with convergent kernels

Consider the following Volterra integral equation
$$
g(t) = \int_0^t K_n(t,s)w_n(s) ds
$$
where g(t) and K_n(t,s) are continuous and $K_n(t,s)\geq K_{n+1}(t,s)$ for all $t,s$. Moreover, $K_n(t,s)$ ...

**1**

vote

**0**answers

145 views

### Reference : Special case of Banach-valued function integration by parts

Let $u \in L^p (0,T; L^1(\Omega))$ with $\partial_t u \in L^p(0,T; L^1(\Omega))$ and $v \in L^q (0,T; L^\infty(\Omega))$ with $\partial_t u \in L^q(0,T; L^\infty(\Omega))$ (with $1/p+1/q=1$ and $p \in ...

**1**

vote

**0**answers

100 views

### Adelic integral factorization

In order to calculate Tamagawa numbers, I need to justify that for a nice (say Schwartz-Bruhat) function, the following identity holds :
$$\int_{\mathbf{A}^2} f(x)dx = \int_{SL_2(\mathbf{A})/SL_2(K)} ...