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1
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36 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
0answers
25 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 ...
7
votes
1answer
258 views

Integral formula for $\int_{0}^{\infty}e^{-3\pi x^{2}}((\sinh \pi x)/(\sinh 3\pi x))\,dx$ by Ramanujan

The following is a re-post from MSE because I did not get any answer even after offering a bounty. Towards the end of G. N. Watson's (one of the joint authors of famous book "A Course of Modern ...
2
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0answers
77 views

What am I missing in this highly oscillatory integral? [closed]

I want to numerically integrate this equation (in python): $\int_{0}^{\infty}{\rm d}k f(k) J_v(r k)J_v(s k) $, where f(k) is a non-smooth function, and $J_v$ are the Bessel function of the fist ...
0
votes
1answer
131 views

Is the following “section-wise” defined function measurable in the product space?

I asked this question in mathstackexchange a couple of days ago. Almost right after posing it a partial (affirmative) answer came to my mind in the following form Proposition: Assume that ...
0
votes
2answers
63 views

Class of analytically-integrable divergence-free vector fields?

Is there an "interesting" class of analytically-integrable, divergence-free vector fields over $\mathbb{R}^2$ and/or $\mathbb{R}^3$? That is, I'm looking for a large class of vector fields given by ...
11
votes
2answers
415 views

On the convexity of certain integrals involving Bessel functions

Let $n\geq 0$ be an integer and let $J_n=J_n(r)$ denote the usual Bessel function (of the first kind) of order $n$ i.e. one of the solutions to Bessel's differential equation ...
2
votes
1answer
78 views

Asymptotic expansion of a sequence given by an integral with reciprocal Gamma function

I would like to know the asymptotic expansion of the sequence of positive numbers given by $$I_{n}:=-\int_{0}^{1}\frac{n^{x-1}}{\Gamma(x-1)}dx,$$ for $n\rightarrow\infty$. One can easily derive an ...
0
votes
1answer
115 views

Unimodality of a certain parametric integral

Suppose $f: [0,1] \to [0,\infty)$ is a smooth, concave and strictly increasing function satisfying $f(0)=0$. Is it true that the map $$ F(y) = \int_0^1 \frac{y^{3/2}}{(y+f(x))^2} dx $$ has exactly one ...
3
votes
2answers
205 views

Monte Carlo integration of Gaussian integrals

I was doing a physical problem, and then it comes to this Gaussian integral. The dimension of the integral is very large (dimension = 300~600), and it is difficult to find the maximum of the ...
9
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0answers
57 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} ...
5
votes
1answer
110 views

Why is it possible to normalize the Haar measure on the quotient?

I just asked a question which is related to the one I'm about to ask, but I realized my question can be reduced to the following: let $G$ be a locally compact abelian group with Haar measure $\mu$, ...
1
vote
0answers
53 views

Two integrals involving Legendre Functions

I have two Integrals which I want to identify with a simpler functions (if possible). Firstly, $$\int_{1}^{\infty} \frac{1}{\sqrt{x^2-1}}Q_{-1/2+k}\left(\frac{x\cdot a-b}{\sqrt{x^2-1}} \right)\cdot ...
0
votes
0answers
148 views

How to evaluate the following integral related to exponential distribution

I would like to evaluate the following integral related to the exponential distribution. Let $\delta>1$, and $0<p<1$ and $0<\epsilon<1/\delta$ be reals. We have that $$ ...
1
vote
0answers
109 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 ...
0
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0answers
38 views

Interchange limit with integral for subsequence in subsequence of general distribution functions

I asked this question on math.stackexchange a few days ago but didn't get any response, so I thought I would try here. I'm trying to find a solution for the following problem: Let ...
0
votes
2answers
181 views

How do I Calculate :$\int_{0}^{1}x^{k}\psi(x)dx$ where $k\geq 3$ is an integer?

How do I Calculate, if possible, in terms of well-known constants the integral : $\int_{0}^{1}x^{k}\psi(x)dx$ , where $k\geq 3$ is an integer ? note: $\psi(x)$ is digamma function. Any help would ...
2
votes
0answers
39 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
3answers
324 views

Positivity of the Coulomb energy in two dimensions

In dimensions $d\geq 3$ the Coulomb energy is always non-negative (since the Fourier transform of $\frac{1}{\|\cdot\|^{d-2}}$ is non-negative). What can one say about positivity properties of the ...
3
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1answer
78 views

Hyperelliptic generalization of Euler's formula

Are there any hyperelliptic generalizations of the following formula, first proved by Euler in 1782, ...
6
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2answers
344 views

Interesting triple integral

Some time ago I stumbled on an alleged identity $$\int\limits_0^\infty \frac{dx}{x} \int\limits_0^x \frac{dy}{y} \int\limits_0^y \frac{dz}{z} [\sin{x}+\sin{(x-y)}-\sin{(x-z)}-\sin{(x-y+z)}]= ...
11
votes
1answer
725 views

Calculate in closed form $\int_0^1 \int_0^1 \frac{dx\,dy}{1-xy(1-x)(1-y)}$

The following question has a 500 points bounty on MSE that soon comes to an end, and no answer as expected was given yet. How would a professional solve the problem? Wish you succcess. ...
5
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3answers
696 views

Action Integral

In the theory of action-angle variables, you wind up having to solve integrals with a characteristic square-root behavior near the endpoints to express the action in terms of the orbital quantities. ...
0
votes
0answers
48 views

differentiating under the integral sign Henstock integral

Does anyone know, where I can find the necessary and sufficient conditions for differentiating under the integral sign in case of Henstock integral? Here is the title that says so, bud i think there ...
0
votes
0answers
75 views

Closed form for convolution of two-dimensional Gaussian with characteristic function of a disk

Is there a closed form expression for the convolution of a two-dimensional (elliptical) Gaussian function with the characteristic function of the interior of an ellipse? The motivation is that I have ...
6
votes
1answer
177 views

Henstock, Differentiation under the integral sign

Does anyone know, where I can find the proof of necessary and sufficient conditions for differentiating under the integral sign in case of Henstock integral? Here are the theorems but not all the ...
4
votes
1answer
120 views

Asymptotics of Fresnel integrals

It is known that \begin{equation*} I(p) = \sqrt { \frac {4 \mathrm{i} p} {\pi}}\int \limits _{-\infty} ^{\infty} \mathrm{e}^{- \mathrm{i} p x^2} \varphi (x) \mathrm{d}x \end{equation*} is a bounded ...
12
votes
1answer
1k views

Naive definition of surface area doesn't work?

A first stab at a definition of surface area might go like this: Let S be a surface. Select finitely many points from S and make a bunch of triangles having these points as vertexes. Add up the ...
9
votes
2answers
226 views

Computing Gauss Legendre Quadrature for Large N

I've been scanning across the web, and haven't found a good method to compute the Gauss Legendre abscisas and weights $\{ x_j, w^j \} _j$ for large N. My question is how to do it, and why should it ...
2
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0answers
44 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. ...
0
votes
0answers
25 views

Rate of convergence of Riemann sum of quasi-regular functions

The following result is well-known (I consider the 3-dimensional case only): Theorem: if $f \in H^s(\mathbb{R}^3)$ with $ s > 3/2$ is compactly supported, then $$ \left| \int_{\mathbb{R}^3} f - ...
2
votes
2answers
127 views

Integrating a barycentric monomial over a simplex

Are there standard formulas for the integral over a simplex of a monomial in the barycentric coordinates? Can someone supply a reference? I think I have seen such formulas, but I am unable to find ...
5
votes
1answer
517 views

Integral of a product of Laguerre polynomials

In order to estimate the non linear term in a particular PDE, I have to decompose $L_k^\alpha(x)^3\cdot x^{-\delta}$ (with $0<\delta<\alpha+1$) into a basis consisting of Laguerre polynomials ...
2
votes
1answer
76 views

Solving complicated equation involving integral of error functions

I've been trying to solve the following equation for $\sigma$ $$\sigma^2 = \int_0^1 \left\{ \frac{1}{2} \operatorname{erfc} \left[ \frac{x + B}{\sqrt{2 (Rp - 1)} \, \sigma} \right] + \frac{1}{2} ...
4
votes
0answers
65 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 ...
2
votes
1answer
304 views

Real and imaginary part of an holomorphic function

I guess this could be a very elementary question. Anyway I can not find an answer in literature. Let $f:U\rightarrow\mathbb{C}$ be an holomorphic function on an upen subset $U\subseteq\mathbb{C}$. ...
18
votes
4answers
3k views

Integrals from a non-analytic point of view

I've mentioned before that I'm using this forum to expand my knowledge on things I know very little about. I've learnt integrals like everyone else: there is the Riemann integral, then the Lebesgue ...
1
vote
0answers
157 views

What does the Riemann–Stieltjes integral measure? [closed]

The Riemann–Stieltjes integral is a generalization of the Riemann integral, and has a definition based on a sum analogous to the Riemann sum: $$ S(P,f,g) =\sum_{k=1}^{n} f(x_k)\Delta g(x_k) $$ where ...
9
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0answers
213 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 ...
2
votes
1answer
134 views

acoustic dipole volume integral w/ dirac delta?

I have an acoustic research problem that leads to the following integral formulation: \begin{align} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}p(\mathbf{y},\tau)\frac{\partial}{\partial ...
6
votes
0answers
88 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: ...
1
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0answers
93 views

What is $\int (1-e^{-x})^n dx$? [closed]

For my purposes, $n$ is a non-negative integer, and $x > 0$. I didn't know how to evaluate this integral, so I plugged it into Mathematica. It told me the solution is $(-1)^n B(e^x; -n, n+1)$ I ...
2
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1answer
67 views

Asymptotic expansion of a Laplace-type integral with a “manifold of maxima”

Moved this here from MathSE because I had no luck there and suspect the question may be harder than I first thought. Consider the integral $$ I(\alpha)=\int_0^1 dx_1 \int_0^1 ...
3
votes
1answer
382 views

Help with the integral $\int_{0}^{\infty}\log\left(1+\frac{s^{2}}{4\pi^{2}} \log^{2}(1+ix)\right ) e^{-2\pi nx}dx$

We have the integral : $$\int_{0}^{\infty}\log\left(1+\frac{s^{2}}{4\pi^{2}} \log^{2}(1+ix)\right ) e^{-2\pi nx}dx$$ Where s is a complex parameter, and n is a positive integer. The integral ...
2
votes
1answer
190 views

A conjecture regarding the integral of the square of an entire function

Can some help me prove or disprove the following assertion which I encountered in research? Thanks! Let $f:\mathbb R\to\mathbb R$ be an analytic function. If for $\forall c > 0$, we can find some ...
4
votes
3answers
317 views

Area of metric spheres in Riemannian manifolds

I am trying to estimate the integral $\int \mathbb{e} ^{-d(x_0,x)^2} \mathbb{d}x$ on a Riemann manifold $(M,g)$, for some arbitrary fixed $x_0 \in M$ and $d$ the usual distance. The only thing that I ...
10
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0answers
196 views

Are there integral representations of the Mertens constant?

It is well-known that the Euler constant $\gamma=\lim\limits_{n\to \infty}\biggl( \sum\limits_{k\le n}\dfrac{1}{k}-\ln{n}\biggr ) $ has a bunch of integral representations, e.g. ...
0
votes
0answers
532 views

Integration involving the complete elliptic integral of the first kind K(k)?

Is there any reference showing how to do definite integrals involving the complete elliptic integral of the first kind K(k)? Something like $\int_0^1 K(k) dk $ $\int_0^1 k^nK(k) dk$ $\int_0^1 ...
4
votes
1answer
223 views

Motivic integration in positive characteristic: how much is known?

It seems that in papers on motivic integration people usually assume the base field to have characteristic $0$ (and algebraically closed?). My question is: how much can one prove over a positive ...
2
votes
0answers
43 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 { ...