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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1answer
41 views

Proving that an integral related to order statistics is increasing in a certain parameter

Let $f$ and $F$ denote, respectively, the pdf and cdf of a probability distribution on $\mathbb R$. Take any natural $n\ge3$ and any real $a$ and $c$ such that $a\le c$. Does it always follow that $$...
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78 views

Summing a series of integrals [on hold]

EDIT: This IS related to my research (investigating representations of the harmonic mean)and I gave the wrong formula for the sum the first time around. The sum formula has been amended below. I ...
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0answers
54 views

The necessary and sufficient condition for exchanging the integration process and the limit process [closed]

We only consider real functions here. It can be shown that \begin{eqnarray*} \int_{0}^{\infty}dx\frac{\sin x}{\sqrt{x}} & = & \int_{0}^{\infty}dx\frac{\cos x}{\sqrt{x}}=\sqrt{\frac{\pi}{2}} \...
6
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2answers
115 views

Legendre Polynomial Integral

How can I evaluate $$ \int_{-1}^1 P_n(x)P_l(x)x^k dx $$ when $k$ is even? Or what might be a source where I could find integrals like this?
10
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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 ...
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1answer
92 views

How to prove this integral equality with limits

The following result is from a paper. The author says it is not hard to show that: $$\lim_{t\to 1}\dfrac{1-t}{\sqrt{1+pt}}\int_{0}^{t}\dfrac{a(1+pa)}{(1-a)^2}\left(4a\left[1-\left(\dfrac{1-t}{1-a}\...
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0answers
226 views

Prove this function is increasing

I'm stuck in showing that the following function is increasing over the domain $\left[0,\hat{b}\right]$: \begin{eqnarray} \Pi\left(z\right) & = & \int_{0}^{\phi\left(z\right)}\int_{x}^{\bar{x}...
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1answer
102 views

Convergence of Riemann integrals that do not hold for Lebesgue integrals

I am interested in convergence results that are true for Riemann-integrable functions but fail for Lebesgue-integrable functions. I know three of these, which happen to be closely related. ...
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20 views

Splitting the region and estimating fractional Sobolev norms

x-post from math.stackexchange (http://math.stackexchange.com/q/1836766/349671), since the question arose from reading through a scientific paper: I've been reading the paper "On the Bourgain, Brezis,...
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0answers
29 views

Integral of Daubechies wavelets [closed]

For Daubechies wavelets according to this paper (above eq 19) this relation holds $$ \int_{-\infty}^{-\infty} \phi(2x-i)\phi(2x-j)dx = \frac{1}{2} \int_{-\infty}^{-\infty} \phi(x-i)\phi(x-j)dx $$ ...
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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^{\...
4
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1answer
66 views

Usable Change-of-Variables Formula for Hausdorff Measure

Let $H^{s}$ be the $s$-dimensional Hausdorff measure, let $D$ be a nonsingular matrix. Consider the change of measure formula: $$ \int\limits_{A} f(Dx) \; \mathrm{d}H^{s}(x) = \int\limits_{ D A} f(y)...
2
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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,...
4
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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 (...
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0answers
33 views

Compute Mixed Volume with Respect to Some Regular Sets

Let $( \mathbb{R}^n, \mathcal{B}, \gamma)$ be a measure space where $\mathcal{B}$ is the Borel sigma algebra and $\gamma$ is a continuous measure. For $A, B\in \mathcal{B}$ that are convex, the mixed ...
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0answers
41 views

Equivalent Definitions of the Gaussian Surface Measure for Regular Sets

I wonder if the following definitions of the Gaussian surface measure are equivalent. First, let $\mathbb{R}^n$ be the Euclidean space and $A \subseteq \mathbb{R}^n$ be a sufficiently regular set, e....
3
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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 ...
11
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1answer
171 views

Harmonic analysis, compute that this integral tends to $0$

We have the following setting. $U$ is a bounded Lipschitz domain in the complex plane. Consider the following classical Dirichlet problem for the Laplace operator: $$\begin{align} \Delta{}u&=0 \...
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2answers
244 views

How can we obtain the $-\frac{4\pi}3\mu(x)$ term?

Given the expression $$K_{ik} := \frac{\partial}{\partial x_k} \int_{\mathscr X} \frac{y_i-x_i}{|y-x|^3} \mu(y) dy,$$ where $\mathscr X=\mathbb R^3$, how does one derive the expression \begin{align} ...
4
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1answer
67 views

Integral Expression in Complex Dynamics

Let $\phi\in \mathbb{C}(z)$ be a degree $d\geq 2$ rational map, which we can write as $\phi = \frac{f}{g}$ for $f,g\in \mathbb{C}[z]$. Let $\omega_{FS}$ denote the Fubini-Study form on $\mathbb{P}^1(\...
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2answers
382 views

Generalizations of the Euler-Maclaurin Summation Formula

I'm using the Euler-Maclaurin formula in a research I'm working on. However brilliant is the elementary proof found here, I need and want to know more about it. Namely Specifically, I would like to ...
3
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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 ...
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0answers
78 views

Integration and Inverse Function Theorem

Apologies if this sounds too silly for advanced math people here. It's long since I moved from mathematics to medicine and this problem appears in my research. For an $f^{-1}\in C^{1}([a,b])$, ...
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34 views

Generalizing Integration by parts for general bounded continous measure

Consider a probability measure $d\mu = w(t) dt$ with $w(t)\in L^1(I)$, $I =\left[ 0,1\right]$. What are the minimal assumption I can take on two functions $f,g:I\ \to \mathbb{R}$ so that an ...
2
votes
1answer
187 views

Extension of a function from almost everywhere to everywhere

The informal general question is: let $f$ be a "sufficiently nice" function, defined "almost everywhere". Can we develop a method to uniquely extend $f$ to the "remaining" points? Example: Let $f(x)=\...
1
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2answers
299 views

Is the singular integral that come up in circle method independnet of the representatin of the equations?

Let $F_1(\mathbf{x}), F_2(\mathbf{x}) \in \mathbb{Z}[x_1, ..., x_n]$ be a degree $d$ homogeneous polynomial. For the system of equations $$F_1(\mathbf{x})= F_2(\mathbf{x}) =0,$$ we have the ...
0
votes
1answer
167 views

Formula for an integration on $\mathbb{Q} \cap [0,1]$

In order to work with functions defined on $\mathbb{Q} \cap [0,1]$ I would like to define an adapted "integration" formula on this set. I though that following definition could be interesting: $$ \...
5
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1answer
127 views

Showing the positivity of a singular integral that came up in circle method

Let $F(\mathbf{x}) \in \mathbb{Z}[x_1, ..., x_n]$ be a degree $d$ homogeneous form. Let $$ I(\alpha) = \int_{[0,1]^n} e^{2 \pi i F(\mathbf{x}) \alpha} dx_1...dx_n. $$ Then the singular integral is ...
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0answers
50 views

First variation on double integral [closed]

Currently I am trying to fully understand the paper of munk1921. In the derivation of the minimum induced drag theorem it is at one point stated (p.378) that in order to minimize drag the following ...
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1answer
79 views

$p$-adic Dirac measure as a weak limit

The standard Dirac delta is a generalised function (or measure, or distribution...) $\delta(x)$ which can be seen as a weak limit functions $\delta_n(x)$ spiked at the the origin, in the sense that: $...
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0answers
21 views

Bivariate integration with the range of one variable shrinking to a point

Let $f(x,y)$ be a measurable function defined on $(\mathbb{R}^2, \mathcal{B}(\mathbb{R}^2))$. Define $C_{\epsilon} = \{y:d(y, y_0)\leq \epsilon\}$, then can we say for sure that the integration $$ \...
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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 $\Omega_i$'...
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0answers
94 views

Integration of Bessel Function of the first kind

I have encountered a problem during my thesis study. I need to find $F(x)$ in terms of $G(y)$. My statement as follows: $$\int_{0}^\infty F(x)[x^3*B*J_0(xy)+x^4*J_1(xy)]dx=G(y)$$ where $B$ is a ...
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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 ...
4
votes
1answer
269 views

Two similar integrals

Let $n$ be a given even positive integer. We have the following integral \begin{align} \int_0^{\infty}\cdots\int_0^{\infty}e^{-(x_1+\cdots+x_n+y_1+\cdots+y_n)}\prod\limits_{i=1}^n\prod\limits_{j=1}^n(...
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0answers
53 views

Complex integration over 1-singular chain

Let $f$ be a continuous function on $\mathbb{C}$ and assume that $\lim_{z\to \infty} zf(z) = \lambda.$ Let us note for all natural $n$ $$C_n = \{z \in \mathbb{C} : |z|=n\}.$$ Then, a usual fact of ...
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0answers
68 views

A complicated integral inequality

How can we bound this integral: $${\displaystyle \int_{-1}^{1}2\left[\dfrac{1}{4}-\dfrac{1}{4\left(1-\xi^{2}\right)}\left(1-\dfrac{\xi^{2}}{2}\right)^{2}\right]\left(\hat{f}\left(\xi\right)\right)^{2}...
4
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1answer
133 views

When this Ad-invariant function on a Lie algebra is zero?

Let $G$ be a compact Lie group with Haar measure $dg$ and (finite-dimensional real) Lie algebra $\frak g$. Endow $\frak g$ with an $\hbox{Ad}$-invariant norm $\|\cdot\|_{\frak g}$ so that $\frak g$ ...
5
votes
1answer
620 views

Is the following integral nonzero?

Recently I met an integral as follow: $$\int_0^{2\pi}\cdots\int_0^{2\pi}\left(\prod\limits_{1\leq i<j\leq9}\sin\frac{\theta_i-\theta_j}{2}\right)\left(\prod\limits_{i=1}^9(1+\cos(\theta_i-\theta_{i+...
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0answers
64 views

Basic Monte Carlo Integral Approximation

On the very first page of a well-known book on Monte Carlo techniques, there is the following statement. Let \begin{equation} I = \int_D g(\textbf{x})d\textbf{x}, \end{equation} where $D \subset \...
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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 ...
13
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1answer
288 views

Summation of series involving $\sinh$ of a square root

Consider the following series: $$ S = \sum_{\text{odd } n} \sum_{\text{odd } m} \frac{(-1)^{(n+m)/2}}{nm} \frac{\sinh( \pi \sqrt{n^2 + m^2}/2)}{\sinh( \pi \sqrt{n^2 + m^2})} $$ From the physical ...
5
votes
2answers
141 views

Integral over the Cantor's set Hausdorff dimension

As can be seen in the David Morin's Classical Mechanics, there are some scaling strategies in order to calculate the moments of inertia of certain fractals, for example, the Cantor's set has a moment ...
5
votes
1answer
138 views

Multidimensional integrals that diverge by oscillation

It's not hard to extend the theory of integration over ${\bf R}$ so that the integral of any compactly supported function is its usual value, while the integral of $f(t) = \cos (at+b)$ (with $a \neq 0$...
1
vote
1answer
72 views

Definite intergal with two K-Bessel functions and x

I would like to calculate the definite integral with K-Bessel funcitons and a and b complex (n and k integers): $$\int_{0}^{\infty} x \;K_{a}(nx) \; K_{b}(kx) \; dx$$ I could not find it in ...
3
votes
1answer
138 views

A linear functional on $C(K)^*$ continuous on each $L_1(\mu)$

I asked this at math.stackexchange, but nobody answered. Let $K$ be a (Hausdorff) compact topological space, ${\mathcal C}(K)$ the usual Banach space of continuous functions $x:K\to{\mathbb C}$, ${\...
6
votes
1answer
323 views

Does every (generalized?) Markov chain admit transition probabilities?

To pose the question let us start by recalling the following notions: Transition Probabilities. A transition probability matrix between two measurable spaces $(S,\mathcal{S})$ and $(V,\mathcal{V})$...
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0answers
37 views

Can there be a nonzero period integral of this form?

I have been trying to compute the following integral: $$\displaystyle \int_{\gamma} \frac{g \Omega_{\mathbb{P}^3}}{f^2}$$ where: $\gamma$ is the torus cycle $\{ |z_1| =|z_2| =|z_3| =|z_4| =1 \}$, $...
3
votes
0answers
91 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 ...
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 ...