# Questions tagged [integration]

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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### Expectation of $\left| \frac{(\textbf{x}+\textbf{y})^{H} \textbf{x} }{\| \textbf{x} + \textbf{y} \|^2} \right|^2$, with complex Gaussians?

Given that following two random variables $\textbf{x} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{x}^{2}\textbf{I}_{M})$ and $\textbf{y} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{y}^{2}\textbf{I}_{M})$ ...
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### Function orthogonal to powers of $1/\left(1+x^2\right)$

Does there exist any continuous function $f:\mathbb{R}\to\mathbb{R}$, $f(x)/(1+x^2)\in L^1(\mathbb R)$, such that $f(0)=1$ and $$\int_{-\infty}^{\infty}\frac{f(x)}{\left(1+x^2\right)^p}dx=0$$ for ...
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### Simplify Wasserstein distance between Gaussians with binary cost function

Let $\mu_1$ and $\mu_2$ be 1D gaussian distributions with means $m_1$ and $m_2$ respectively and common variance $\sigma$. Let $\Omega$ be a closed subset of $\mathbb R^2$, and consider the cost ...
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### On $\zeta(7)$ as the integration of the product of an indefinite integral due to Lobachevskii by a power of the inverse Gudermannian function

In this post I invoke certain function from a post of this site MathOverflow it is [1] (please see further references from the post, authors from the Springer link of the cited literature and answers ...
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### Exterior derivative independence from coordinate systems

In the book Mathematical Methods of Classical Mechanics by V.I. Arnold, the author introduces (p.189) the concept of exterior derivative as "the principal linear part of the increment" of the function ...
343 views

### Closed form $\int_{0}^{\frac{r}{2}} {\binom{n}{p} \binom{n-p}{r-2p} 2^{r-2p}}{\binom{2n}{r}^{-1}} \ \text{d}p$

Note: This is exact copy of my Math.SE question, which I am reposting here, as despite bounty it did not receive any answers. Let there be $n$ pairs of shoes in a box. The the probability that from ...
77 views

### Invariant measure on coset space and integrable functions

Let $G$ be a locally compact abelian group, and $H$ a closed subgroup. Let $C_c(G)$ be the space of continuous, compactly supported complex valued functions on $G$. General theory of Haar measure ...
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### Intrinsic volume - is there a simplified formula?

I'm struggling to understand how can I compute the instrinsic volumes (or Minkowski functionals) of a submanifold $\Omega$ of $\mathbb{R}^N$. I found a formula, but I really can't understand it... It ...
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### About a multiple integral [closed]

In my current research, I'm confronted with the justification of some facts, and I don't know how to proceed in proving them, so I need to know if there exist some theorems (precisely three theorems) ...
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### Elements of vector-valued $L^1$-spaces

Let $E$ be a complete locally convex space and let $(X, \Sigma, \mu)$ be a measure space where $\mu$ is a Radon measure. Then the space $L^{1}(X,E)$ is defined as a the completion of the space $S(X,E)$...
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### Why is the integral of the tautological 1-form equal to the action?

I am having a hard time to understand why the integral of the tautological 1-form is the action of the system. The tautological one form is defined by : \begin{align} \theta_{(q,p)} : T_{(q,p)}T^*Q &...
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### Closed Poincaré dual, why $\int_M \omega \wedge \eta_S$ and not $\int_M \eta_S \wedge \omega$?

My book is Differential Forms in Algebraic Topology by Loring W. Tu and Raoul Bott of which An Introduction to Manifolds by Loring W. Tu is a prequel. The characterization of the closed Poincaré dual ...
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### The integrals of things looking like $e^{(\frac{a}{z}+\frac{b}{z-c})}$ on closed contours

I have recently encountered a truely terrible integral which I need to compute. I am not sure it's doable but before throwing the whole project in the bin I thought I would ask here. At the moment, a ...
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### Are we better in computing integrals than mathematicians of 19th century?

When I started to learn mathematics, I was fascinating by legendary «Демидович»: problems in mathematical analysis. Fifteen years later, when I open chapters about integrals, I see a long list of ...
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### A reduction problem from $\mathbb{R}^2$ to $\mathbb{R}$

Let $f,g \in L^1_\text{loc}(\mathbb{R})$, with $g \geq 0$, and such that for almost every $(x,y) \in \mathbb{R}^2$, at least one of the following equations is true : \begin{align*} f(x) + f(y) + g(...
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### Integrate $1/(x_1x_2\cdots x_n)^k$ for $1\le x_i \le a$, where product of coordinates satisfies $b\le x_1\cdots x_n\le c$

I need to integrate $$\int\limits_{[1,a]^n} \frac {\chi(\{ b \le x_1 \cdots x_n \le c \})} {( x_1 x_2 \cdots x_n)^k} \,dx_1 \cdots dx_n,$$ where $\chi(E)$ is the characteristic function of a set $E$....
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### Differential equation with Fresnel integral

We have $\frac{y'(x)}{\cos(x)}=C(x)$ and need to find y(x). Generally we should express $y(x)$ through $C(x)$ and elementary functions. I can only do it through $C(x)$ and $S(x)$, or through $\Phi(x)$....
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### How can one integrate over the unit cube, subject to certain (quantum-information-theoretic) constraints?

To begin, we have two constraints $$C1=x>0\land z>0\land y>0\land x+2 y+3 z<1$$ and C2=x>0\land y>0\land x+2 y+3 z<1\land x^2+x (3 z-2 ...