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, ...

Filter by
Sorted by
Tagged with
4
votes
1answer
155 views

Functions orthogonal to powers of $1/{\left(1+x^2\right)}$

Let $f,g:\mathbb{R}\to\mathbb{R}$ be continuous functions with the following properties: $f(x)$ and ${g(x)}/x$ are bounded; ${g(x)}/{\left(1+x^2\right)}\in L^1\left(\mathbb{R}\right)$; $\lim_{x\to0}f(...
0
votes
1answer
100 views

Proof of $\sum\limits_{k\in\mathbb{Z}}\int_{\mathbb{R}}|f(x+k)f'(x)|dx<\infty$ for Schwartz function $f$

For a function $f$ from the Schwartz space $\mathcal{S}(\mathbb{R})$ do we have that $$\sum\limits_{k\in\mathbb{Z}}\int_{\mathbb{R}}|f(x+k)f'(x)|dx$$ converges?
2
votes
1answer
136 views

Does exist a variant of Frullani's theorem valid for $f(x)=\pi(x)/x$ or $f(x)=\psi(x)/x$, where the numerators are prime-counting functions?

Frullani's theorem is a deep theorem in real analysis with applications, see the Wikipedia Frullani integral and other uses and contexts (see [2]). I wrote two imaginative examples of what can be ...
0
votes
0answers
120 views

A complex integration formula

I'm trying to integrate a formula but I can’t figure it out. Its calculation involves an error function. Here's the formula: $f(a, b, c)=\int_{0}^{+\pi} d \theta \exp (a \cos \theta) \operatorname{...
1
vote
2answers
66 views

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 $\textbf{y} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{y}^{2}\textbf{I}_{M})$ ...
11
votes
2answers
582 views

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 ...
1
vote
2answers
461 views

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 ...
1
vote
1answer
315 views

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 ...
4
votes
3answers
322 views

Legendre Polynomial Integral over half space

I need to compute the following integral $$ I_{n,m} := \int_0^1 P_n(x) P_m(x) \; \mathrm{d}x $$ where $P_n$ is the Legendre polynomial. For an even sum $n+m=2l$ it is easy to show that $$ I_{n,m} = \...
2
votes
0answers
214 views

Calculating $\int_1^{\infty}\frac{\operatorname{ali}(x)}{x^3}dx$, where $\operatorname{ali}(x)$ is the inverse function of the logarithmic integral

It is well-known that we can compute the closed-form of the integrals $$\int_1^{\infty}\frac{\log x}{x^2}dx$$ and $$\int_1^{\infty}\frac{\operatorname{li} (x)}{x^3}dx,$$ where $\operatorname{li} (x)$ ...
0
votes
0answers
114 views

Express $\zeta(n)$, in particular the Apéry's constant, in terms of series involving Gregory coefficients or the Schröder's integral

The idea and motivation of this post is to know if it is possible to express integer arguments of the Riemann zeta function $\zeta(n)$, in particular $\zeta(3)$, in terms of series involving the so-...
7
votes
1answer
450 views

Change of variables for $p$-adic integral

Say $p$ is an odd prime. Suppose I have a measure $\mu$ on $\mathbf{Z}_p$. As in II.4.3 in Colmez - Fonctions d'une variable $p$-adique, I can restrict $\mu$ to $1+p\mathbf Z_p$, and there is a ...
7
votes
1answer
242 views

Integration with values in a topological vector space

Is there a general theory of integration of functions with values in a topological vector space (not necessarily locally convex)? Browsing through mathoverflow posts, I came across a discussion ...
2
votes
1answer
319 views

Help in proving, that $\int_{0}^{\infty} \frac{1}{\Gamma(x)} d x=e+\int_{0}^{\infty} \frac{e^{-x}}{\pi^{2}+(\ln x)^{2}} d x$ using real methods only

I hope this does not seem like a too easy question for Overflow. I would like to find an easier method than mine to prove the above statement for the Fransén-Robinson Constant. My first method was to ...
1
vote
1answer
227 views

Limits of a family of integrals

Assume $\lambda_1+\lambda_2=1$ and both $\lambda_1$ and $\lambda_2$ are positive reals. QUESTION. What is the value of this limit? It seems to exist. $$\lim_{n\rightarrow\infty}\int_0^1\frac{(\...
2
votes
1answer
291 views

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 ...
4
votes
3answers
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 ...
4
votes
0answers
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 ...
2
votes
0answers
119 views

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 ...
3
votes
1answer
123 views

Asymptotic behaviour of function using Fox $H$-function representation

In equation (9) of this paper, it is claimed that the limiting behaviour $$ \int_0^\infty \frac{1-\cos(kx_0)}{s+Dk^\alpha}dk \sim \frac{\Gamma(2-\alpha)\sin(\pi(2-\alpha)/2)x_0^{\alpha-1}}{(\alpha-1)D}...
5
votes
0answers
288 views

Homeomorphism between $L^p$-spaces on metric spaces and $L^p$-spaces on Euclidean space

Setup: Fix $p \in [1,\infty)$. Let $(X,d_X,x_0)$ and $(Y,d_Y,y_0)$ be complete pointed metric spaces and $\mu$ be Borel. Let $E^n,E^D$ be Euclidean spaces of respetive dimensions $n$ and $D$ and ...
2
votes
0answers
319 views

Normal multivariate orthant probabilities

(Previously I posted a similar question on math.SE, hoping that this question would have an easy answer. As the question appears hard, I am hoping I can perhaps get more feedback here.) Let $\mathbf{...
6
votes
2answers
410 views

Does $\int_0^{2\pi} e^{i\theta(t)} (\phi(t))^n dt=0$ $\forall \; n\in\mathbb{N}_0$ imply $\phi$ periodic?

PROBLEM. Let $\theta(t)$ and $\phi(t)$ be two real analytic non-constant functions $[0,2\pi]\rightarrow \mathbb{R}$. I am trying to prove the following claim If the integral $$ \int_0^{2\pi} e^{...
0
votes
1answer
77 views

An extension of the mean-value theorem for integrals? [closed]

The mean value theorem for integrals states that if $f$ and $g$ are continuous on $[a, b]$ and $g$ never changes sign on $[a, b]$, then there exists some $c\in [a, b]$ such that $$\int_{a}^{b} f(x)g(...
2
votes
0answers
78 views

Distribution of a linear pure-birth process' integral

I stumbled across the following random variable, defined as the integral of a linear pure-birth process i.e. a Yule process: $$ Z_t = \mathbb{E}\bigg[\int_0^t Y_s ds \bigg| Y_t=k , Y_0=1\bigg] $$ ...
2
votes
0answers
176 views

Orthogonality relation in $L^2$ implying periodicity

Let $\theta(t)$ and $\phi(t)$ be two real $C^1$ functions $[0,2\pi]\rightarrow \mathbb{R}$. Let us assume $\theta$ has the properties $$ \int_0^{2\pi} e^{i\theta(t)} dt=0. $$ Geometrically this means ...
-1
votes
1answer
113 views

Is there a name for this family of integral?

This one: $\int_{0}^{\bar{x}}e^{-x^{a}}x^{b}(1-x)^{c}dx,a,b,c\ge0$. When $a=1,c=0,\bar{x}=\infty$ it is the gamma function.
2
votes
0answers
93 views

exact form of the integral of x^(-x) between 0 and infinity [closed]

I have searched the internet for an exact answer and although I have found many decimal approximations, https://www.wolframalpha.com/input/?i=integrate+x%5E(-x)+from+0+to+infinity I have not been able ...
0
votes
0answers
39 views

Specify modified error function in form of error functions

How can we express $\mathrm{erf}(\frac{t-a}{m})$ as a sum of functions of the form $\mathrm{erf}(t)$? I am developing a fitting routine and I encounter this integral: $$\int_{0}^{x}\mathrm{erf}\left(\...
-1
votes
1answer
151 views

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) ...
3
votes
0answers
124 views

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 $S(X,E)$...
2
votes
0answers
316 views

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 &...
1
vote
1answer
447 views

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 ...
3
votes
2answers
235 views

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 ...
35
votes
0answers
1k views

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 ...
3
votes
2answers
219 views

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(...
4
votes
2answers
495 views

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 $E$....
1
vote
0answers
76 views

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 $\Phi(x)$....
1
vote
1answer
136 views

How can one integrate over the unit cube, subject to certain (quantum-information-theoretic) constraints?

To begin, we have two constraints \begin{equation} C1=x>0\land z>0\land y>0\land x+2 y+3 z<1 \end{equation} and \begin{equation} C2=x>0\land y>0\land x+2 y+3 z<1\land x^2+x (3 z-2 ...
1
vote
0answers
117 views

Is this integral zero?

I'd like to know if one integral expression I have can be shown to be zero for all possible cases. Let me introduce some notation. Consider $\mathfrak{g}=C^{\infty}(M)$ and the dual $\mathfrak{g}^*=\...
1
vote
1answer
232 views

Integration by parts on manifold with corners

Suppose that $M$ is a compact manifold with corners, where each boundary hypersurface is an embedded submanifold. Then, do we have an integration by parts identity? i.e. \begin{align*} \int_M g(\nabla ...
1
vote
1answer
275 views

$L^2$ norm of fractional Laplacian

Is it possible to calculate the $L^2$ norm associated to fractional Laplacian of $u$ and $s\in (0, 1).$ $$\|(-\Delta)^{s/2} u\|_2^2=\int_{\mathbb R^N}|(-\Delta)^{s/2} u|^2dx=C_{N,s}\int_{\mathbb R^{...
2
votes
0answers
143 views

A bound using Cauchy formula

Let $0<t_0<1$ fixed number , $ n_0$ integer $ \geq 2$ fixed and let $\forall 0<u<1, f(u)= \displaystyle \frac{(1-u)^{n_0} \log(1-u)}{(1-ut_0)^{n_0+1}} $. Let $0<u_0<1 $ be given. ...
0
votes
0answers
51 views

Integral involving legendre (as Beukers integral) [duplicate]

let $\forall n $ integer $p_n(t)=\frac{1}{n!}(t^n(1-t)^n)^{(n)} $ i 'm looking for an explicit constant $0<c<1$ ( very good small c ) independant of $n$, and a constant $b$ ( non explicit) ...
3
votes
1answer
150 views

Looking for bound in integral involving Legendre polynomial

I'm looking for an upper bound to the following integral or equivalent when $n$ leads to $ +\infty $ to the following expression $$I_n:=\left|\int_{0}^1 \int_{0}^1 \frac{p_n(x) p_n(y)}{(1-xy)} dx dy ...
5
votes
3answers
375 views

Limit of an integral vs limit of the integrand

I have a simple Fourier transform problem, originating from mathematical physics (system of linear PDEs), which reduces to taking the integral $$ I(\alpha)\equiv\int_{-\infty}^\infty e^{ikr} \cfrac{\...
7
votes
3answers
663 views

$\int_{-\infty}^\infty \frac{e^{-y^2/2}}{((y+y_0)^2+x_0^2)^r} dy$

I have to estimate the integral $$\int_{-\infty}^\infty \frac{e^{-y^2/2}}{((y+y_0)^2+x_0^2)^r} dy,$$ for $r\in \mathbb{R}^+$. I am a little amazed that Sage and Wolfram Alpha have nothing to say about ...
10
votes
3answers
599 views

Nonnegativity of an integral over the unitary group

For an $n$-by-$n$ unitary matrix $U$ and a permutation $\sigma\in S_n$, let $$w_\sigma=(-1)^\sigma\det(U^*)\prod_{i=1}^n U_{i,\sigma(i)}.$$ Is $\int_{U(n)}\mathrm{Re}(w_{\sigma_1})\mathrm{Re}(w_{\...
1
vote
1answer
89 views

The expectation of binary logistics regression with respect to Gaussian distribution

I am trying to compute the expectation of $g(s,x)=s \ln \sigma(x)+(1-s)\ln(1-\sigma(x))$ with respect to the normal distribution $\mathcal{N}(x;m,v)$, where we have $\sigma(x)=\frac{1}{1+e^{-x}}$. If ...
3
votes
1answer
185 views

Ratio of Selberg integral

I'm considering a ratio of incomplete Selberg integral: $$f_n(a,b)=\frac{\int_{\Delta_a}\prod_{i=1}^nx_i^{\alpha-\frac{n+1}{2}}\prod_{i=1}^n(1-x_i)^{-1/2}\prod_{i<j}|x_i-x_j|}{\int_{\Delta_b}\prod_{...

1
3 4
5
6 7
20