# 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, ...

981
questions

**4**

votes

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**2**answers

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

**2**answers

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

**2**answers

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

**1**answer

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

**3**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**3**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**2**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**2**answers

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

**0**answers

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

**2**answers

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

**2**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**3**answers

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

**3**answers

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

**3**answers

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

**1**answer

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

**1**answer

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_{...