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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102
votes
6answers
8k views

Source and context of $\frac{22}{7} - \pi = \int_0^1 (x-x^2)^4 dx/(1+x^2)$?

Possibly the most striking proof of Archimedes's inequality $\pi < 22/7$ is an integral formula for the difference: $$ \frac{22}{7} - \pi = \int_0^1 (x-x^2)^4 \frac{dx}{1+x^2}, $$ where the ...
2
votes
0answers
675 views

What's the probability distribution of a deterministic signal or how to marginalize dynamical systems? (functional integrals in probability theory)

In many signal processing calculations, the (prior) probability distribution of the theoretical signal (not the signal + noise) is required. In random signal theory, this distribution is typically a ...
14
votes
5answers
3k views

Why do I need densities in order to integrate on a non-orientable manifold?

Integration on an orientable differentiable n-manifold is defined using a partition of unity and a global nowhere vanishing n-form called volume form. If the manifold is not orientable, no such form ...
3
votes
0answers
235 views

When does this method for integrals of fractional/integer parts work?

In a question Agno suggested an interesting way to compute $\{x\}$ and $\zeta(s)$. Define $$ F(x) = \{x\} = x - \lfloor x \rfloor = \frac{i \, \log\left(-e^{\left(-2 i \, \pi x\right)}\right)}{2 \, ...
2
votes
0answers
220 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 ...
2
votes
3answers
8k views

Integration of the product of pdf & cdf of normal distribution [closed]

Denote the pdf of normal distribution as $\phi(x)$ and cdf as $\Phi(x)$. Does anyone know how to calculate $\int \phi(x) \Phi(\frac{x -b}{a}) dx$? Notice that when $a = 1$ and $b = 0$ the answer is ...
88
votes
17answers
20k views

Why is differentiating mechanics and integration art?

It is often said that "Differentiation is mechanics, integration is art." We have more or less simple rules in one direction but not in the other (e.g. product rule/simple <-> integration by ...
59
votes
1answer
4k views

A hard integral identity on MATH.SE

The following identity on MATH.SE $$\int_0^{1}\arctan\left(\frac{\mathrm{arctanh}\ x-\arctan{x}}{\pi+\mathrm{arctanh}\ x-\arctan{x}}\right)\frac{dx}{x}=\frac{\pi}{8}\log\frac{\pi^2}{8}$$ seems to be ...
12
votes
4answers
9k views

Visualization of Riemann–Stieltjes Integrals

The Riemann–Stieltjes integral $\int_a^b f(x)\,dg(x)$ is a generalization of the Riemann integral. It is e.g. heavily used as a starting point for stochastic integration. The approximating ...
17
votes
2answers
408 views

Non-negative polynomials on $[0,1]$ with small integral

Let $P_n$ be the set of degree $n$ polynomials that pass through $(0,1)$ and $(1,1)$ and are non-negative on the interval $[0,1]$ (but may be negative elsewhere). Let $a_n = \min_{p\in P_n} \int_0^1 ...
24
votes
3answers
2k views

Can we simplify $\int_{0}^{\infty}\frac{{\sin}^px}{x^q}dx$?

We know the followings : $$\int_{0}^{\infty}\frac{{\sin}x}{x}dx=\int_{0}^{\infty}\frac{{\sin}^2x}{x^2}dx=\frac{\pi}{2},\int_{0}^{\infty}\frac{{\sin}^3x}{x^3}dx=\frac{3\pi}{8}.$$ Also, we can get ...
13
votes
5answers
1k views

An algebra of “integrals”

When discussing divergent integrals with people, I got curious about the following: Is there an $\mathbb{R}$-algebra $A$ together with a map (could be defined on just a subspace) $$\int_0^{\infty}: ...
4
votes
1answer
454 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 ...
12
votes
3answers
411 views

Is there a closed form of $\int_0^\frac12\dfrac{\text{arcsinh}^nx}{x^m}dx$?

For naturals $n\ge m$, define $$I(n,m):=\int_0^\frac12\dfrac{\text{arcsinh}^nx}{x^m}dx$$ with $\text{arcsinh}\ x=\ln(x+\sqrt{1+x^2} )$, so $\text{arcsinh} \frac12=\ln \frac{\sqrt{5}+1}2 $. Is it ...
12
votes
7answers
4k views

Numerical integration over 2D disk

I have a real-valued function $f$ on the unit disk $D$ that is fairly well behaved (real-analytic everywhere) and would like to find the integral $\int_D f(x,y)dxdy$ numerically. After much searching, ...
11
votes
1answer
434 views

$\pi e$ and an unfamiliar polynomial

Ever since my exposure to this integral involving $\pi e$, I've conjectured and set about evaluating the possible nature of the following integral $$\int_0^1 x^m \sin(\pi x) x^x (1-x)^{1-x} \ dx, ...
9
votes
2answers
511 views

Rain droplets falling on a table

Suppose you have a circular table of radius $R$. This table has been left outside, and it begins to rain at a constant rate of one droplet per second. The drops, which can be considered points as they ...
8
votes
2answers
539 views

$\mathrm{Bessel}^3$ Integral

I'm trying to calculate the following integral: $\int_0^\infty \mathrm{BesselJ}[l_0,k_0r] \cdot \mathrm{BesselJ}[l_1,k_1r] \cdot \mathrm{BesselJ}[l_0-l_1,kr] \cdot r\,dr$ ($\mathrm{BesselJ}[n,x]$ is ...
7
votes
2answers
467 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)}]= ...
5
votes
1answer
145 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 ...
3
votes
2answers
404 views

How to integrate an exponential function of an exponential function?

Does any one know how to calculate the following integration? $$ \int_{\mathbb{R}} \left(\exp(z \: e^{-y^2})-1\right)^2 dy=?,\quad z>0. $$ This post is related to my previous question here , ...
3
votes
3answers
335 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
votes
2answers
700 views

Riesz's representation theorem for non-locally compact spaces

Every version of Riesz's representation theorem (the one expressing linear functionals as integrals) that I have found so far assumes that the underlying topological space is locally-compact. (For ...
10
votes
2answers
359 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 $\{ x_j, w^j \} _{j=1}^N$ for large $N\in\mathbb{N}$. My question is how to do it, ...
8
votes
2answers
343 views

Convergence of an oscillatory integral

Given $f\in H^1(\mathbb{R}^3)$ and $t>0$, consider the following integral: $$I_f(t):=\int_{\mathbb{R}^3}\int_0^{+\infty}e^{-s+i\frac{(s+|x|)^2}{t}}f(x)dsdx$$ I need to show that $I_f(t)$ is finite, ...
8
votes
1answer
285 views

When is this multiple integral finite?

Consider the following integral: $$ I_k(\alpha)=\int_{[0,1]^k}|x_1-x_2|^{\alpha}|x_2-x_3|^{\alpha}\ldots|x_{k-1}-x_k|^{\alpha}|x_k-x_1|^{\alpha}d\mathbf{x}. $$ where $k=2,3,4,\ldots$ The question is ...
7
votes
1answer
365 views

Improper integral $\int_0^1 \frac{\exp(ctx)}{\sqrt{(\exp(bt)-1)(1-\exp(atx))-(1-\exp(at))(\exp(btx)-1)}} dx$ with $-a$ and $b$ positive

Is the following function real analytic in $t>0$: $$F(t)=\int_0^1\frac{\exp(ctx)}{\sqrt{(\exp(bt)-1)(1-\exp(atx))-(1-\exp(at))(\exp(btx)-1)}} dx,$$ where $-a$ and $b$ are positive, and $c\not=a$? ...
4
votes
1answer
128 views

Mean value of a function associated with continued fractions

Suppose that an irrational $x$ in $(0,1)$ has convergents $c(k,x)$, and let $$d(x) = \sum_{k=0}^{\infty} \mid x - c(k,x)\mid.$$ What is the mean value of $d$?
2
votes
2answers
325 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 ...
2
votes
0answers
262 views

Definite integral probably equal to zeta with known (but unusable) closed form for the indefinite integral

Related to this and this questions. Basically got definite integral that experimentally equals $\zeta(s)$ both numerically and symbolically. Closed form for the indefinite integral is known, but I ...
1
vote
0answers
347 views

convolution integral involving modified Bessel functions of the first kind

I'm stuck with this convolution integral ($z \geq 0$)... \begin{equation} f_{Z}(z)=\int^{\infty}_{-\infty}f_{1}(x)f_{2}(z-x)dx = \mbox{ } ??? \end{equation} which represents the pdf of the sum $Z = ...
0
votes
1answer
164 views

$\int_{R^2}\varphi(x)d\mu(x)=0$ $\Leftrightarrow$ $\sum_{n\in \mathbb Z^2} d\mu(x-2\pi n)=0$

Let $\mu$ be a finite measure supported by $\Gamma $ (a smoth finite curve) and absolutely continuous with respect to the length measure on $\Gamma$ such that $\Gamma \cap (\Gamma+x)$ is a finite ...
0
votes
1answer
131 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 ...