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, ...
1,589 questions
1 vote
0 answers
105 views
Reference for absolutely continuous version of differentiation under the integral
There is an absolutely continuous version of the measure theory statement of Leibniz's rule (see https://math.stackexchange.com/questions/1683350/differentiability-under-the-integral-sign-of-...
- 33
-1 votes
0 answers
32 views
Gradient of indicator function under the integral
Suppose a smooth positive function $f$ on $\mathbb R^d$. Suppose further subsets $A,B\subseteq\mathbb R^d$ of finite Lebesgue measure. I would like to understand whether there exists something like $$\...
0 votes
1 answer
70 views
Global Integrability from local integrable function and scaling degree
I am asking myself the following question: $f \in C^{\infty}(0,1]$ and $\text{sd(f)}=\inf_{s \in \mathbb{R}} \{ \lim_{\lambda\to 0} \lambda^s f(\lambda x)=0 | \forall x \in (0,1]\}< 1$ (Scaling ...
- 1
6 votes
0 answers
133 views
Contour integration with poles that coincide with branch points
I would like to compute the following integral: $$\int_{-\infty}^{\infty}\frac{dx}{2\pi} \frac{e^{-2d\sqrt{-x^2+\alpha^2+i\epsilon}}}{(-x^2+\alpha^2+i\epsilon)^2}$$ where $d, \alpha, \epsilon > 0$ ...
- 61
2 votes
0 answers
262 views
Integral of product of $\sin(x/n)/(x/n)$
I recently watched a video of 3b1b's about Borwein integral. I got interested when the integral product goes to infinity What is the integral of $$\int_0^{\infty} \prod_{n=1}^{\infty}{\sin(x/n)\over x/...
- 29
-1 votes
0 answers
75 views
Closed form formula for $\int \frac{dx}{x(\sin x+1)}$ [migrated]
I am trying to solve the following differential equation: $$ x \frac{dy}{dx} = y \sin\left(\frac{y}{x}\right) + 2y. $$ Using the substitution $v = \frac{y}{x}$, we have $y = vx$ and $\frac{dy}{dx} = v ...
- 107
2 votes
1 answer
265 views
Asymptotics of a Gaussian integral
Let us consider a sequence of iid, standard Gaussian random variables $\{X_i\}_{i\geq 1}$. Let $Y_n = \max_{2 \leq i \leq n} |X_i|$. I am interested in the asymptotic behavior of $$ E_n(t) = \mathbb{E}...
- 542
1 vote
1 answer
150 views
Reference for multidimensional steepest descent with higher order correction terms
I am looking for a reference which proves a multidimensional steepest descent with higher order correction terms. The integral that I consider is of the form \begin{equation} I(\lambda)=\int_\Gamma e^{...
- 39
2 votes
1 answer
456 views
Show that:$\int_0^{\frac{\pi}{2}} x\frac{\ln^2(2\cos x) - \pi + x^2}{f(x)}\sin(2x) dx=-\frac{1}{4}$ [closed]
Prove that: $$\mathcal{I}=\int_0^{\frac{\pi}{2}} x\frac{\ln^2(2\cos x) - \pi + x^2}{(\pi-x^2)^2 +2(x^2+\pi)\ln^2(2\cos(x)) +\ln^4(2\cos(x)) }\sin(2x) dx=-\frac{1}{4}$$ Let $f(x) = (\pi-x^2)^2 +2(x^2+...
- 131
2 votes
0 answers
193 views
Rigorous analysis of an elliptic integration arising from physics
Let $\delta_+,\delta_-$ be two complex variables. Denote $e_i, i = 1,...,6$ be (multi-valued) holomorphic functions of $\delta_+,\delta_-$ defined as follows: $$ \begin{align} e_1 & = -2\left(1 - \...
- 635
10 votes
0 answers
115 views
Does the set of integrability-preserving rotations of a two-dimensional Henstock-Kurzweil integrable function have measure zero?
A known drawback of Henstock-Kurzweil integration of real functions on $\mathbb R^2$ is the lack of a satisfactory change of variables formula. Even the composition of a function with a rotation can ...
- 707
1 vote
0 answers
106 views
Asymptotic double integral Airy functions
I am back with some tough asymptotic expansion that I would like to share with experts. I suspect the following identity is true (at least is some sense, maybe as a distribution): \begin{equation} ...
- 61
6 votes
1 answer
451 views
Lebesgue vs Riemann numerically
Let $f : [0,T] \to \mathbb R$ be a continuous function. We are interested in computing the integral $$ I_{\mathrm{Riemann}} := \int_0^T f(t)\,dt, $$ which is the standard Riemann integral. ...
- 161
1 vote
0 answers
177 views
An inequality for the sum of integrals in Bourgain's paper
I'm now (quickly) reading Bourgain's paper: J. Bourgain, Decoupling, exponential sums and the Riemann zeta function, J. Amer. Math. Soc. 30 (2017), 205-224 (https://www.ams.org/journals/jams/2017-30-...
- 393
5 votes
1 answer
253 views
Random tetrahedron inscribed in a sphere: expectation of angle between faces?
The vertices of a tetrahedron are independent and uniform random points on a sphere. What is the expectation of the internal angle between faces? Simulation suggests $\frac{3\pi}{8}$ I simulated $10^...
- 5,019
-1 votes
1 answer
171 views
Can we make assumptions such that $K(\cdot, x) \ll \lambda K$ for a.e. $x$ where $K$ is a markov kernel
Let $(X, \Sigma_X)$ and $(Y, \Sigma_Y)$ be two measurable spaces. Let $K: X \times \Sigma_Y \rightarrow [0,1]$ be a markov kernel. Let $\lambda$ be a $\sigma$-finite measure on $X$ and let $\lambda K (...
- 155
6 votes
1 answer
286 views
Are unitary representations of compact Lie groups Bochner integrable?
Let $G$ be a compact Lie group endowed with the bi-invariant Haar measure $\mu$ of total mass $1$. Let $\Phi \colon G \to \mathcal{B}(H)$ be a unitary representation of $G$ on a Hilbert space $H$, ...
- 600
1 vote
0 answers
148 views
A generalized comparison principle - is it true?
Consider $v_1,v_2\in H^1([0,T];L^2(\Omega))\cap L^2(0,T;H^1(\Omega))$ (Bochner spaces) be two weak solutions of the following doubly-nonlinear parabolic problems $$\begin{cases}\dfrac{\partial v_2^2}{\...
- 2,029
3 votes
1 answer
193 views
Residual theorem analogue with branches of the complex logarithm
Let $\log_+$ denote a branch of the complex logarithm, which is holomorphic on $\mathbb{C} \setminus [0,\infty)$. Since $\log_+$ is clearly not holomorphic on $B_1^{\mathbb{C}}(0)$, the Residual ...
- 1,389
0 votes
0 answers
128 views
How to solve any lemnatomic equation?
A cyclotomic equation is the equation obtained by setting the minimal polynomial, over $\mathbb{Q}$, of $\mathrm{e}^{2\pi i/n}$ equal to zero. Gauss wrote about solving cyclotomic equations by ...
- 331
6 votes
2 answers
349 views
A general result about measurable function
$f\geq 0$ measurable on $[0,1]$ with $\forall a \in \mathopen]0,1\mathclose[, \lim a^{n}f(a^{n})=1$. Is it true that $\forall e>0, \int_0^e f =+\infty$ ?
- 5,883
0 votes
1 answer
169 views
Shape and Integration of a Sum of Gaussian PDFs over Independent Variables
I’m analyzing a function defined as the sum of multivariate Gaussian PDFs evaluated at different independent variables: $$ f(x_1, \dots, x_M) = \sum_{i=1}^{M} N(x_i; \mu, \Sigma) $$ where each $x_i \...
- 19
1 vote
1 answer
206 views
Interchange order of integration over conditional and marginal distributions
Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space and $X:\Omega\rightarrow \mathcal{X}$ and $Y:\Omega\rightarrow\mathcal{Y}$ random variables. Let $f:\mathcal{X}\times{\mathcal{Y}}\...
- 155
-3 votes
1 answer
209 views
Is Re(m)=Re(n) the only condition for $\displaystyle \int_0^{\infty} \frac{t^m-t^n}{e^t-1} \, dt = 0$ to hold? [closed]
For such an integral: $$\displaystyle \int_0^{\infty} \frac{t^m-t^n}{e^t-1} \, dt = 0$$ Given that m and n are constants and we suppose that m and n are equal, it can clearly be observed that the ...
- 11
0 votes
0 answers
133 views
Who first discovered the formula for the volume of an ellipsoid?
Was it Newton or Leibniz in one of their works on volumes of rotation? Or was it known by some earlier Islamic or Greek mathematicians?
- 9
0 votes
1 answer
136 views
Potential theoretic integral
In a paper that I am reading, the following equality is stated ($s,p>0$ and $|S^{n−1}|$ the measure of the $(n−1)$-dimensional sphere) $$ \left(s \int_{\mathbf{R}^n} \int_{|y| \geqslant|x|} \frac{d ...
- 410
3 votes
1 answer
182 views
Equivalence between sum and integral of regular functions over positive real axis
Let $f$ be a function regular in $\Re z \geq 0$ whose indicator function satisfies: $$h(\theta) = \limsup_{r \to \infty} \frac{\log |f(r e^{i \theta})|}{r} \leq c < \pi$$ in this half plane. Let $(\...
- 127
0 votes
0 answers
52 views
Differentiating Coulomb Integrals with respect to a parameter
Suppose you have the following integral $$V_{pq}(t):=\int_{\mathbb{R}^3}\frac{\chi_p(x,t)\chi_q(x,t)}{\|x-R_c(t)\|}\mathrm{d}x,$$ where $\chi_p(x,t):= (x_1-R_{p,x_1}(t))^\ell(x_2-R_{p,x_2}(t))^m(x_3-...
1 vote
0 answers
135 views
How to compute this multi-dimensional integral?
I’m trying to evaluate the following integral, which appears in a decision-theoretic context: $$P = \int_{\Lambda(Z)< \eta} \left(a +\sum_{i=1}^{M} p(z_i \mid s)\right) dz$$ where: $p(z_i \mid s)$ ...
- 19
1 vote
0 answers
136 views
Does second quantization imply the need for distributions?
In classical field theory the Hamiltonian $H$ is given as the integral of a Hamiltonian density $\mathcal{H}$: $$H = \int_\Omega \mathcal{H}(x) dx$$ To quantize the field (going from classical field ...
- 873
2 votes
1 answer
252 views
Equality of two integrals when $1/p + 1/q=1$
Assume that \begin{align*} \pi_p &= \frac{2}{p} \int_0^1 \left[ u^{1-p} + (1-u)^{1-p} \right]^{1/p} \mathrm{d}u \end{align*} How to prove that if $\frac{1}{p} +\frac{1}{q}=1$ then $\pi_p=\pi_q$?
- 61
3 votes
1 answer
126 views
Closed-form Expression for Spherical Integral $I(n, m)$
I am investigating the following integral over the unit sphere $\mathbb{S}^{N-1}$: $ I(n, m) = \int_{\mathbb{S}^{N-1}} (1 + \langle u, x \rangle)^n (1 + \langle v, x \rangle)^m \, d\omega(x), $ where: ...
3 votes
1 answer
140 views
Comparing Trace of two functions who agree a.e. except on a small set
Suppose $U$ is an open bounded set in $\mathbb{R}^n$ ($n \geq 3)$ with Lipschitz boundary. Let $f \in H^1(U)$ with $f>c$ a.e in $U$ for some $c \in \mathbb{R}$. Suppose for any $\epsilon >0$ ...
- 193
0 votes
0 answers
129 views
Asymptotic behaviour of the analytic integral
Given $m \ge 0$ and $n$ odd, $ s >0 $ is there a way, for computing asymptotic expansion of following integral: $$ \int_{\mathcal{C}} \int_{\mathcal{C}} \frac{ \Gamma(m + s_1) \Gamma(m + s_2) \...
- 1
0 votes
1 answer
83 views
Confusion in creating bounds over step size in numerical integration for error calculation?
This question is linked to: Computational complexity of integration in two dimensions Here in a suggested answer by John Gunnar Carlsson it is mentioned that $|E| \leq K_1 h^2 |\Omega| M_2$ for some ...
- 103
3 votes
1 answer
234 views
An integral of Bessel function
The question is related to Improper integrals of Bessel function I wonder if there is a closed form for the integral: $$\int_0^\infty e^{-cx^2} I_\nu(ax)I_\nu(bx) dx,$$ where $I_\nu$ is the modified ...
- 181
7 votes
2 answers
596 views
Positivity of integral
We consider the function $F:(0,\infty) \to \mathbb R$. I am trying to show that for all $\alpha>0$ the integral $$F(r):=\int_0^{\infty} \frac{1-e^{- q}}{1+\alpha q} J_0(rq) \ dq$$ is positive for ...
3 votes
1 answer
846 views
Missing factor of 10 in derivation for integral form of ζ(3)
While playing around with the divergence theorem to find a new form for $ζ(3)$, I stumbled across this form: $$\iint_{[0,1] \times [0,1]} \frac{\ln(1-xy)+\ln(1-\sqrt{xy})}{xy} \ dx \ dy$$ The main ...
- 149
21 votes
1 answer
553 views
Bochner integral within or without a subspace
Let $X,V$ be Banach spaces and assume that $V$ is continuously embedded into $X$. Let $f: [0,1] \to X$ be Bochner integrable and let $h: [0,1] \to [0,\infty)$ be measurable and integrable such that, ...
- 12.9k
5 votes
2 answers
320 views
Asymptotics of a two-dimensional integral
I am interested in determining the behavior of the following double integral $$ I_N = \int\limits_{0}^{1} \int\limits_{0}^{2 \pi} \Big[ \big( (2x-1)(1+\cos t) -i \sin t \big) x (1-x) \Big]^{N} \, dt \,...
- 2,583
1 vote
1 answer
356 views
Closed form of an integral
I asked the following question on Math Stack Exchange about a week ago, with no response: https://math.stackexchange.com/questions/5054949/closed-form-of-an-improper-integral I ask the question here ...
- 29.9k
2 votes
1 answer
349 views
Asymptotics of integral involving Bessel functions
How to study large $r \gg 1$ asymptotics of $$I(r):=\int_0^{\infty} \frac{1-e^{-q}}{1+q} J_0(rq) \ dq,$$ where $J_0$ is the zeroth order Bessel function of the first kind. I did some numerics and it ...
8 votes
1 answer
272 views
set of diffuse probability measures that concentrates mass
Let $\lambda$ be the Lebesgue measure. I am wondering wether it is possible to construct a family of diffuse probability (ie without atom) measures $\mu_x$ such that for all $A \in \mathcal{B}(\mathbb{...
- 666
0 votes
0 answers
103 views
Surjective Operator from L1(ν) to L1(μ) for Discrete and Diffuse Probability Measures
I am looking for, if it exists, an example of a discrete probability measure $ \nu $ and a diffuse (non-atomic) probability measure $ \mu $, such that there exists a continuous linear surjective ...
- 666
0 votes
1 answer
171 views
Improper integrals of Bessel function
Let $I_\nu$ be the modified Bessel function of first kind. I wonder if there is some formulas for the integrals: $\int_0^\infty e^{-cx} I_\nu(a \sqrt{x})I_\nu(b \sqrt{x})dx$, $\int_0^\infty x e^{-cx}...
- 181
2 votes
1 answer
166 views
Concentration of conditional densities over sets of positive measure
Let $\lambda$ be the Lebesgue measure. Does there exist a family of non-negative measurable functions $(f_y)_{y \in \mathbb{R}}$ from $[0,1]$ to $\mathbb{R}$ such that $(x,y) \mapsto f_y(x)$ is ...
- 666
1 vote
1 answer
120 views
Is the Doléans measure zero on null sets?
Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space, and let $\lambda$ be Lebesgue measure. Consider a square integrable continuous martingale $M$ with quadratic variation $\langle M\rangle$. ...
- 708
-5 votes
1 answer
187 views
Can there be a "universal" deductive system? [closed]
In "PRINCIPLES AND IMPLEMENTATION OF DEDUCTIVE PARSING" by STUART M. SHIEBER, YVES SCHABES, AND FERNANDO C. N. PEREIRA, they say "We present a system for generating parsers based ...
- 521
6 votes
1 answer
400 views
Existence of a family of Lebesgue densities concentrating on every Borel set
Let $\lambda$ be the Lebesgue measure. Does there exist a family of non-negative measurable functions $(f_y)_{y \in \mathbb{R}}$ such that $ (x,y)→f_y(x) $ is mesurable, $ \lambda(dy)\text{-a.e., }\...
- 666
0 votes
0 answers
83 views
Asymptotics of Laplace-type integral with moving maxima
I am looking at the integral \begin{align}I(b,\epsilon)&=\int_{x_0}^{x_T} \frac{1}{x} e^{-\frac{1}{\epsilon}\left(b+\frac{x}{1+x}\right)^2} dx\\ &=\int_{x_0}^{x_T} e^{-\frac{1}{\epsilon}\left[\...
- 111