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, ...
1,594 questions
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Stochastic dominance $ \tanh Y\tanh Z \succeq \tanh X $
Let $X,Y,Z$ independent Gaussian r.v.'s with mean=variance. Let's denote these mean/variance parameters by $g_X,g_Y,g_Z>0$ respectively.
Set $T_1:=\tanh X$, $T_2:=\tanh Y\tanh Z$.
My question. ...
- 363
4
votes
2
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$\sum_{j=2}^n\sum_{p=n}^\infty \frac1{p j^p} + \sum_{j=n+1}^\infty \sum_{p=2}^\infty \frac1{p j^p}<\frac1n$ for all $n\ge3.$
How do I prove the following inequality for all $n\ge3$?
$$\sum_{j=2}^n\sum_{p=n}^\infty \frac1{p j^p} + \sum_{j=n+1}^\infty \sum_{p=2}^\infty \frac1{p j^p}<\frac1n$$
I've tried to bound it with ...
- 1,089
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votes
3
answers
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Integrating over Feynman parameters
Inspired by the answers in a past question I asked on techniques for computing integrals using the Feynman parametrization, I became interested in computing integrals over Feynman parameters like so:
...
- 169
1
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1
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Bounds on integrals of $1/(1+|\arccos(a+b\cos(t))|^k)$
In my research, I have come across a particularly nasty integral. Let $a$ and $\delta$ be such that $-1 \le a+\delta\cos(\psi) \le 1$ for all $\psi \in [0,\pi]$ and $\delta>0$. I would like to have ...
- 13
0
votes
0
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Indefinite integral of the product of two Chebyshev polynomials
Can someone tell me if the following two integrals have generic formulas?
(1) $\int T_m(ax+b)*U_n(cx+d)\ dx$;
(2) $\int x*T_m(ax+b)*U_n(cx+d)\ dx$;
where $T_m()$ and $U_n()$ are the m-th and n-th ...
0
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65
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Successive Riemann integrability of products of successively Riemann integrable functions
In teaching multivariable Riemann integration, I was trying to develop the theory of successive Riemann integrals (so all start with the one-dimensional case familiar to the students) as far as ...
- 1,118
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122
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Continuous analogue of the discrete simple continued fraction
Background
The classical Riemann integral of a function $f : [a,b] \to \mathbb{R}$ can be defined by setting $$\int_{a}^{b} f(x) \ dx := \lim_{\Delta x \to 0} \sum f(x_{i}) \ \Delta x. $$ Here, the ...
- 5,065
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About duality of Bochner spaces
I have a question about Bochner spaces, especially valued in Lebesgue $L^p$ spaces.
I have been reading Analysis in Banach Spaces Volume I by Tuomas Hytönen, Jan van Neerven, Mark Veraar and Lutz ...
- 133
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Solution of integral of integral times a function using integration by parts
I am trying to find a simpler solution to the following integral:
$$\int^\infty_{-\infty} \bigg [ \int^\infty_t \frac{dm(R_0(\tau))}{dr} \frac{1}{R_0(\tau)} d\tau \bigg ] m(R_0(t)) dt$$
where:
$$ m(r) ...
- 87
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Reference for absolutely continuous version of differentiation under the integral
There is an absolutely continuous version of the measure theory statement of Leibniz's rule (see https://math.stackexchange.com/questions/1683350/differentiability-under-the-integral-sign-of-...
- 33
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80
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Global Integrability from local integrable function and scaling degree
I am asking myself the following question:
$f \in C^{\infty}(0,1]$ and
$\text{sd(f)}=\inf_{s \in \mathbb{R}} \{ \lim_{\lambda\to 0} \lambda^s f(\lambda x)=0 | \forall x \in (0,1]\}< 1$ (Scaling ...
- 1
9
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1
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393
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Contour integration with poles that coincide with branch points
I would like to compute the following integral:
$$\int_{-\infty}^{\infty}\frac{dx}{2\pi} \frac{e^{-2d\sqrt{-x^2+\alpha^2+i\epsilon}}}{(-x^2+\alpha^2+i\epsilon)^2}$$
where $d, \alpha, \epsilon > 0$ ...
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Integral of product of $\sin(x/n)/(x/n)$
I recently watched a video of 3b1b's about Borwein integral. I got interested when the integral product goes to infinity
What is the integral of
$$\int_0^{\infty} \prod_{n=1}^{\infty}{\sin(x/n)\over x/...
- 29
2
votes
1
answer
272
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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
$$
E_n(t) = \mathbb{E}...
- 582
1
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1
answer
160
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Reference for multidimensional steepest descent with higher order correction terms
I am looking for a reference which proves a multidimensional steepest descent with higher order correction terms.
The integral that I consider is of the form
\begin{equation}
I(\lambda)=\int_\Gamma e^{...
- 39
2
votes
1
answer
474
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Show that:$\int_0^{\frac{\pi}{2}} x\frac{\ln^2(2\cos x) - \pi + x^2}{f(x)}\sin(2x) dx=-\frac{1}{4}$ [closed]
Prove that:
$$\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}$$
Let $f(x) = (\pi-x^2)^2 +2(x^2+...
- 131
2
votes
0
answers
195
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Rigorous analysis of an elliptic integration arising from physics
Let $\delta_+,\delta_-$ be two complex variables. Denote $e_i, i = 1,...,6$ be (multi-valued) holomorphic functions of $\delta_+,\delta_-$ defined as follows:
$$
\begin{align}
e_1 & = -2\left(1 - \...
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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 ...
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0
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116
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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}
...
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6
votes
1
answer
469
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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
187
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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
256
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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^...
- 5,059
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votes
1
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175
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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 (...
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1
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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 $H$, ...
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1
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0
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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
$$\begin{cases}\dfrac{\partial v_2^2}{\...
- 2,029
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votes
1
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Residual theorem analogue with branches of the complex logarithm
Let $\log_+$ denote a branch of the complex logarithm, which is holomorphic on $\mathbb{C} \setminus [0,\infty)$. Since $\log_+$ is clearly not holomorphic on $B_1^{\mathbb{C}}(0)$, the Residual ...
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0
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129
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How to solve any lemnatomic equation?
A cyclotomic equation is the equation obtained by setting the minimal polynomial, over $\mathbb{Q}$, of $\mathrm{e}^{2\pi i/n}$ equal to zero.
Gauss wrote about solving cyclotomic equations by ...
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2
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352
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A general result about measurable function
$f\geq 0$ measurable on $[0,1]$ with $\forall a \in \mathopen]0,1\mathclose[, \lim a^{n}f(a^{n})=1$.
Is it true that $\forall e>0, \int_0^e f =+\infty$ ?
- 6,010
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1
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Shape and Integration of a Sum of Gaussian PDFs over Independent Variables
I’m analyzing a function defined as the sum of multivariate Gaussian PDFs evaluated at different independent variables:
$$
f(x_1, \dots, x_M) = \sum_{i=1}^{M} N(x_i; \mu, \Sigma)
$$
where each $x_i \...
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1
vote
1
answer
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Interchange order of integration over conditional and marginal distributions
Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space and $X:\Omega\rightarrow \mathcal{X}$ and $Y:\Omega\rightarrow\mathcal{Y}$ random variables. Let $f:\mathcal{X}\times{\mathcal{Y}}\...
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Is Re(m)=Re(n) the only condition for $\displaystyle \int_0^{\infty} \frac{t^m-t^n}{e^t-1} \, dt = 0$ to hold? [closed]
For such an integral: $$\displaystyle \int_0^{\infty} \frac{t^m-t^n}{e^t-1} \, dt = 0$$
Given that m and n are constants and we suppose that m and n are equal, it can clearly be observed that the ...
- 11
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0
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133
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Who first discovered the formula for the volume of an ellipsoid?
Was it Newton or Leibniz in one of their works on volumes of rotation? Or was it known by some earlier Islamic or Greek mathematicians?
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1
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140
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Potential theoretic integral
In a paper that I am reading, the following equality is stated ($s,p>0$ and $|S^{n−1}|$ the measure of the $(n−1)$-dimensional sphere)
$$
\left(s \int_{\mathbf{R}^n} \int_{|y| \geqslant|x|} \frac{d ...
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3
votes
1
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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 $(\...
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0
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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
137
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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
137
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 ...
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2
votes
1
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261
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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
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1
answer
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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
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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
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0
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135
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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:
$$
\int_{\mathcal{C}} \int_{\mathcal{C}}
\frac{
\Gamma(m + s_1) \Gamma(m + s_2)
\...
- 1
0
votes
1
answer
84
views
Confusion in creating bounds over step size in numerical integration for error calculation?
This question is linked to:
Computational complexity of integration in two dimensions
Here in a suggested answer by John Gunnar Carlsson it is mentioned that
$|E| \leq K_1 h^2 |\Omega| M_2$
for some ...
- 103
3
votes
1
answer
254
views
An integral of Bessel function
The question is related to Improper integrals of Bessel function
I wonder if there is a closed form for the integral:
$$\int_0^\infty e^{-cx^2} I_\nu(ax)I_\nu(bx) dx,$$
where $I_\nu$ is the modified ...
- 181
7
votes
2
answers
607
views
Positivity of integral
We consider the function $F:(0,\infty) \to \mathbb R$. I am trying to show that for all $\alpha>0$ the integral
$$F(r):=\int_0^{\infty} \frac{1-e^{- q}}{1+\alpha q} J_0(rq) \ dq$$
is positive for ...
3
votes
1
answer
847
views
Missing factor of 10 in derivation for integral form of ζ(3)
While playing around with the divergence theorem to find a new form for $ζ(3)$, I stumbled across this form:
$$\iint_{[0,1] \times [0,1]} \frac{\ln(1-xy)+\ln(1-\sqrt{xy})}{xy} \ dx \ dy$$
The main ...
- 149
21
votes
1
answer
569
views
Bochner integral within or without a subspace
Let $X,V$ be Banach spaces and assume that $V$ is continuously embedded into $X$.
Let $f: [0,1] \to X$ be Bochner integrable and let $h: [0,1] \to [0,\infty)$ be measurable and integrable such that, ...
- 13k
5
votes
2
answers
324
views
Asymptotics of a two-dimensional integral
I am interested in determining the behavior of the following double integral
$$
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
376
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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 ...
- 30.7k
2
votes
1
answer
368
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
273
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{...
- 696