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, ...

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5
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2answers
2k views

Elementary calculus estimate or not?

Does there exist a constant $C>0$ such that for all $f \in H^3(\mathbb R)$ $$\int_{\mathbb R} \vert x f''(x) \vert^2 \ dx \le C \int_{\mathbb R} \vert f'''(x) \vert^2 + \vert x^3f(x) \vert^2 + \...
14
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1answer
519 views

Almost all non-negative real numbers have only finitely many multiple lies in a measurable set with finite measure

I do not know whether this is the right place for posting this problem. But for several months I have no solution to this problem. Let $A$ be Lebesgue measurable subset of $[0,\infty)$ such that ...
2
votes
0answers
31 views

Solving the inverse of a matrix under a uniform distribution

I am looking to solve the following equation: $$\left(\begin{array}{cc}{ g_{11}} & { g_{12}} \\ { g_{21}} & { g_{22}}\end{array}\right)=\int_{a}^{b} \frac{1}{b-a}\left(\begin{array}{cc}{- g_{...
0
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1answer
171 views

Logarithm of an integral involving generalized real binomial coefficients

I could not find a closed form for this integral although I think it should have been studied. What is a good approximation to $I$ in $$I=\ln\Bigg(\int_{0}^y\binom{2m}{m(1+x)}dx\Bigg)$$ where $0\...
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0answers
57 views

Derivative of a convolution integral of the following type?

I'm looking to find the derivative of a convolution integral of the following form: \begin{equation} \frac{d}{dr}((G(r,t)*f(t)) = \frac{d}{dr} (\int_{-\infty}^{\infty} G(r,t-\tau)f(\tau) d\tau) \end{...
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0answers
204 views

Integral of matrix determinant with respect to Lebesgue measure

$\newcommand\norm[1]{\lVert#1\rVert} \newcommand\opnorm[1]{\norm{#1}_{\text{op}}} \newcommand\Frnorm[1]{\norm{#1}_{\text F}}$Define \begin{align*} S_t=\{ (A,B)\in\mathbb{R}^{n\times n}\times\...
1
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0answers
142 views

Prove the following property about natural integral

Natural integral is the distinguished antiderivative of a function that can be understood as an analytic continuation of consecutive derivatives of a function towards $-1$th order. It is defined as $...
3
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2answers
143 views

A definite integral related to sample variances of bivariate Gaussians

This integral is needed to obtain the joint distribution of the sample variances of a random sample from a bivariate Gaussian distribution. For details on the joint distribution of the sample means, ...
3
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0answers
89 views

Comparing an integral to zero, by slicing and stacking

Let $f \colon [0,1] \rightarrow {\mathbb R}$ be a nice function -- real-analytic, or maybe definable in some o-minimal structure, let's say. Let $0 < \alpha_1 < \cdots < \alpha_n < 1$ be ...
2
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3answers
359 views

A Curved/Warped Version of Fubini's Theorem

I will think of $ \mathbb{R}^{n+m}$ as $\mathbb{R}^n \times \mathbb{R}^m$. Let $ V \subset \mathbb{R}^{n+m}$ be open and $g:V \to U \subset \mathbb{R}^{n+m} $ be a $C^1$ diffeomorphism. For a fixed ...
1
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0answers
98 views

|Evaluating integral on $ \mathbb S^{d-1}$

I am trying to evaluate the following integral: $$ \int_{\mathbb S^{d-1}} \exp \bigg(-\frac{(1+x\cdot y)^2}{\|x+y\|^2} \bigg) \ dx $$ for $x,y \in \mathbb R^d$. Does anyone know a solution or an ...
1
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1answer
185 views

Integration on sphere $\mathbb{S}^{d-1}$ for $d$ large — Change of variables

I'm trying to integrate a function over two vectors which lie on the surface of the unit sphere in D dimensions. The function depends only on the difference between the two vectors, and their dot ...
4
votes
3answers
311 views

Meaning of divergent integrals

In quantum field theory, one usually encounters divergent integral when one calculates loop functions, such as the integral $\int_{0}^{\infty}dkk^{3}\frac{1}{(k^{2}-m^{2})^{2}}$ which is divergent. ...
2
votes
1answer
195 views

An integral of $|\sin(x)\cos(nx)^{-2/n}|$ from $-\pi$ to $\pi$

For an integer $n \geq 3$, define $$A_n = \int\limits_{-\pi}^\pi\frac{|\sin(x)|}{|\cos(nx)|^{2/n}}dx.$$ It is a fact that $A_n$ is finite for all such $n$. I am interested in the behaviour of a ...
1
vote
1answer
404 views

What are the consequences if we could express tangent via logarithm in an algebraic system? [closed]

Working on an algebra of divergent integrals I came to the following relation: If $\tau=\int_0^\infty dx$ then $$\ln (\tau+a)=\int_{0}^\infty \psi'(x+1/2+a)dx$$ and this directly gives the following ...
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0answers
140 views

Solving an equation of function

How to solve, or at least how to proceed to solve, the following equation for $g(u)$ $$\int_0^{\infty} \{1-\cos(2\pi uh)\} g(u)du = (1+h^{\alpha})^{\beta/\alpha} -1?$$ Here $0<\alpha\leq2$ and $-\...
6
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1answer
480 views

Possible application of divergence Theorem?

suppose that $f \in C^1 (\mathbb{R}^{N+1},\mathbb{R})$. It's well known that if all his points are regular points i.e. $$\nabla f (x) \neq 0 \; \; \; \forall x \in \mathbb{R}^{N+1}$$ then, for every ...
49
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4answers
5k views

A historical mystery : Poincaré’s silence on Lebesgue integral and measure theory?

Lebesgue published his celebrated integral in 1901-1902. Poincaré passed away in 1912, at full mathematical power. Of course, Lebesgue and Poincaré knew each other, they even met on several occasions ...
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0answers
67 views

Does $L^1$ convergence preserve the regularity of this sequence of functions?

Let $f_n$ be a sequence of $L^1(]0,1[)$ functions such that $f_n$ is non-decreasing, at least left-continuous, $f_n(0^+) <0$, $f_n(1^-) >0$, for all $n \in \mathbb N$. This sequence converges $...
4
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2answers
145 views

Is it true that the quantile function of an $L^1$ random variable is $L^2(]0,1[)$?

Let $(\Omega, \mathcal A, P)$ be a probability space. Let $X:\Omega \rightarrow \mathbb R$ be an $L^1(\Omega, \mathcal A, P)$ random variable. We define the distribution function of $X$ by $$F(x) = ...
1
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2answers
67 views

Opial type inequalities

Let $x(t)\in C^1[0,h]$ be such that $x(0)=x(h)=0$ and $x(t)$ in (0,h) ,then the following inequality $ \int^h_0 |x(t)x^{'}(t)|dt \leq \frac{h}{4}\int^h_0(x^{'}(t))^2dt$ my question: I would like ...
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0answers
107 views

Bayesian Bandits - What's the probability that choice K is the best?

I have $K$ very unfair coins. I don't know how unfair they are, but they all seem to have different probabilities of landing heads. I'd like to figure out which one is best as quickly as possible. ...
0
votes
1answer
171 views

All roots of a polynomial are inside the unit circle [closed]

How to compute the following integral? $\int_0^1 P_j(x)dx$, where $P_j(x) = \prod\limits_{i=0,i\ne j}^n(x+i)$? I met this problem when doing estimation on the roots of the following polynomial. All ...
1
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0answers
55 views

Daniell integral of “generalized (of some sort)” functions?

Let $E$ be a (Dedekind $\sigma$-complete) Riesz space and $H\subseteq E$ a subspace. A Daniell integral $I\colon H\to\mathbb R$ is defined to be a positive linear functional which is continuous with ...
5
votes
1answer
278 views

Can I cover a compact set by balls {B} such that {2B} has bounded overlap?

Suppose I have a compact set $K \subset B_1(0) \subset \mathbb{R}^n$. Can I always find a family of open balls $\{B_{r_j}(x_j)\}$ such that $x_j \in K$ and $B_{r_j}(x_j) \subset B_1(0)$ for each $j$; ...
1
vote
1answer
228 views

If $h$ is a decreasing function then $\psi$ is an increasing function

Let $h : [0,1] \to [0,1]$ be a $\mathcal{C}^1$ function such that $h'(x)<0$ for all $x \in (0,1)$. Consider the function $$ \psi(x) = \frac{\int_0^1yh(|x-y|) dy}{\int_0^1 h(|x-y|)dy} $$ I am trying ...
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0answers
132 views

An integral involving many exponential terms with quadratic exponents in the denominator

Given $k$ points $\{p_1,\cdots, p_k\}$ in $\mathbb{R}^n$ and positive constants $r_1, ..., r_k$ and another positive constant $\alpha>0$. Is there a way to compute/approximate the following (...
1
vote
1answer
256 views

Is this relation between divergent intergals justifiable?

Graf's book on hyperfunction theory says (page $36$) that $$\frac1{(x-i0)^n}=\frac{(-1)^{n-1}\pi i}{(n-1)!}\delta^{(n-1)}(x)+\operatorname{fp}\frac1{x^n},$$ while the table of Fourier transforms ...
3
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0answers
164 views

Dominated convergence Theorem

I am struggling to understand the proof in the paper, Learning Temporal Evolution of Spatial Dependence with Generalized Spatiotemporal Gaussian Process Models. Theorem 2.1 in the page 33 uses ...
5
votes
0answers
178 views

Is there a practical application of natural integral or differintegral?

The following formulas give natural differintegral (that is one with naturally fixed integration constant): $$f^{(s)}(x)=\sum_{m=0}^{\infty} \binom {s}m \sum_{k=0}^m\binom mk(-1)^{m-k}f^{(k)}(x)$$ $$...
1
vote
1answer
58 views

Perform certain constrained integrations over an ordered subsection of a 3-simplex, yielding “absolute separability” probabilities

Let us order the four points $\lambda_1 \geq \lambda_2 \geq \lambda_3 \geq \lambda_4 \geq 0$ of a 3-simplex, $\lambda_1+\lambda_2+\lambda_3+\lambda_4=1$, giving us a subsection $L$. Integration over $...
0
votes
1answer
144 views

Taylor expension of a simple integral [closed]

I'm trying to derive some weights expression for a boosting algorithm on a L2-ISE loss function, and i have trouble with the taylor expension. Suppose that $f$ and $g$ are two densities from $\...
1
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0answers
90 views

Change variable in integration with symmetry

Not sure if I can ask such fundamental problem here. Let $G$ be a finite group, $\sigma \in G$. Consider linear group actions fo $G$ on $\mathbb{R}^n$. $\sigma(K)=\{x\in \mathbb{R}^n: \sigma^{-1}(...
2
votes
1answer
106 views

Example of evaluation of $\int_0^1\left(\sum_{k=0}^n (f(x))^k\right)^{\alpha}dx$, for some choice of $f(x)$ satisfying certain requirements

Let $0<\alpha\leq\frac{1}{2}$ a fixed real number. I wondered if it is possible to evaluate the sequence of definite integrals $$\int_0^1\left(\sum_{k=0}^n (f(x))^k\right)^{\alpha}dx\tag{1}$$ for ...
3
votes
3answers
296 views

Density of a functional space

Let $D$ be a bounded domain of $\mathbb{R}^n$ with smooth boundary $\partial D$. Is the following subspace dense in space $L^2(D)\times L^2(\partial D)$: $$\{(f,f\rvert_{\partial D}) : f\in C^\infty(\...
3
votes
1answer
419 views

Risch's algorithm for symbolic integration and its variations

I want to explore symbolic integration, but for this I initially need to imagine what are the algorithmic achievements in this area today, so I have some questions about Risch's algorithm and all its ...
2
votes
1answer
102 views

Sufficient condition for function of conditional probability density to be increasing

Let $Y$ and $W$ be two jointly distributed random variables; $Y$ takes values on $(y_1,y_2)$ and $W$ takes values on $(w_1,w_2)$. The conditional probability density of $W$ given $Y$ is given by $f_{W|...
4
votes
2answers
222 views

Express $\int_0^{\pi/2}\{ \operatorname{gd}^{-1}(x)\}dx$ as series of special functions, with $\operatorname{gd}^{-1}(z)$ the inverse Gudermannian

I know that in the literature there are a lot of integrals involving the fractional part (and other floor and ceiling functions). For some of these integrals is provide its evaluation as a series or ...
1
vote
1answer
68 views

Inequalities for the Gudermannian function of the type $\operatorname{gd}(x)\operatorname{gd}\left(\frac{1}{x}\right)<\text{upper bound}$, where $x>0$

After I've read the solution of Problem 4327 (see [1]) I wondered if an inequality for a similar upper bound in the RHS is feasible for the known as Gudermannian function $$\operatorname{gd}(x)\...
2
votes
1answer
161 views

Uniform sampling on a Riemannian manifold via tangent space and exponential map

Given a Riemannian manifold $(\mathcal{M}, \{g_x\}_{x \in \mathcal{M}})$ and a fixed point $x \in \mathcal{M}$, does the following procedure yield uniform samples from $\{y \in \mathcal{M} : d_\...
1
vote
1answer
221 views

Examples of Steffensen's inequality at undergraduated level studies

I've known few days ago the known as Steffensen's inequality, see the article Steffensen's inequality from Wolfram MathWorld and the cited bibliography. It seems that there are applications (I don't ...
13
votes
2answers
919 views

Challenge: Non-Gaussian quartic integral and path integral in Quantum field theory

(1) It is well-known that we can get a Gaussian integral of this type, where $x$ is in $\mathbb{R}$: $$ \int_{-\infty}^{\infty} dx e^{-ax^2}=\sqrt{(2\pi)/a}. \tag{i} $$ We can generalize this ...
1
vote
1answer
145 views

A marginal space splitting $\{ \psi \}^{\perp}$

Let $\psi \in L^2(\mathbb R^2,\mathbb C)$. Is there a continuous projection from $\{ \psi \}^{\perp}$ onto $$ \left\{ \varphi \in L^2(\mathbb R^2) \:\:\Big| \int \overline{\psi}(x,y) \varphi(x,y)\...
1
vote
1answer
87 views

On a type of sequence of integrals inspired in the Borwein integral and an integral due to Furdui

This post is inspired in the Borwein integral, and in a problem proposed by Ovidiu Furdui in Crux Mathematicorum, that is the Problem 3707, in page 151. I've considered integrals of the form $$\int_0^...
5
votes
1answer
2k views

A rather curious equality: is this true?

I came across (coincidentally) two integral evaluations, which seem to agree according to numerical tests. It did not seem easy to convert one into the other. QUESTION. Is this true? $$\int_0^1\...
1
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0answers
90 views

Reparametrization of a closed curve that balances sum of first derivatives

(Question in the yellow box below.) A few weeks ago I was wondering about the existence of a scalar function $f(s): S^1 \rightarrow \mathbb{R}$ and a turning angle $\phi(s):S^1 \rightarrow \mathbb{R}/...
1
vote
1answer
109 views

Limiting value of definite integral [closed]

$$I = \int_{-4}^4 e^{in\pi x/4}\frac{\sinh(b\pi/4)}{\sin^2\left(\frac{a-x}{8/\pi}\right)+\sinh^2(b\pi/8)}\,dx$$ I am unable to integrate the above equation when when $b$ tends to $0$, because of a ...
1
vote
2answers
357 views

Prove that a certain integration yields the value $\frac{7}{9}$

Numerical methods surely indicate that $\int_0^{\frac{1}{3}} 2 \sqrt{9 x+1} \sqrt{21 x-4 \sqrt{3} \sqrt{x (9 x+1)}+1} \left(4 \sqrt{3} \sqrt{x (9 x+1)}+1\right) \, dx= \frac{7}{9}$. Can this be ...
5
votes
1answer
177 views

Triangle inequality for Ito integral?

For Lebesgue integrals one has the triangle inequality saying that for continuous functions let's say $$\left\vert\int_0^t f(s) \ ds\right\vert \le \Vert f \Vert_{\infty} \int_0^t \ ds$$ Now if ...
0
votes
0answers
141 views

Weak derivative under the integral sign

Let $\Omega$ be a bounded and regular open subset $\Omega$ of $\mathbb{R}^N$ and $u:[0,\infty)\times \Omega\to \mathbb{R}$ be a smooth function (for example a smooth solution to a PDE). Thus the ...

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