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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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 $...
17
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5answers
888 views

Closed-form expression for certain product

$\mathrm G$ is Catalan's constant. I recently found the product $$ \alpha=\prod_{n=1}^{\infty}\frac{E_n(\frac12)E_n(\frac7{12})E_n(\frac1{20})E_n(\frac{13}{20})}{E_n(\frac14)E_n(\frac1{12})E_n(\...
13
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1answer
763 views

Was Cantor aware of Lebesgue theory of integration?

Georg Cantor died in 1919, more than ten years after appearance of the Lebesgue theory of measure and integration at the beginning of the twentieth century. Lebesgue theory has a deep connection with ...
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0answers
22 views

How do I solve this using the cylindric shell method? [closed]

I'm having trouble figuring out how I can solve this: ![1]: https://i.stack.imgur.com/rSB1d.png
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0answers
45 views

Solve: dy/dx = 1/ (e^(y) - x)? [closed]

This question is related to integrals. I'm try to solve it but I stuck in between can somebody help me out with this
0
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0answers
63 views

Real integrals with complex analysis [closed]

I don't have a clear formal viewpoint on this problem. Resolving the Euler-Lagrange equations for the string with a point mass perturbation: $$ \frac{\partial^2 \phi }{\partial x^2} = \delta (x-a)$$ I ...
0
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1answer
53 views

Looking for a family of random variables such that only the second clause is fulfilled [closed]

Working with the epsilon-delta-criterium, a family $(X_i)_{i \in I}$ on $(\Omega,A,P)$ is uniformly integrable if i) $sup_{i \in I} E(X_i) <\infty$ ii) $\forall \epsilon>0$ ex. $\delta>0$ s.t....
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0answers
50 views

Gaussian integral on a compact set

I am trying to compute a Gaussian integral on a compact set $\mathcal{R}\subset\mathbb{R}^n$. Through Taylor expansion of the exponential function, we could obtain, $$\int_{\mathcal{R}} e^{a\mathbf{x}^...
3
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1answer
189 views

Exchanging series and integrals

I know that I can use Lebesgue or monotone convergence theorem to exchange limit of partial sums and a Lebesgue integral, given a power series or a generic function series. But in general given a ...
34
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2answers
2k views

Does there exist a complete implementation of the Risch algorithm?

Is there a generally available (commercial or not) complete implementation of the Risch algorithm for determining whether an elementary function has an elementary antiderivative? The Wikipedia article ...
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 ...
3
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1answer
249 views

Is this operator continuous?

Let $I=[0,1]$ and $E$ a Banach space. We note by $X:=\mathcal {C}(I,E), $ the space of all continuous functions from $I$ to $E$, with $\left \| x \right \|_X=\sup_{t\in I }\left \| x(t) \right \|_E $. ...
2
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1answer
77 views

Integral inequality for Schwartz function

Let $s\in(0,1)$, $u\in\mathcal{S}({\mathbb{R}^n})$, $x\in\mathbb{R^n}$ with: $|x|\geq1$, i have to prove that: $$ \int_{B_{|x|/2}(0)} \frac{|u(x+y)+u(x-y)-2u(x)|}{|y|^{n+2s}}\,dy\leq c|x|^{-n-2s}, $$ ...
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0answers
36 views

Integral and derivative inequality [closed]

If $f(x)$ is derivative in the $[0,1]$ and $|f'(x)|\le M$ for all $x\in[0,1]$, the prove that $$\left|\frac1n\sum_{k=0}^{n-1} f(\frac kn) - \int_0^1 f(x) du\right| \le \frac M{2^n}$$
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0answers
44 views

Haar measure decomposition using orbital integrals

Let $G$ be a unimodular locally compact group, $N,A \le G$ be unimodular closed subgroups. Suppose that $A$ normalizes $N$. Let $N_0 \le N$ be a compact open subgroup. Suppose that a function $f : N \...
2
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1answer
86 views

An inequality involving fractional Laplacian

I have to prove that for $s\in(0,1)$, $u\in\mathcal{S}(\mathbb{R}^n)$, (i.e. $u$ is a Schwartz function): $$ |(-\Delta)^su(x)|\leq c_{n,s}|x|^{-n-2s},\quad\forall x\in\mathbb{R}^n\setminus B_1(0), $$ ...
5
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0answers
66 views

Extending gauge integral to higher dimensions/spaces and analogue of Riemann rearrangement theorem for it

The gauge/Henstock-Kurzweil integral allows for the integration of a very large set of functions in $\Bbb R$, at the cost of many of the nice properties of Lebesgue integration, of which it is a ...
8
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1answer
126 views

Integral representation of product of two Whittaker functions

Does anyone know anything about the following formula involving special functions: $$ W_{\kappa,\mu}(z)W_{\lambda,\mu}(w)=\frac{e^{-(z+w)/2}(zw)^{\mu+1/2}}{\Gamma(1-\kappa-\lambda)}\int_0^\infty e^{-t}...
13
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5answers
9k views

The Fundamental Theorem of Calculus in Lebesgue Theory

I am interested to what extent the famous identity $$ \int_a^b f'(x) \ dx=f(b)-f(a) $$ is true for a function $f:[a,b]\to \mathbb C$ continuous on $[a,b]$ and differentiable on $(a,b)$. One famous ...
34
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6answers
5k 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}: ...
5
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0answers
176 views

More or less universal formula for regularization of divergent integrals?

Is there a simple formula that would produce the regularized value for the most common divergent integrals? I know, there is a formula for Cesaro integration, but it is applicable only to Cesaro-...
4
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1answer
78 views

Family of Pettis integrals functions “uniformly approximated” by sums

In this book (proof of $4.1.3.$ Lemma. exactly), one can find this passage, that I tried to rephrase here: Let $f:I\times E\rightarrow E$ a Pettis integrable function, where $I:=[0,T]\subset \mathbb{...
5
votes
1answer
166 views

Is there a standard way of defining the integral of an extended real function with respect to a finitely additive probability measure?

Let $X$ be a set, and let $\mu$ be a finitely additive probability measure defined on $2^X$. Let $\Phi$ be the set of functions from $X$ to $\mathbb R \cup \{-\infty, \infty\}$. Is there a standard ...
3
votes
1answer
67 views

Integration on quasi-Banach spaces and Schatten ideals

Let $[a,b]$ be an interval and $X$ a Banach space (for starters). We know that continuous functions $f:[a,b]\to X$ are Riemann integrable. Suppose now that $X$ is a quasi-Banach space, that is, its ...
2
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0answers
53 views

How to define Lebesgue Integrability of functions assuming values in an arbitrary topological vector space over an arbitrary topological field?

I have already asked this question in this MSE thread, but some people suggested me to ask to the MO community also. Preliminaries An algebra of sets in a set $X$ is an $\mathcal{X}\subseteq\mathcal{P}...
3
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2answers
126 views

Conditions for continuity of an integral functional

Let $X$ be a metric space, $\nu,\mu$ be Borel measures on $X$, $f:X\times \mathbb{R}\rightarrow [0,\infty)$ be a measurable function. Under what conditions is the integral functional $F_f$, defined ...
2
votes
1answer
288 views

Asymptotic behaviour of an integral. How should I proceed?

Let us consider the following SDE: $$dY_t=b(Y_t)dt+\sigma(Y_t)dW_t\tag{1}$$ with $b, \sigma: (l, r)\to\mathbb{R}$, $−\infty \leq l < r \leq \infty$ bounded functions on compact intervals of $(l, r)$...
0
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1answer
109 views

Formally confirm a formula for a certain three-dimensional constrained integral over the unit cube

The result of the three-dimensional constrained integration (for the Hilbert-Schmidt two-qubit absolute separability probability) over the unit cube $[0,1]^3$ \begin{equation} \label{one} \int_0^1 \...
1
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1answer
156 views

A Bessel-like integral

I encounter the following integral when trying to find the inverse Fourier transform of the characteristic function of a certain sum of random variables. Here, $0\le\lambda\le1$, $p\ge0$, $q\ge0$ are ...
5
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0answers
147 views

Modified energy method for transformed Fokker-Planck equation (tricky integration by parts…)

I came across Villani's paper titled "Hypocoercive diffusion operators" and couldn't figure out a computation that is skipped in that paper. Specifically, consider the following transformed ...
2
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0answers
65 views

Haar measure and Integral

I am wondering whether the following integral over Haar measure has explicit form(edit: say $U$ is $d\times d$ unitary, orthogonal or symplectic matrix) $$ \int dU [(U\otimes U^*)X(U^{+}\otimes U^T)]^{...
7
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1answer
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 ...
0
votes
1answer
44 views

Laplace transforms of fractional equation

is there a finite expression of the Laplace transforms of the function \begin{align} L\left[ {\frac{{{x^n}}}{{{{(1 + x)}^m}}}} \right]\quad n \ge m \end{align}
4
votes
1answer
124 views

Integration theory for functions and values with values in topological rings

I am curious whether somebody ever tried to generalize the classical theory of Lebesgue integral to functions and measures with values in Hausdorff topological rings. The generalization of a measure ...
4
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0answers
120 views

Length of the arc of a Fourier series

I'm working modeling the behavior of periodic variable stars and I have a question about reducing the expression of a parameter involved in this analysis. Let $f(t)$ be a Fourier series define as: $$f(...
5
votes
1answer
264 views

Is there a closed-form expression for these integrals?

I am computing the following integrals by numerical integration and this takes a lot of time, although I'm sure there is a general closed-form formula but I can't find it. Let $t$ be a vector of $\...
-1
votes
1answer
60 views

Asymptotic expansion / analysis of this integral

As $M \to +\infty$, how could I make a good asymptotic analysis of this integral? $$\int_0^1 \dfrac{\cos(M x)}{1 + x^2} e^{-M (x^2 - 1/9)}\ \text{d}x$$ The exponential term shall dominate, yet I ...
2
votes
1answer
79 views

fractional moments of multivariate normal distributions

is there an analytic formula for fractional moments of multivariate normal distribution? $E(\prod_{i=1}^k x_i^{\nu_i})=?$ where $X=(x_1,\ldots,x_k) \sim N_k(\mu, \Sigma)$, $\nu_i\in \mathcal{R}$ and $\...
3
votes
0answers
541 views

On properties on a certain functional

Consider the following function: $$F(z) = \omega(z)\sin^2\left(\frac{c\Gamma(z)}{z}\right)$$ Here, $\omega(z)$ is a weight we have to construct and $c$ is a constant. The following three conditions ...
2
votes
1answer
84 views

Definite integral of modified Bessel function of second kind

How do I integrate a modified Bessel function of the second kind as shown below? A good approximation of the definite integral is also ok, I do not need an exact solution. $\int_\frac{1}{\lambda}^{\...
5
votes
3answers
1k views

Value of an integral

I need to verify the value of the following integral $$ 4n(n-1)\int_0^1 \frac{1}{8t^3}\left[\frac{(2t-t^2)^{n+1}}{(n+1)}-\frac{t^{2n+2}}{n+1}-t^4\{\frac{(2t-t^2)^{n-1}}{n-1}-\frac{t^{2n-2}}{n-1} \} \...
0
votes
0answers
55 views

Solving $\int_0^\infty N(1-F(t))^{N-1}tf(t)dt$ when the expected value is known

Suppose that $f:\mathbb R_{\geq 0} \rightarrow \mathbb R_{\geq 0}$ is a probability density function, and $F$ is a cumulative distribution function (i.e. $F(t)=\int_0^t kf(k)dk$). Also, assume that ...
3
votes
0answers
180 views

An “elementary” inequality

The following is left unproven in a monograph. The author refers to it as "elementary exercise" but I am unable to prove it. Any insight is appreciated. $$ \int f \log f d\mu \le 2 \left[\...
0
votes
1answer
130 views

Bounds on expectation of $X/(X^2 + c)$ with $X$ ~ Gaussian and $c > 0$

I'm trying to compute expectation of $X / (X^2 + c)$ when $X$ is normally distributed with mean $\mu$ and variance $\sigma^2$, and $c$ is some positive constant. I think this cannot be solved ...
2
votes
1answer
161 views

explicit computation of fractional Laplacian of a function

For $x\in\mathbb R$ let $$ u(x)=\begin{cases} |x|^{2s-1}-1 &\mbox{if } |x|>1,\\ 0 & \mbox{otherwise}. \end{cases} $$ Is it possible to calculate explicitly the fractional Laplacian $(-\...
1
vote
1answer
266 views

On Riemann integration of stochastic processes of order $p$

Let $x:[a,b]\times\Omega\rightarrow\mathbb{R}$ be a stochastic process, where $\Omega$ is the sample space from an underlying probability space. Let $L^p$ be the Lebesgue space of random variables on $...
5
votes
1answer
285 views

Spherical average of $\frac{1}{x}$

Let $X_1,...,X_n$ be points on $\mathbb S^1.$ We then define the expectation value $E(X)=\frac{1}{n}\sum_{i=1}^n X_i.$ Let $\frac{dS(X_1)}{2\pi}$ be the normalized surface measure of $\mathbb S^1,$ i....
2
votes
0answers
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 $\operatorname{li} (x)$ ...
2
votes
1answer
89 views

Non-zero, bounded, continuous, differentiable at the origin, compactly supported functions with everywhere non-negative Fourier transforms

Do there exist functions $F(x) \! : \, \mathbb R \to \mathbb R$ which are non-zero and bounded: $$ \mathrm {Range} (F) = [l, u] \, , \quad \mathrm {where} \quad l, u \in \mathbb R \land u > l \, ; \...
1
vote
1answer
138 views

Optimization problem with definite integral inequality constraints

Question: How can we prove that there exists a real constant $c\ge 1$ such that the following inequality holds for all integers $d>1$ and all real numbers $r\in\left[1,\sqrt{d}\right]$? $$\int_{-1}^...

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