# 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, ...

981
questions

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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**

votes

**5**answers

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**

votes

**1**answer

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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**0**answers

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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votes

**0**answers

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

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votes

**0**answers

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**

votes

**1**answer

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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votes

**0**answers

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**

votes

**1**answer

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**

votes

**2**answers

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 ...

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**4**answers

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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votes

**1**answer

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**

votes

**1**answer

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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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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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**

votes

**1**answer

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**

votes

**0**answers

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 ...

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votes

**1**answer

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}...

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votes

**5**answers

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**

votes

**6**answers

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**

votes

**0**answers

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**

votes

**1**answer

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

**1**answer

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

**1**answer

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 ...

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**0**answers

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}...

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votes

**2**answers

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 ...

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votes

**1**answer

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 $(l, r)$...

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**1**answer

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**

vote

**1**answer

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 ...

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votes

**0**answers

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 ...

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votes

**0**answers

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**

votes

**1**answer

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

**1**answer

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

**1**answer

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 ...

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**0**answers

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

**1**answer

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 $\...

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votes

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**3**answers

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

**0**answers

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 ...

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**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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}^...