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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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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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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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 ...
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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 ...
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 ...
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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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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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): $$ |(-\Delta)^su(x)|\leq c_{n,s}|x|^{-n-2s},\quad\forall x\in\mathbb{R}^n\setminus B_1(0), $$ ...
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2answers
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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 ...
5
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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 ...
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 $. ...
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}...
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{...
3
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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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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}...
5
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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 ...
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)]^{...
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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}
2
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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 $(l, r)$...
4
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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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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
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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 $\...
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 \...
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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
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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}^{\...
0
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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 ...
5
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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} \} \...
3
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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[\...
5
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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
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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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0answers
48 views

Extension to all dimensions of complex line integral

Let $\Gamma$ be a smooth curve in $\mathbb{C}^d$. Since $\mathbb{C}^d$ can be seen as $\mathbb{R}^{2d}$, one can define the line integral of functions $f:\Gamma\to \mathbb{C}$ using for instance ...
2
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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 $(-\...
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-...
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0answers
123 views

Does the following sequence $\{g_n\}$ converge?

Consider a function sequence $\{f_n(t)\}$ ($n\in\mathbb{N}^+$) defined on the interval $(\frac{1}{2},1)$, where \begin{eqnarray}\label{eqn:constraint1} f_n(t)=\frac{\exp\left(n\left(\log R(h_t) - th_t\...
2
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1answer
128 views

When is it possible to use the Parseval-Plancherel identity to solve an integral?

The integral is of the form $\int_{-\infty}^\infty \sigma(x)\mu(x)\,\mathrm{d}x$. Where the Fourier transform of the $\sigma$ function is $\tilde \sigma(p)= e^{-iap}\frac{1}{1+e^{-c|p|}}$ and the ...
5
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0answers
182 views

Riesz Representation Theorem for $L^2(\mathbb{R}) \oplus L^2(\mathbb{T})$?

The spaces $L^2(\mathbb{R})$ (square-integrable functions) and $L^2(\mathbb{T})$ (1-periodic square-integrable functions, considered over the real line $\mathbb{R}$) are two subspaces of the space of ...
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1answer
172 views

Prove convergence of the integral $\int_0^{\infty} xe^{-x^6\sin^2x}dx$ [closed]

I would perhaps to this post add problems of a similar kind. Problem 1. Prove convergence of the integral $\int_0^{\infty} xe^{-x^6\sin^2x}dx$, From https://www.desmos.com/calculator/sjizhbtbhp we see ...
2
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0answers
78 views

Integral of Legendre's function

Is there any formula for computing the following integral $$\int_a^1(P^m_l)^2(x)\,dx \, ,\text{with} -1<a<1$$ where $P^m_l$ is the associated Legendre's function (of the first kind) of order $m\...
2
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0answers
79 views

inequality for two integral expressions

Given bounded positive functions $f, g, h \in L^1$, with $g, h$ finitely supported, I would like to compare the following two expressions: $$\begin{aligned} a_1 &= \int_{0}^{\infty} \! dx \, f(x) \...
4
votes
1answer
88 views

Deriving integral in Gaiotto-Tommasiello theory

I was looking at a paper by Takao Suyama on GT theory, and I couldn't figure out how he derived his formula (3.59): $$\frac{1}{\pi}\int_a^bdx\frac{1}{z-x}\frac{\sqrt{(z-a)(z-b)}}{\sqrt{|(x-a)(x-b)|}}\...
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 ...
0
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0answers
26 views

Verification of an Cauchy's contour Integral of Complementary Error function?

I tried to find an integral of the following,$\DeclareMathOperator{\erfc}{erfc}$ $\int\limits_0^{2\pi} \erfc(a + b\cos(\theta))\erfc(c + d\sin(\theta))\,d\theta $ Where, $a,b,c,d \in \Bbb R$ Now, $\...
4
votes
1answer
80 views

An integral of composite function of triangle functions [closed]

I expected the following formula to hold: $\int^{2n\pi}_0\cos(\sin t+t/n)dt=0$, for ${}^\forall n\in\mathbb{N},\ n\geq2$ But I can't prove it. Could you please tell me.
8
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1answer
381 views

Deep applications of the Pettis integral?

In the Notes section of chapter 2 of Diestel and Uhl's Vector Measures they make the comment: "Presently the Pettis integral has very few applications. But our prediction is that when (and if) ...
4
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1answer
177 views

How to estimate the order of this integral with parameter

Some introduction: Given a homogeneous structure called "dilation" in $R^n$: For $t\geq 0$ $$D_t: R^n\rightarrow R^n$$ $$D_t(x)=(t^{a_1}x_1,...,t^{a_n}x_n)$$ where $1=a_1\leq...\leq a_n$, ...
1
vote
2answers
99 views

Inaccurate results for the analytical expression of $\mathbb{E}\left[ a \mathcal{Q} \left( \sqrt{b } \gamma \right) \right]$

I'm trying to plot a graph for the following expectation $$\mathbb{E}\left[ a \mathcal{Q} \left( \sqrt{b } \gamma \right) \right]=a 2^{-\frac{\kappa }{2}-1} b^{-\frac{\kappa }{2}} \theta ^{-\kappa } \...
3
votes
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 ...

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