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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4
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0answers
182 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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0answers
73 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
49 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 ...
-5
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0answers
47 views

Proving $ \sum_{n=1}^{ \infty } f_n(1) $ converges when $ \sum_{n=1}^{ \infty } f_n(x) $ converges uniformly [closed]

I have a question which I tried solving for a few hours. My only solid direction so far is trying using cauchy. Let $(f_n(x))$ be a series of continuous functions in $[0,1]$. Prove; If $ \sum_{n=1}^{ \...
2
votes
1answer
65 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
51 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
174 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
votes
1answer
278 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....
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0answers
20 views

Why does the integral for surface area require an expression for arc length? (Solid of Revo.) [migrated]

I'm a lowly Calculus II student here and I noticed something interesting/confusing. The integral formula for the volume of a solid of revolution (SoR) with the disk method is pi*f(x)^2 dx, the same as ...
2
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1answer
84 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
132 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}^...
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
votes
1answer
154 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 $(-\...
4
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0answers
141 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-...
0
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0answers
122 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
votes
1answer
125 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
votes
0answers
180 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 ...
0
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1answer
169 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
74 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
77 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
votes
0answers
114 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
25 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
votes
1answer
375 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
votes
1answer
169 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
95 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
1answer
57 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 ...
0
votes
2answers
88 views

Finding the expectation of $a \mathcal{Q} \left( \sqrt{b } \gamma \right) $, where $\gamma$ is a Gamma r.v

I'm trying to analytically find the following expectation $$\mathbb{E}\left[ a \mathcal{Q} \left( \sqrt{b } \gamma \right) \right],$$ where $a$ and $b$ are constant values, $\mathcal{Q}$ is the ...
3
votes
1answer
113 views

Closed form for the integral of a squared Legendre function

Is there a closed form for the integral $$\int_0^{\pi/2}(P_\nu^\mu(\cos\theta))^2\,\mathrm d\theta,\quad\mu>\nu\gt-\frac12$$ where $P_\nu^\mu(x)$ is the associated Legendre function of the first ...
1
vote
1answer
205 views

Computing the integral $\int_{-1}^1 dx \, |x| J_0(\alpha \sqrt{1 - x^2}) P_\ell(x)$

I would like to compute the following integral: $$ I_\ell(\alpha) := \int_{-1}^1 dx \, |x| J_0(\alpha \sqrt{1 - x^2}) P_\ell(x) \tag{1} \label{1} $$ where $\alpha \geq 0$, $J_0$ is the zeroth-order ...
2
votes
0answers
39 views

Removing integral from norm by inequality

My first question on Math Overflow. For my Mathematics Bachelor thesis I am looking at a paper called "Deep Limits of Residual Neural networks" by Matthew Thorpe and Yves van Gennip. (arxiv....
6
votes
2answers
221 views

An abstract characterization of line integrals

Let $M$ be a smooth manifold (endowed with a Riemann structure, if useful). If $\omega \in \Omega^1 (M)$ is a smooth $1$-form and $c : [0,1] \to M$ is a smooth curve, one defines the line integral of $...
-2
votes
1answer
188 views

Is it possible to express $\int_{0+\epsilon}^{1-\epsilon}\left(\sqrt{1-x^2}^{\sqrt{1-x^2}^{\cdots}}\right) dx$ in elementary functions?

let $\epsilon >0$, I tried to evaluate $\int_{0+\epsilon}^{1-\epsilon}\left(\sqrt{1-x^2}^{\sqrt{1-x^2}^{\cdots}}\right) dx$ , using the fact $x= \cos t$ yield to have integrand using $\sin $ ...
5
votes
2answers
408 views

Compute $ \int_{0}^{+\infty} \left( \frac{\ln(x)}{e^x}\right)^2 dx $ [closed]

How can I compute this integral? $$ \int_{0}^{+\infty} \left( \frac{\ln(x)}{e^x}\right)^2 dx $$
-1
votes
1answer
199 views

What is the integral of $r \frac{2^{r-1} \log (2) e^{-\frac{\sqrt{2^r-1}}{b}} \left(2^r-1\right)^{\frac{d}{2}-1}}{b^d \Gamma (d)}$?

I have been trying to solve a research problem for a while now and in doing so, I stumbled upon the following integral: $$\int_0^{\infty } r \frac{2^{r-1} \log (2) e^{-\frac{\sqrt{2^r-1}}{b}} \left(2^...
0
votes
1answer
47 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....
1
vote
0answers
83 views

A Fredholm equation with non-separable kernel [closed]

I'm trying to solve this form of Fredholm equation: $$ g(v)=f_1(v)+\int\limits_{0}^{v_\mathrm{th}} g(v_s)\frac{e^{-\tfrac{\big[(v-v_\mathrm{init})-(1-v_\mathrm{leak})(v_s-v_\mathrm{init})\big]^2}{2v_\...
5
votes
1answer
216 views

Three integral expressions for integer values of $\zeta(s)$. Could these be further reduced to known integrals?

In this MSE-question I've asked about three, similarly shaped, integrals for integer vales of $\zeta(s)$ that I found numerically: $$\zeta \left( 3 \right) =\frac12{\int_{0}^{1} \frac{1}{x}\big(\zeta(...
3
votes
2answers
295 views

From Zurab's integral representation for the Apéry's constant to almost impossible integrals

I would like to know if the following integrals are known, or in case that aren't in the literature we can calculate these in closed-form (in terms of elementary and standard functions). I wondered ...
0
votes
0answers
92 views

Solving integral equation with an unknown probability distribution

Considering this system of integral equations, where $\gamma \in \mathbb{R} $ and $\alpha\in \mathbb{C}$ are the unknown to solve : $$ 1=\int_{-\infty}^{\infty} p(u) \frac{ -1}{\gamma-\left(u-z^{*}+\...
7
votes
0answers
146 views

A function with unexpectedly simple Legendre transformation

Let $I(x) = \frac{1}{2\pi} \int_{-2}^2 \sqrt{4-y^2}\ln|x-y|dy$. Then $I(x)$ is a concave function and \begin{equation} I(x)= \begin{cases} \frac{1}{4}x^2-\frac{1}{2}, &\text{if } |x|\leq2 \\ \...
1
vote
0answers
52 views

Differentiable dependence on the initial condition of the solution of a SDE

Let $b,\sigma:\mathbb R\to\mathbb R$ be differentiable and Lipschitz continuous $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge0}$ be a complete and right-...
4
votes
1answer
292 views

Two definitions of $L^p$ spaces that are not always equivalent

There are two definitions of $L^p(S, \Sigma,\mu)$ in the literature. (Here $S$ is a set, $\Sigma$ is a $\sigma$-algebra of subsets of $S$ and $\mu$ is a positive measure.) The two definitions are ...
2
votes
0answers
66 views

Constructing a small Radon-Nikodym derivative

Let $u:\mathbb{R}^n\to\mathbb{R}^n$ be a $C^1$ function. Is it possible to (explicitly construction) a function $h$ such that: $0<h(x)$. $\int_{x \in \mathbb{R}^n} |h(x)|<\infty$, $\sup_{x \in ...
0
votes
0answers
37 views

Bounding the absolute value of a complex integral with itself

I already asked a similar question on this topic, but after a small discussion, I noted that I did must boil down the problem such that the solution space so to say to maybe have a concrete answer. I ...
0
votes
0answers
51 views

Bounding the absolute value of a complex integral

I'm working on some problems involving Fourier transforms and convolution problems and there is one problem I cannot solve. In my situation we have a complex valued function $f(ix)$, with $x\in\mathbb{...
0
votes
1answer
52 views

Variant of modified Bessel functions

Consider the integral \begin{align*} g_f(x)=\int_{\phi=0}^{2\pi} f(\phi) ~e^{x cos(\phi)}~\mathrm{d}\phi, \end{align*} where $f(\phi)$ is a probability density functions defined over $[0,2\pi]$, \...
0
votes
0answers
43 views

Oscillatory integral independent of a parameter

Let $h: \mathbb{R} \mapsto \mathbb{C}$ be a positive definite function, continuous at the origin. (In fact, $h$ is the Fourier transform of a finite measure). Define the oscillatory integral $$Q(t) :=...

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