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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Integral of the deconvolution kernel density estimator

Let $Y_i = m(X_i) + \eta_i$, $W_j = X_j + U_j$, $E[\eta_i | X_i] = 0$ with $X_i \sim f_X$, $U_i \sim f_U$ be an errors-in-variable problem and $K_{U}(x) = \dfrac{1}{2 \pi} \int \mathrm{e}^{-itx} \...
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Is the integral functional analytic?

Let $E = C^k(\overline{U})$, $k \geq 1$, be the Banach space of real functions of class $C^k$ on a bounded open set $U \subset \mathbb{R}^n$, with its usual norm. Let $I : E \to \mathbb{R}$ be the ...
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1 vote
3 answers
200 views

Squeezing more convergence from the convergence in all $L^p$ spaces

Let $X$ be a space endowed with a finite measure $m$. Let $f_n : \to \mathbb C$ be measurable functions such that $|f_n| \le 1$ for all $n$ and $f_n \to 0$ in every space $L^p (X)$ with $p \in [1, \...
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Can the 4-dimensional volume integral be converted into a surface integral? [closed]

Given two-dimensional location and velocity fields $x,u : \mathbb{R}^2 \to \mathbb{R}^2$$, I am trying to convert the volume integral of the cross-product between position and velocity vector into ...
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Difference between summation for "$\aleph$" terms and summation for "$\aleph_0$" terms

Addition: Could we say that the dimension of a space is "$\aleph_0$" or"$\aleph$"? I guess that every elementary functions can be uniquely expanded as integer order power series ...
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1 vote
2 answers
134 views

Is $\int_{\mathbb{R}} \int_{\mathbb{R}^n} \alpha w(t) e(\alpha (a_1t_1 + \dotsb + a_n t_n)) dt\,d \alpha = 0$?

Let $a_i$ be a nonzero real number for each $1 \leq i \leq n$. $w$ a smooth nonnegative with compact support. I would like to understand the following integral. $$ I = \int_{\mathbb{R}} \int_{\mathbb{...
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2 votes
1 answer
87 views

Probability density of a hyperplane for a Gaussian distribution

I have a vector $\mathbf{x}$ with a multivariate Gaussian distribution $$P[\textbf{x}\in S] =\int_{\textbf{x}\in S} \det(2\pi H^{-1})^{-1/2}\exp(-\frac{1}{2} \textbf{x}^T H\textbf{x}) \, d\textbf{x}$$...
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1 answer
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Weak lower semicontinuity of a sequence of Riemann sums

Let us have a sequence of functions $\{f^K\}_{K \in \mathbb{N}} \in C([0,1],\mathbb{R})$ which is uniformly bounded in $L^2((0,1))$. We observe a sequence of Riemann sums $$R^K=\frac{1}{K} \sum_{k=0}^{...
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Integration of matrix form of Vasicek variance (Python/Matlab)

$X_t$ is a vector and follows the following Vasicek process. $$ dX_t=(mu-K\cdot X_t)dt+Sigma_x\cdot dZ_t \\ $$ What is the variance of $X_t$? In scalar form the answer is $\frac{Sigma_x^2}{2\cdot K}\...
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How to relate this integration with the integral expansion of special functions?

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, $p\ge0$, $q\ge0$ are real, and $n,a,b$ ...
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4 votes
1 answer
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Can the integral inherit the Lipschitz continuity of its integrand?

Let $C$ be the set of continuous functions on $[0,T]$ taking values in $[0,1]$. Denote $\|f-g\|_t:=\max_{0\le u\le t}|f(u)-g(u)|$ for $f,g \in C$ and $t\in [0, T]$. Let $\phi: C\times C\times \{(s,t): ...
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Integral of $f \in L^{\infty}(\mathbb{T}^2)$

Let $f \in L^{\infty}(\mathbb{T}^2)$, where $\mathbb{T}$ is the torus. Can we somehow compare the integral $$\int_{\mathbb{T}^2}\int_{\mathbb{T}^2} \left( \int_{3x+y=z} f(w+x)f(w+y)dm_{\mathbb{T}^2}(x,...
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5 votes
2 answers
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Inverse Mellin transform of 3 gamma functions product

I want to calculate the inverse Mellin transform of products of 3 gamma functions. $$F\left ( s \right )=\frac{1}{2i\pi}\int \Gamma(s)\Gamma (2s+a)\Gamma( 2s+b)x^{-s}ds$$ Above contour integral has ...
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How was this heat semigroup estimate made in a paper on reaction–diffusion systems?

In Yamauchi - Blow-up results for a reaction–diffusion system, in the proof of Lemma 3.3, there is the passage $$S(t-s)|x|^{\sigma/1-k} \geq C_1(t-s)^{\sigma/2(1-k)}.$$ Here $S(t)$ denotes the heat ...
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What is the justification for using Wiener integrals to integrate over a space of differentiable functions?

In the literature on stiff/semiflexible polymer chains modelled as continuous chains rather than as discrete links, the partition function (among other things) is taken to be an integral over the ...
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1 vote
0 answers
112 views

Radon-Nikodym derivative of vector-valued measure with respect to another vector-valued measure

Let $(X, | \cdot |)$ be a Banach space. I am investigating whether one can extend the definition of the Kullback-Leibler divergence $$ \text{KL}(\mu \ \Vert \ \nu) := \int_{\Omega} \ln\left(\frac{\...
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3 votes
1 answer
177 views

A specific integration with Grassmann variables

I have recently read (for example, here) that this relation below is true $$ \int dz \: e^{\frac{1}{2} \sum_{ij} z_i A_{ij} z_j} = Pf(\mathbf{A}), $$ where $Pf(\mathbf{A})$ is the Pfaffian of an even ...
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22 votes
2 answers
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$\mathbf{P} = \mathbf{NP}$, what's the problem?

Let's take the problem of the backpack: $A_1,\ldots ,A_n$ the weights that are integers, and we want to know if we can achieve a total weight of $V$. We take $$I=\dfrac{1}{2\pi}\int_0^{2\pi} \exp(-iVt)...
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2 votes
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Birationally equivalent elliptic curves and singularities

I got the following cubic elliptic curve from some physical problem $$E_c(\mathbb{C}): w^2=4 z^3-zG_2-G_3,$$ where $G_2=3 \alpha ^2+\gamma$ and $G_3=\alpha ^3-\alpha \gamma -\beta ^2$ for known ...
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3 votes
1 answer
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Questions on the integral of pseudo Gaussian kernel and its derivative on $(0,\infty)$

Consider pseudo Gaussian densities for $0<s<t$ and $x,y\in\mathbb R$ $$f(s,x,t,y):=\frac{1}{\sqrt{2\pi A(s,t,y)}}\exp\left(-\frac{(y-x)^2}{2A(s,t,y)}\right)\quad\mbox{and} \quad g(s,x,t,y):=\...
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1 vote
1 answer
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Convergence of oscillatory integrals

I'm considering integrals of the (Hilbert transform) type $$p.v.\int_{-\infty}^\infty\frac{f(r)}{r}\,dr$$ where $f(r)$ is periodic, say, with period $2\pi$. I'm assuming very little regularity on $f$. ...
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7 votes
1 answer
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Best approximation of L1 function by Lipschitz function

Fix constant $L,C>0$ and $k\geq 1$ and let $f\in W^{1,k}(\mathbb{R}^d,\mathbb{R}^n)$ with $\|f\|_{W^{1,k}}\leq C$. Is there a known estimate on the distance $$ \|f - \operatorname{Lip}_L(\mathbb{R}^...
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1 vote
0 answers
56 views

How to define integration for general functions?

In general, the integration is defined on a measurable space $(X, \mathcal{A}, \mu)$, where $X$ is the whole space, $\mathcal{A}$ is a $\sigma$-algebra on $X$, and measure $\mu$. Usually integration ...
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3 votes
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Gaussian integral with Vandermonde determinant

I want to compute the following integral, which contains a Gaussian piece and a Vandermonde determinant: $$ \int d^Nx \,e^{-\frac{1}{2} \sum_{k=1}^N a_k x_k^2 + \sum_{k=1}^N b_k x_k} \Delta(x), $$ ...
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5 votes
1 answer
123 views

Asymptotics of error function integral with square root

I am interested in the asymptotics of the integral $$I(a):=\int_0^\infty \sqrt{x}\operatorname{Erfc}(x+a)\,\mathrm{d}x$$ for $a>0$. I think that $I(a)$ should be decaying exponentially as $I(a)\...
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35 votes
5 answers
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Looking for some interesting complex integration contours

I am currently working on some tools to make contour integration in a proof assistant less painful and I'm looking for interesting examples of contours in the complex plane used in the literature. I ...
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2 votes
1 answer
77 views

Radon transform of the function $h(x_1,\ldots,x_n) = x_1 g(x_1,\ldots,x_n)$, where $g$ is the density of multivariate Gaussian $N(\mu,\Sigma)$

Given an absolutely integrable function $f:\mathbb R^n \to \mathbb R$, let $R[f]$ be its Radon transform defined for every $(w,b) \in (\mathbb R^n \setminus \{0\}) \times \mathbb R$ by $$ R[f](w,b) := ...
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2 votes
1 answer
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As-closed-as-possible formula for an integral and/or sum

I need to find the solution of this integral: $$\int^\pi_{-\pi}{d\varphi\,{}\frac{\Gamma(n,2\pi\varphi)}{(1+ia\varphi)^n}},\tag{1}\label{1}$$ where $a\in(0,1)$ and $n$ is a positive integer (not zero)....
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-1 votes
1 answer
66 views

Closed-form CDF for bivariate normal distribution in point $(\Phi^{-1}(p),\,\Phi^{-1}(p))$

Let $\Phi(x)$ be a CDF of standard normal distribution and $\Phi^{-1}(p),\,p\in(0,1)$ its inverse. It is evident that $$ \mathbb{P}(X<\Phi^{-1}(p))=\Phi(\Phi^{-1}(p))=p, $$ where $X\sim N(0,1)$. Is ...
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3 votes
2 answers
149 views

Bochner integral over convex sets lies in the convex set?

Let $(\Omega,\Sigma,\mu)$ be a probability space, $E$ be a separable Banach space, $f:\Omega\rightarrow E$ be Borel-measurable and consider the Bochner-integral $$ \bar{\mu}:=\int_{\omega\in \Omega}\, ...
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1 vote
0 answers
56 views

Prove or disprove the positivity of the ess inf of a singular function

Consider a measurable radial function $u:\Bbb R^d\to(0,\infty)$ such that $$\int_{B_\delta(0)} u(x) d x=\infty\quad\forall\,\, \delta>0.$$ I would like to prove or to disprove that there exists $r&...
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4 votes
0 answers
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Integrating a function of distance between a set and its neighbourhood

I am aware of the isoperimetric inequality, which states that if you fix the Lebesgue measure of a measurable set $A \subset \mathbb R^d. d \geq 2$ then the smallest possible value of the perimeter of ...
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3 votes
1 answer
111 views

On an asymptotic integral

Let $\phi, a \in C^{\infty}([0,1])$ and assume $a(0)=1$. Suppose that $$ \int_0^1 e^{\tau \,\phi(t)}\,a(t)\,dt =0 \qquad \text{for all $\tau \in \mathbb R$}. $$ Does it follow that $\phi$ is a ...
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1 vote
0 answers
52 views

Fourier transform of inverse of determinant of 1+ skew-symmetric matrix

I have asked the following question in math stackexchange(https://math.stackexchange.com/questions/4389626/fourier-transform-of-inverse-of-determinant-of-1-skew-symmetric-matrix), but did not receive ...
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7 votes
1 answer
116 views

Is there a purely constructive presentation of the HK integral?

Treating the Riemann integral in a constructive setting is easy and straightforward. Treating the closely related but much more powerful Henstock-Kurzweil integral constructively is almost easy, ...
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1 vote
0 answers
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Does this book use non-mainstream stochastic analysis constructions and is thus perhaps not a good start?

I'm attempting to read a book on stochastic calculus by D.H. Fremlin, which is the 6th volume of his treatise on measure theory encompassing all kinds of topics related it. Before I make a very ...
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2 votes
2 answers
127 views

Lower bound for integrals like $\int_1^{t+1}e^{-\sqrt{s}}s^{-1}ds$

Let $$I(t) = \int_{1}^{t+1}\exp\left\{-c\frac{s^{1-\beta}}{1-\beta}\right\}s^{-2\beta}ds,$$ where $c$ is some positive constant and $\beta\in(0, 1)$. Since the integral $I(t)$ given above could not be ...
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1 vote
1 answer
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Exponential decay bound on integral

I have an integral of the form $$ \int_R^{\infty} e^{-x} x^n \vert L_m^{\alpha}(x) \vert^2 \ dx,$$ where $L_m^{\alpha}$ is the generalized Laguerre polynomial and $n \ge 0.$ I would to get a nice ...
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0 votes
0 answers
32 views

Integral involving the product of Kummer Confluent Hypergeometric and exponential functions

I am having trouble with calculating the following integral: $$\int_{0}^{z} x^{b - 1} (z - x)^{\alpha - 1} e^{-\beta x} {}_1 F_1{(a;b;x)} \, dx,$$ where ${}_1F_1$ is the Kummer Confluent ...
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0 votes
1 answer
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Characterizing subsets of integrable functions

Let us consider the space $L^1(0,1;\mathbb{R})$ of real-valued, Lebesgue integrable functions defined on the interval $(0,1)$ (where we only distinguish functions which are not equal almost everywhere)...
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26 votes
0 answers
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Is $\int_0^\infty{dx\over x^{x^{x^x}}}<\int_0^\infty{dx\over x^{x^{x^{x^{x^x}}}}}<\int_0^\infty{dx\over x^{x^{x^{x^{x^{x^{x^x}}}}}}}<\cdots$ true?

On MSE I asked whether each of $\int_0^\infty\frac{dx}{x^x},\int_0^\infty\frac{dx}{x^{x^{x^x}}},\int_0^\infty\frac{dx}{x^{x^{x^{x^{x^x}}}}},\cdots$ was less than $2$ and received answers on bounding ...
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1 vote
1 answer
112 views

Volume of a frustum knowing the volume and height of the pyramid and the height of the frustum [closed]

Can I calculate the volume of a frustum if all I know is the volume of the pyramid the height of the pyramid and the height of the frustum?
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1 vote
0 answers
55 views

Is the set of functions between two arbitrarily close L^1 functions closed?

Let $(X, {\cal \tau}, \mu)$ be a probability space and $F$ some vector subspace of integrable functions (defined everywhere). I can look at the set $$\hat{F} = \{f \hbox{ integrable } \mid \forall \...
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Computing $\int^{4b}_0 {e^{-tx}\biggl(\frac{\sqrt{4bx-x^2}}{(2b-2c)^2+4cx}\biggr) dx}$

I am a PhD student working on complex analysis. After integrating over a keyhole contour to obtain the inverse of a particular Laplace transform, I ended up with the following integral: $$\frac{1}{\pi}...
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1 vote
1 answer
76 views

How to prove that $\int (1-z)^{u} z^{v} dz$ is equal to $\frac{z^{v+1}}{v+1}_2F_1(-u, v+1; v+2; z)$?

How to prove that $$\int (1-z)^{u} z^{v} dz = \frac{z^{v+1}}{v+1} \, _2F_1(-u, v+1; v+2; z)?$$
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1 vote
1 answer
152 views

Two trigonometric integrals: looking for a transformation

I have two integrals of trigonometric functions and I would like to ask: QUESTION. Is there a transformation rule (or general principle) to show this equality? $$\int_0^{\frac{\pi}2}\frac{d\theta}{\...
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0 votes
1 answer
69 views

Integral form of expectation with respect to complex random variables [closed]

Let $h$ be a random variable and $g(h)$ be a real-valued function of $h$. We know that if h is a real-random variable then: $E_h[g(h)] = \int_{-\infty}^{\infty} f(h) g(h) dh$ where f(h) is the PDF of ...
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0 votes
1 answer
135 views

What are the steps involved in the solution to $\int{x^{-a} (b -cx^{-d})^e }dx$? [closed]

Mathematica gives me the following solution to $\int{x^{-a} (b -cx^{-d})^e }$: $$\int{x^{-a} (b -cx^{-d})^e dx} = -\frac{b^{e}x^{1-a} \, _2F_1\left(\frac{a-1}{d},-e;\frac{a+d-1}{d};\frac{c x^{-d}}{b}\...
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4 votes
1 answer
105 views

Antiderivative of $f\left( \xi \right) =\xi^{a}\left( b\xi ^c+K\right) ^d$

I need to know the primitive function (Antiderivative) of this function: $$f\left( \xi \right) =\xi^{a}\left( b\xi ^c+K\right) ^d$$ where $K$ is an integration constant, $d=-\frac{1}{2p}$ with $p<...
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0 votes
1 answer
79 views

Solution or approximation to $\int x^{-a} \text{erf}\left( b - c x^{-d} \right) dx$?

I'm looking for a closed solution or an approximation to $$\int x^{-a} \text{erf}\left( b - c x^{-d} \right) dx,$$ where $a, b, c, d > 0$.
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