All Questions
Tagged with integral or integration
1,506 questions
0
votes
1
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123
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Could variable be still function in x and y after performing Reynolds averaging over area
All,
Let $S(x,y,t)$ be a variable function in $x$, $y$, and $t$. After
performing Reynold averaging over area $\frac{1}{A}\int S(x,y,t) dA$, could $S$ still be a function in $x$, and $y$?
Equations (1-...
- 101
2
votes
0
answers
154
views
Applying 1D integral to matrix integral
In the proof for finding an analytic solution to the propagation of a Hermite-Gaussian beam though a paraxial system given in the paper "The elliptical Hermite–Gaussian beam and its propagation ...
- 83
2
votes
0
answers
155
views
Second differential of total variation
I am trying to give meaning to the notion of second differential of total variation.
For sufficiently regular $u:\Omega \subset \mathbb{R}^2 \to \mathbb{R}$ let the total variation be given by
$$TV(u)=...
- 319
1
vote
1
answer
293
views
Expressing the integral over boundary of a domain as an integral over the domain
Let $\Omega \subset \mathbb{R}^2$ be a domain which is "well behaved" (has all "wishable" properties), so as its boundary. For every $u \in C^\infty(\Omega,\mathbb{R})$, I would ...
- 319
7
votes
3
answers
662
views
Asymptotics for $\int\exp( -x t / \log t)dt$
What is the asymptotic growth rate of $$f(x) = \int_e^\infty e^{ - x t / \log t} dt$$ as $x \to 0$?
As an example of what is meant by "growth rate" consider $$g(x) = \int_e^\infty e^{-x t} ...
- 457
3
votes
2
answers
85
views
Properties of functions provided that the integral equation $\int_x^1 a(y-x)a(y) \, dy = \int_x^1 b(y-x)b(y) \, dy$ holds for $x \in [0,1]$
Let $a,b : [0,1] \to \mathbb R$ are two functions (e.g. suppose that they are in $L^2[0,1]$ or are $N$-times continuously differentiable). Now suppose that
$$
\int_x^1 a(y-x)a(y) \, dy = \int_x^1 b(y-...
- 437
1
vote
0
answers
66
views
Properties regarding Poisson non-integrability [closed]
Let $a>0$ and $f:[a,\infty]\to [0,\infty)$ be a continuous increasing function. We call $f$ to be "Poisson non-integrable" if $f$ satisfies
$$\int_a^\infty \frac{f(x)}{x^2}dx=\infty.$$
...
- 73
2
votes
1
answer
326
views
Evaluation of Gaussian multivariable integral
In the context of evaluating the propagation of a flattened Gaussian beam, the following integral appears:
\begin{equation}
\int (\mathbf x^T \mathbf F \mathbf x)^n \exp \left [ - \mathbf x^T \mathbf ...
- 83
7
votes
2
answers
1k
views
How would you work out this integral as a series?
The integral is:
$$f(a) = \int\limits_{-\infty}^\infty \frac{x e^{-a^2 x^2}}{\tanh(x)}dx$$
which seems to converge for all $a>0$. But I don't know how to get a sense of the function $f(a)$ such as ...
- 255
0
votes
1
answer
127
views
upper bound for infimum of integral
I was reading this post , where the following question is discussed:
Let $h:[0,1] \rightarrow[0,1]$ be a $C^{1}$ function such that $h^{\prime}(x)<0$ for all $x \in(0,1)$. Then,
$$
\inf _{f \in \...
3
votes
1
answer
95
views
On an integral of Gaussian CDFs
Let $c>0$ and $T>0$ be fixed. Denote by $F$ the Gaussian CDF, i.e. $F:\mathbb R\to\mathbb R$ is defined by
$$F(x):=\int_{-\infty}^x \frac{1}{\sqrt{2\pi}} e^{-z^2/2}dz.$$
For every $a\in [0,1)$, ...
- 1,334
4
votes
1
answer
405
views
Why is it difficult to define a direct integral of Banach spaces or Banach algebras?
In the relevant Wikipedia entry, I can read about how to define a direct integral on Hilbert spaces and Von-Neumann algebras.
Suppose that I want to define a direct integral on either Banach spaces or ...
- 1,054
4
votes
1
answer
339
views
Mikusiński's approach to Bochner integrals; replace absolute by unconditional?
In the book The Bochner Integral, Mikusiński described an approach to Lebesgue and Bochner integrals via absolutely convergent series corresponding to step functions:
Defn. Let $X$ be a Banach space. ...
- 39k
1
vote
3
answers
580
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 $p \in [1, \...
- 5,407
0
votes
0
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145
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Difference between summation for "$\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 ...
1
vote
2
answers
152
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.
$$
I = \int_{\mathbb{R}} \int_{\mathbb{...
- 3,625
2
votes
1
answer
330
views
Probability density of a hyperplane for a Gaussian distribution
I have a vector $\mathbf{x}$ with a multivariate Gaussian distribution
$$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}$$...
- 162
1
vote
1
answer
61
views
Weak lower semicontinuity of a sequence of Riemann sums
Let us have a sequence of functions $\{f^K\}_{K \in \mathbb{N}} \in C([0,1],\mathbb{R})$ which is uniformly bounded in $L^2((0,1))$. We observe a sequence of Riemann sums
$$R^K=\frac{1}{K} \sum_{k=0}^{...
- 319
0
votes
0
answers
67
views
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}\...
- 1
1
vote
0
answers
35
views
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 $n,a,b$ ...
- 11
1
vote
1
answer
158
views
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): ...
- 1,334
1
vote
0
answers
76
views
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 $$\int_{\mathbb{T}^2}\int_{\mathbb{T}^2} \left( \int_{3x+y=z} f(w+x)f(w+y)dm_{\mathbb{T}^2}(x,...
- 195
5
votes
2
answers
288
views
Inverse Mellin transform of 3 gamma functions product
I want to calculate the inverse Mellin transform of products of 3 gamma functions.
$$F\left ( s \right )=\frac{1}{2i\pi}\int \Gamma(s)\Gamma (2s+a)\Gamma( 2s+b)x^{-s}ds$$
Above contour integral has ...
- 59
0
votes
0
answers
53
views
How was this heat semigroup estimate made in a paper on reaction–diffusion systems?
In Yamauchi - Blow-up results for a reaction–diffusion system, in the proof of Lemma 3.3, there is the passage
$$S(t-s)|x|^{\sigma/1-k} \geq C_1(t-s)^{\sigma/2(1-k)}.$$
Here $S(t)$ denotes the heat ...
- 677
2
votes
0
answers
118
views
What is the justification for using Wiener integrals to integrate over a space of differentiable functions?
In the literature on stiff/semiflexible polymer chains modelled as continuous chains rather than as discrete links, the partition function (among other things) is taken to be an integral over the ...
- 33
3
votes
0
answers
278
views
Radon-Nikodym derivative of vector-valued measure with respect to another vector-valued measure
Let $(X, | \cdot |)$ be a Banach space.
I am interested in whether one can extend the definition of the Kullback-Leibler divergence
$$
\text{KL}(\mu \ \Vert \ \nu)
:= \int_{\Omega} \ln\left(\frac{\...
- 67
3
votes
1
answer
428
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 ...
- 31
22
votes
2
answers
6k
views
$\mathbf{P} = \mathbf{NP}$, 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 $$I=\dfrac{1}{2\pi}\int_0^{2\pi} \exp(-iVt)...
- 4,074
3
votes
0
answers
254
views
Birationally equivalent elliptic curves and singularities
I got the following cubic elliptic curve from some physical problem $$E_c(\mathbb{C}): w^2=4 z^3-zG_2-G_3,$$ where $G_2=3
\alpha ^2+\gamma$ and $G_3=\alpha ^3-\alpha \gamma
-\beta ^2$ for known ...
- 231
1
vote
1
answer
87
views
Questions on the integral of pseudo Gaussian kernel and its derivative on $(0,\infty)$
Consider pseudo Gaussian densities for $0<s<t$ and $x,y\in\mathbb R$
$$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):=\...
user478492
1
vote
1
answer
251
views
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 $f$. ...
- 287
6
votes
1
answer
346
views
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
$$
\|f - \operatorname{Lip}_L(\mathbb{R}^...
- 5,405
1
vote
0
answers
66
views
How to define integration for general functions?
In general, the integration is defined on a measurable space $(X, \mathcal{A}, \mu)$, where $X$ is the whole space, $\mathcal{A}$ is a $\sigma$-algebra on $X$, and measure $\mu$. Usually integration ...
- 162
3
votes
0
answers
346
views
Gaussian integral with Vandermonde determinant
I want to compute the following integral, which contains a Gaussian piece and a Vandermonde determinant:
$$
\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),
$$
...
- 133
6
votes
2
answers
270
views
Asymptotics of error function integral with square root
I am interested in the asymptotics of the integral
$$I(a):=\int_0^\infty \sqrt{x}\operatorname{Erfc}(x+a)\,\mathrm{d}x$$
for $a>0$. I think that $I(a)$ should be decaying exponentially as $I(a)\...
- 623
35
votes
5
answers
3k
views
Looking for some interesting complex integration contours
I am currently working on some tools to make contour integration in a proof assistant less painful and I'm looking for interesting examples of contours in the complex plane used in the literature. I ...
- 1,241
2
votes
1
answer
173
views
Radon transform of the function $h(x_1,\ldots,x_n) = x_1 g(x_1,\ldots,x_n)$, where $g$ is the density of multivariate Gaussian $N(\mu,\Sigma)$
Given an absolutely integrable function $f:\mathbb R^n \to \mathbb R$, let $R[f]$ be its Radon transform defined for every $(w,b) \in (\mathbb R^n \setminus \{0\}) \times \mathbb R$ by
$$
R[f](w,b) := ...
- 6,853
2
votes
1
answer
181
views
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)....
0
votes
1
answer
370
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 ...
3
votes
2
answers
247
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
$$
\bar{\mu}:=\int_{\omega\in \Omega}\, ...
- 5,405
1
vote
0
answers
70
views
Prove or disprove the positivity of the ess inf of a singular function
Consider a measurable radial function $u:\Bbb R^d\to(0,\infty)$ such that
$$\int_{B_\delta(0)} u(x) d x=\infty\quad\forall\,\, \delta>0.$$
I would like to prove or to disprove that there exists $r&...
- 1,101
4
votes
0
answers
136
views
Integrating a function of distance between a set and its neighbourhood
I am aware of the isoperimetric inequality, which states that if you fix the Lebesgue measure of a measurable set $A \subset \mathbb R^d. d \geq 2$ then the smallest possible value of the perimeter of ...
3
votes
1
answer
140
views
On an asymptotic integral
Let $\phi, a \in C^{\infty}([0,1])$ and assume $a(0)=1$. Suppose that
$$
\int_0^1 e^{\tau \,\phi(t)}\,a(t)\,dt =0 \qquad \text{for all $\tau \in \mathbb R$}.
$$
Does it follow that $\phi$ is a ...
- 4,143
1
vote
0
answers
173
views
Fourier transform of inverse of determinant of 1+ skew-symmetric matrix
I have asked the following question in math stackexchange(https://math.stackexchange.com/questions/4389626/fourier-transform-of-inverse-of-determinant-of-1-skew-symmetric-matrix), but did not receive ...
- 63
10
votes
2
answers
513
views
Is there a purely constructive presentation of the HK integral?
Treating the Riemann integral in a constructive setting is easy and straightforward. Treating the closely related but much more powerful Henstock-Kurzweil integral constructively is almost easy, ...
- 1,947
0
votes
0
answers
131
views
Does this book use non-mainstream stochastic analysis constructions and is thus perhaps not a good start?
I'm attempting to read a book on stochastic calculus by D.H. Fremlin, which is the 6th volume of his treatise on measure theory encompassing all kinds of topics related it.
Before I make a very ...
- 41
2
votes
2
answers
150
views
Lower bound for integrals like $\int_1^{t+1}e^{-\sqrt{s}}s^{-1}ds$
Let
$$I(t) = \int_{1}^{t+1}\exp\left\{-c\frac{s^{1-\beta}}{1-\beta}\right\}s^{-2\beta}ds,$$
where $c$ is some positive constant and $\beta\in(0, 1)$.
Since the integral $I(t)$ given above could not be ...
- 63
1
vote
1
answer
276
views
Exponential decay bound on integral
I have an integral of the form
$$ \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 ...
- 73
0
votes
1
answer
125
views
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)...
- 820
43
votes
1
answer
2k
views
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 ...
- 1,459