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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Compute the two-fold partial integral, where the three-fold full integral is known
I have the following trivariate ($\rho_{11}, \rho_{22}, \mu$) function
\begin{equation}
4 \mu ^{3 \beta +1} \rho_{11}^{3 \beta +1} \left(-\rho_{11}-\rho_{22}+1\right){}^{3 \beta +1}
\rho_{22}^{3 \...
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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 ...
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2
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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 ...
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1
answer
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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. ...
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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 $a\in [0,1)$, ...
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3
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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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Generalizing contour integration to quaternions and bicomplex numbers
I am interested in the possibility of generalizing the notion of contour integration to the quaternions or bicomplex numbers. I am aware that the Frobenius theorem prevents the construction of a true ...
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4
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Change of variables in a Gaussian integral in matrix form
I have a problem in which I have to compute the following integral: $$\mathop{\idotsint\limits_{\mathbb{R}^k}}_{\sum_{i=1}^k y_i=x} e^{-N^2r(\sum_{i=1}^k y_i^2-\frac{1}{k}x^2)} dy_1\dots dy_k,$$
where ...
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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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Probability density of a hyperplane for a Gaussian distribution
I have a vector $\mathbf{x}$ with a multivariate Gaussian distribution
$$P[\textbf{x}\in S]
=\int_{\textbf{x}\in S}
\det(2\pi H^{-1})^{-1/2}\exp(-\frac{1}{2} \textbf{x}^T H\textbf{x}) \, d\textbf{x}$$...
CommunityBot
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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.
$$
I = \int_{\mathbb{R}} \int_{\mathbb{...
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vote
1
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Weak lower semicontinuity of a sequence of Riemann sums
Let us have a sequence of functions $\{f^K\}_{K \in \mathbb{N}} \in C([0,1],\mathbb{R})$ which is uniformly bounded in $L^2((0,1))$. We observe a sequence of Riemann sums
$$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 prove that $\int _0^\infty\frac{\text{arcsinh}^nx}{x^m}dx$ is a rational combination of zeta values?
For $n\ge m\ge 2$, define $$I(n,m):= \int _0^\infty\dfrac{\text{arcsinh}^nx}{x^m}dx$$ Computer algebra systems say that the indefinite integral can be expressed in terms of polylog functions (of ...
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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): ...
CommunityBot
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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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0
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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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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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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 $$\int_{\mathbb{T}^2}\int_{\mathbb{T}^2} \left( \int_{3x+y=z} f(w+x)f(w+y)dm_{\mathbb{T}^2}(x,...
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How was this heat semigroup estimate made in a paper on reaction–diffusion systems?
In Yamauchi - Blow-up results for a reaction–diffusion system, in the proof of Lemma 3.3, there is the passage
$$S(t-s)|x|^{\sigma/1-k} \geq C_1(t-s)^{\sigma/2(1-k)}.$$
Here $S(t)$ denotes the heat ...
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Numerical methods for evaluating singular integrals
The Helmholtz decomposition for a vector field B contains both volume integrals and two boundary integrals (https://en.wikipedia.org/wiki/Helmholtz_decomposition). For brevity I show just one of the ...
CommunityBot
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What is the justification for using Wiener integrals to integrate over a space of differentiable functions?
In the literature on stiff/semiflexible polymer chains modelled as continuous chains rather than as discrete links, the partition function (among other things) is taken to be an integral over the ...
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A specific integration with Grassmann variables
I have recently read (for example, here) that this relation below is true
$$ \int dz \: e^{\frac{1}{2} \sum_{ij} z_i A_{ij} z_j} = Pf(\mathbf{A}),
$$
where $Pf(\mathbf{A})$ is the Pfaffian of an even ...
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$\mathbf{P} = \mathbf{NP}$, what's the problem?
Let's take the problem of the backpack: $A_1,\ldots ,A_n$ the weights that are integers, and we want to know if we can achieve a total weight of $V$.
We take $$I=\dfrac{1}{2\pi}\int_0^{2\pi} \exp(-iVt)...
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Questions on the integral of pseudo Gaussian kernel and its derivative on $(0,\infty)$
Consider pseudo Gaussian densities for $0<s<t$ and $x,y\in\mathbb R$
$$f(s,x,t,y):=\frac{1}{\sqrt{2\pi A(s,t,y)}}\exp\left(-\frac{(y-x)^2}{2A(s,t,y)}\right)\quad\mbox{and} \quad g(s,x,t,y):=\...
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Birationally equivalent elliptic curves and singularities
I got the following cubic elliptic curve from some physical problem $$E_c(\mathbb{C}): w^2=4 z^3-zG_2-G_3,$$ where $G_2=3
\alpha ^2+\gamma$ and $G_3=\alpha ^3-\alpha \gamma
-\beta ^2$ for known ...
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answer
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Convergence of oscillatory integrals
I'm considering integrals of the (Hilbert transform) type
$$p.v.\int_{-\infty}^\infty\frac{f(r)}{r}\,dr$$
where $f(r)$ is periodic, say, with period $2\pi$. I'm assuming very little regularity on $f$. ...
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Topological properties of complex valued Riemann sum limit curve and a particular integral inequality
I am studying under what conditions the following integral inequality would hold ($a$ real, $a>0$):
$$ \int_{-\infty} ^{\infty} \frac{f(ix)}{a\pm ix}dx\ = 0 \ \ \ \ \Rightarrow \ \ \ \int_{-\...
CommunityBot
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2
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On the search for an explicit form of a particular integral
Let $f$ be integrable over the interval $(0, 1)$, and
$$I_n = \int_0^{1} x^n f(x) \, \mathrm{d}x.$$
Suppose $f(x) = f(1-x)$; we can then show that
$$I_n = \sum_{k=0}^{n} \binom{n}{k} (-1)^k \, I_{k}...
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0
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How to define integration for general functions?
In general, the integration is defined on a measurable space $(X, \mathcal{A}, \mu)$, where $X$ is the whole space, $\mathcal{A}$ is a $\sigma$-algebra on $X$, and measure $\mu$. Usually integration ...
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Gaussian integral with Vandermonde determinant
I want to compute the following integral, which contains a Gaussian piece and a Vandermonde determinant:
$$
\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),
$$
...
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Looking for some interesting complex integration contours
I am currently working on some tools to make contour integration in a proof assistant less painful and I'm looking for interesting examples of contours in the complex plane used in the literature. I ...
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Asymptotics of a 1D integral, or the orthant probability of an equicorrelated random Gaussian vector
Problem: Let $\phi(x)$ be the normal probability density function (pdf), and $\Phi(x)$ the normal cumulative distribution (cdf). I'm interested in the asymptotic behavior of the following integral
$I(...
CommunityBot
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Radon transform of the function $h(x_1,\ldots,x_n) = x_1 g(x_1,\ldots,x_n)$, where $g$ is the density of multivariate Gaussian $N(\mu,\Sigma)$
Given an absolutely integrable function $f:\mathbb R^n \to \mathbb R$, let $R[f]$ be its Radon transform defined for every $(w,b) \in (\mathbb R^n \setminus \{0\}) \times \mathbb R$ by
$$
R[f](w,b) := ...
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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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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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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
$$
\bar{\mu}:=\int_{\omega\in \Omega}\, ...
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Integrating a function of distance between a set and its neighbourhood
I am aware of the isoperimetric inequality, which states that if you fix the Lebesgue measure of a measurable set $A \subset \mathbb R^d. d \geq 2$ then the smallest possible value of the perimeter of ...
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Prove or disprove the positivity of the ess inf of a singular function
Consider a measurable radial function $u:\Bbb R^d\to(0,\infty)$ such that
$$\int_{B_\delta(0)} u(x) d x=\infty\quad\forall\,\, \delta>0.$$
I would like to prove or to disprove that there exists $r&...
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1
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On an asymptotic integral
Let $\phi, a \in C^{\infty}([0,1])$ and assume $a(0)=1$. Suppose that
$$
\int_0^1 e^{\tau \,\phi(t)}\,a(t)\,dt =0 \qquad \text{for all $\tau \in \mathbb R$}.
$$
Does it follow that $\phi$ is a ...
- 5,632
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vote
0
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Fourier transform of inverse of determinant of 1+ skew-symmetric matrix
I have asked the following question in math stackexchange(https://math.stackexchange.com/questions/4389626/fourier-transform-of-inverse-of-determinant-of-1-skew-symmetric-matrix), but did not receive ...
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2
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Lower bound for integrals like $\int_1^{t+1}e^{-\sqrt{s}}s^{-1}ds$
Let
$$I(t) = \int_{1}^{t+1}\exp\left\{-c\frac{s^{1-\beta}}{1-\beta}\right\}s^{-2\beta}ds,$$
where $c$ is some positive constant and $\beta\in(0, 1)$.
Since the integral $I(t)$ given above could not be ...
CommunityBot
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0
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Does this book use non-mainstream stochastic analysis constructions and is thus perhaps not a good start?
I'm attempting to read a book on stochastic calculus by D.H. Fremlin, which is the 6th volume of his treatise on measure theory encompassing all kinds of topics related it.
Before I make a very ...
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1
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Exponential decay bound on integral
I have an integral of the form
$$ \int_R^{\infty} e^{-x} x^n \vert L_m^{\alpha}(x) \vert^2 \ dx,$$
where $L_m^{\alpha}$ is the generalized Laguerre polynomial and $n \ge 0.$
I would to get a nice ...
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0
answers
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Nature of function as $x\rightarrow\infty$
I'm studying the limits and applicability of Abel Plana summation for different test functions (class of functions). In doing so this just pops out and couldn't handle the said integral so asked here (...
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Characterizing subsets of integrable functions
Let us consider the space $L^1(0,1;\mathbb{R})$ of real-valued, Lebesgue integrable functions defined on the interval $(0,1)$ (where we only distinguish functions which are not equal almost everywhere)...
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votes
1
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When is this multiple integral finite?
Consider the following integral:
$$
I_k(\alpha)=\int_{[0,1]^k}|x_1-x_2|^{\alpha}|x_2-x_3|^{\alpha}\ldots|x_{k-1}-x_k|^{\alpha}|x_k-x_1|^{\alpha}d\mathbf{x}.
$$
where $k=2,3,4,\ldots$
The question is ...
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0
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Optimal growth of an oscillating integral
Let $f\in H^1(\mathbb{R}^3)$, with $f\equiv0$ inside a ball around the origin. For $t>0$, consider the following integral
$$I(t):=\int_{\mathbb{R}^3}e^{i|x|^2/t}\frac{f(x)}{|x|}dx$$
It`s easy to ...
- 18.5k
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1
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Volume of a frustum knowing the volume and height of the pyramid and the height of the frustum [closed]
Can I calculate the volume of a frustum if all I know is the volume of the pyramid the height of the pyramid and the height of the frustum?
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What are the steps involved in the solution to $\int{x^{-a} (b -cx^{-d})^e }dx$? [closed]
Mathematica gives me the following solution to $\int{x^{-a} (b -cx^{-d})^e }$:
$$\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}\...
