All Questions
Tagged with integral or integration
1,506 questions
1
vote
1
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135
views
Integrability of $\exp\left(p\int_0^t |w(s,x(s,y))| \mathrm{d}s\right)$ for $w\in L^\infty(0,T;BMO(\mathbb{T}^d))$
Let $w\colon [0,T]\times\mathbb{T}^d \to \mathbb{R}^n$ be such that
$$ \|w\|_{L^\infty(BMO)} := \sup_{t\in[0,T]}\|w(t,\cdot)\|_{BMO} \leq C $$
and $\int_{\mathbb{T}^d} w(t,x)\mathrm{d}x = 0 $ for all $...
- 123
4
votes
0
answers
221
views
Can Differential Geometry aid in comparing the close contour integrals of $f(z)/z$ and $f(z) / \bar{z}$?
Let us consider a function $f(z)$ holomorphic along and inside a contour $\Gamma$ not surrounding the origin. With reference to the following contour integrals:
$$ \oint\limits_{\Gamma} \frac{f(z)}{z}\...
- 362
1
vote
1
answer
853
views
Quadrature methods for high-dimensional Gaussian integration
Suppose that $f$ is the density of a high(-$d$)-dimensional Gaussian measure with mean $\mu$ and non-singular covariance matrix $\Sigma$. Let $g:\mathbb{R}^d\rightarrow \mathbb{R}$ be a continuous ...
- 5,405
3
votes
0
answers
101
views
Pettis vs. Dunford integrability of operator valued functions
Given a Banach space $X$ and a measure space $(\Omega ,\mu )$, one says that a function
$$
f:\Omega \to X
$$
is Dunford integrable, or scalarly integrable if, for every $\varphi $ in the ...
- 2,263
1
vote
0
answers
74
views
Complex integration related to finite temperature number density correlation function of 1d free fermion
I am looking for an explicit formula for this complex integral.
$$\oint_C \frac{d z}{2 \pi i} \frac{z^{-(x+1)}}{1+e^{-\beta(z+1 / z-\mu)}},$$
where $x\in \mathbb{Z},\ \beta,\mu\in \mathbb{R}\ $. The ...
- 73
2
votes
3
answers
506
views
how to numerically evaluate $\int_{0}^{\infty} \frac{1}{x!} dx$ [closed]
So I was graphing the equation $ y=\frac{1}{x!} $ for $ x \geq 0$ and tried the integral:
$$\int_{0}^{\infty} \frac{1}{x!} dx$$
$$\int_{0}^{\infty} \frac{1}{\Gamma(x+1)} dx$$
$$\int_{0}^{\infty} \frac{...
- 57
6
votes
3
answers
700
views
How to estimate the integral involving the distance function
Let $\Omega\subset\mathbb{R}^n$ be an open bounded domain with smooth boundary. Consider the following integral:
$$I(t)=\int_{\Omega}e^{-\frac{d^2(y,\partial\Omega)}{t}}{\rm d}y.$$
My problem is how ...
- 561
7
votes
0
answers
155
views
Henstock–Kurzweil integral for unbounded domain of $\mathbb{R}^n$
I am working through the textbook Analyse : fondements, techniques, évolution by Jean Mawhin (in French). It was published in 2002 and its main characteristic is to introduce integration to ...
- 71
2
votes
1
answer
137
views
Approximating a limit of an integral
How can we prove the following asymptotic lower bound for the regularized Beta function when $n\rightarrow\infty$?
$$\int_0^{1} I_{2 t - t^2}\left(\frac{n - 1}{2}, \frac{1}{2}\right) dt=\Omega\left(\...
- 1,677
1
vote
1
answer
677
views
Solving integrals without the Fundamental Theorem of Calculus [closed]
Every time I see people attempt to solve or catalogue integrals, the approach ends up being to simplify and reduce the integrand using various techniques to a point where the integrand is simple ...
- 2,902
6
votes
0
answers
257
views
What is the expected value of the volume of a tetrahedron inscribed in the unit sphere?
Four (non-coincident) points on the unit sphere determine a tetrahedron. What is the expected value of the volume of such a tetrahedron--the volume of the sphere itself being $\frac{4 \pi}{3} \approx ...
- 723
2
votes
1
answer
297
views
Analytic continuation of convergent integral
I was trying to solve the following integral:
$$I = \oint _{|z|=1}\frac{dz}{2 \pi i z}\int_{0}^{\infty} dr \dfrac{e^{-\tfrac{r^2}{z^2}}r^{2n+1}}{z^2(z-1)} $$
The singular structure in the $z$ ...
- 121
3
votes
2
answers
262
views
Asymptotic bound for $\sum_{x=0}^\infty \sum_{y=0}^\infty (x+y)^m e^{-\frac{x^2}{2i} - \frac{y^2}{2j}}$ for $i$ and $j$ large
Note: This question relates to two previous questions on math.stackexchange (1 and 2), neither of which had satisfactory answers after posting bounties.
Whilst trying to count certain types of ...
- 472
2
votes
0
answers
84
views
A problem of uniqueness
Let $f\in\mathcal{S}(\mathbb{R}^n)$, $a\in(-1,1)$, how i can prove that the following problem:
$$\text{div}(t^a\nabla u)=0,\quad\text{in }\mathbb{R}^n\times(0,\infty),$$
$$ u(x,0)=f(x),\quad\forall x\...
- 339
1
vote
1
answer
666
views
Definite integral of 2d Gaussian
Is there some analytic expression or even an approximation of the definite 2D Gaussian integral of the form: $$E=\int_a^b Dg \int_{cg+d}^\infty Dh$$ where $Dg=\frac{dg}{\sqrt{2 \pi}} e^{-g^2/2}$ and a,...
- 373
1
vote
0
answers
74
views
Existence of efficiently computable integrals for "spiky" functions
$\DeclareMathOperator\spikify{spikify}$Apologies if I'm misusing the word spiky, I mean it only as a visual description of a function, not in any technical mathematical sense!
We define the function
$\...
- 13
1
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0
answers
161
views
Justify $\int_0^\infty e^{-ax^2}\ \mathrm{d}x$ for complex $a$ and zero real part [closed]
(Reposted from math stack exchange)
I have searched and failed to find a rigorous proof showing that
$$\int_{0}^\infty e^{-ax^2}\ \mathrm{d}x = \frac{\sqrt{\pi}}{2\sqrt{a}}$$
is true for $\Re(a)=0$ ...
user161698
1
vote
0
answers
82
views
How should I proceed to solve this kind of integral equation?
Given $a>0$, $b>0$, I am trying to find the function $f_{a,b} : \mathbb{R}_+ \rightarrow \mathbb{R}_+$ such that for all $u \in \mathbb{R}_+$,
$$\exp\left\{\;\int\limits_{\mathbb{R}_+} \ln\left(...
- 686
1
vote
0
answers
144
views
Recursive formula for integral of Chebyshev-type integral
Define
$$
I_{m,n}(x,y,r) = \int_a^b T_m(x + r \sin(\gamma)) T_n(y-r \cos(\gamma)) d\gamma
$$
where $T_m(x)$ are the Chebyshev polynomials of the first kind, and $a$ and $b$ are constants. Assume that ...
- 11
0
votes
0
answers
267
views
Can we improve the error bounds for spline interpolation if the interpolated function is smooth?
Let me first state the original problem I want to solve:
Given a closed curve $C:[a,b]\to\mathbb R^2$ that is smooth ($C^\infty$), a partition in the parameter space $a=t_0<t_1<\cdots<t_n=b$,...
- 283
2
votes
1
answer
468
views
Are solutions to linear second-order ODEs always expressible by integrals?
The solution of a linear first-order ODE, $y'+P(x)y+Q(x)=0$, is expressible by integrals involving elementary functions, $P(x)$ and $Q(x)$. This can be proved e.g. by the applying the method of ...
- 3,738
0
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0
answers
62
views
Integration question dealing with several variables and Taylor theorem
Dealing with one-variable and smooth function $f$ on a real interval $I$ such that $D^m f\in\mathcal{C}^2$, we have by Taylor theorem centered at $a\in I$
$$ D^mf(y)= D^mf(a) + D^{m+1}f(a)(y-a) + \...
- 383
1
vote
1
answer
141
views
Does the following integral converge?
Let $a\in(-1,1)$, let:
$$ P(x,y)=C_{n,a}\frac{y^{1-a}}{(|x|^2+y^2)^{(n+1-a)/2}},\quad\forall (x,y)\in \mathbb{R}^n\times(0,\infty),$$
let $f\in \mathcal{S}(\mathbb{R}^n)$, i.e. $f$ is a Schwartz ...
- 339
1
vote
1
answer
497
views
Sufficient and necessary conditions for decomposing the sum of random variables
Given two $n$-tuple vectors $\vec{\alpha}=(\alpha_1,\cdots,\alpha_n)$ and
$\vec{h}=(h_1,\cdots,h_n)$, where $h_i\ge0$, $\sum_{i=1}^nh_i=1$, and $\alpha_i\in(0,1)$, we consider a random variable $S$ on ...
- 550
0
votes
0
answers
40
views
To show a analytic map is zero from a property regarding logarithmic integral
Let $F$ be analytic on $\mathbb{H}=\{z\in\mathbb{C}:Im(z)>0\},$ continuous upto $\overline{\mathbb{H}}$ and bounded on each of the half plane $\{Im(z)\geq h>0\}.$ How to show that if $F$ ...
- 173
1
vote
1
answer
474
views
Convolution, Fourier transforms, and area preservation [closed]
Consider the convolution of two functions, f * g. And let us assume, for practicality, some example case where an integral of f or g can be interpreted as the "area under the curve" (or the ...
- 111
0
votes
1
answer
421
views
A "simple" surface-integral over the unit-sphere [closed]
Calculate
$$
I=\iint_{x^2+y^2+z^2=1}{e^{x-y} \mathbb{d}S}
$$
Parameterization is not helpful:
$$
I=\int_0^{2\pi}{\mathbb{d}\varphi\int_0^\pi{e^{\sin\theta(\cos\varphi-\sin\varphi)}\sin\theta\mathbb{d}...
- 111
1
vote
1
answer
206
views
Numerical methods for evaluating singular integrals
The Helmholtz decomposition for a vector field B contains both volume integrals and two boundary integrals (https://en.wikipedia.org/wiki/Helmholtz_decomposition). For brevity I show just one of the ...
- 11
0
votes
0
answers
62
views
Fractional laplacian on $H^s(\mathbb{R}^n)$ and symmetry
Let $s\in(0,1)$, $u\in\mathcal{S}(\mathbb{R}^n),$ i define the fractional laplacian of $u$ in the following way:
$$(-\Delta)^su(x)=C(n,s)P.V.\int_{\mathbb{R}^n}\frac{u(x)-u(y)}{|x-y|^{n+2s}}\,dy,\quad\...
- 339
0
votes
0
answers
82
views
Reverse Inequality
I was doing some numerical integration when I figured the function I was dealing with (i.e., the function I was integrating) evaluated to big numbers on a tiny portion of the interval (over which I ...
- 369
3
votes
1
answer
2k
views
Quadrature for numerical integration over infinite intervals
I am looking for book recommendations or hints on numerical integration over infinite intervals. I am particularly interested in integrals of the form
$\int\limits_{-\infty}^{+\infty} g(x) \exp(p_d(x))...
- 131
0
votes
1
answer
176
views
Symmetry of fractional laplacian
Let $\Omega\subset\mathbb{R}^n$, let $s\in [1/2,1)$, let $u\in C^{1,2s-1+\epsilon}(\Omega)$ such that: $u=0$ on $\mathbb{R}^n\setminus\Omega$, and: $u\in C^{0,s}(\mathbb{R}^n)$, is true that:
$$\int_{\...
- 339
1
vote
0
answers
47
views
Another uniform estimation of an integral involving an Hölder function with derivative that is Hölder
Let $\Omega\subset\mathbb{R}^n$, let $s\in [1/2,1)$, let $u\in C^{1,2s-1+\epsilon}(\Omega)$ such that: $u=0$ on $\mathbb{R}^n\setminus\Omega$, and: $u\in C^{0,s}(\mathbb{R}^n)$, is true that there ...
- 339
2
votes
1
answer
324
views
Uniform estimation of an integral involving a Hölder-continuous function
Let $\Omega\subset\mathbb{R}^n$ be open and bounded, let $s\in(0,1)$, let $u\in C^{0,2s+\epsilon}(\Omega)$ bounded with $u\in C^{0,s}(\mathbb{R}^n)$ and such that: $u=0$, on $\mathbb{R}^n\setminus\...
- 339
0
votes
1
answer
124
views
Uniform estimation of an integral
Let $\Omega\subset\mathbb{R}^n$ be open and bounded, let $s\in(0,1)$, let $u\in C^{0,2s+\epsilon}(\Omega)$ bounded and such that: $u=0$, on $\mathbb{R}^n\setminus\Omega$, is true that there exist a ...
- 339
1
vote
0
answers
210
views
The translation is continuous in $L^1(\mathbb{R}^n,d\mu)$, $d\mu=\frac{1}{1+|y|^{n+a}}dy$,$ a>0$
For any function $f\colon\mathbb{R}^n\to\mathbb{R}$, set: $\tau_hf(x):=f(x+h)$, $x,h\in\mathbb{R}^n$. Consider the following finite measure on $\mathbb{R}^n$:
$$\mu(A):=\int_A\frac{1}{1+|y|^{n+a}}\,dy$...
- 339
0
votes
1
answer
276
views
Estimate for computing the $L^2$-norm of a function from its data
Let $f:\mathbb{T}^m \to \mathbb{R}$ is a function of bounded variation(BV). Let $D=\{\boldsymbol{p}_i,i=1,2,3\ldots\}$ be a countable dense subset of $(0,1)^m$. Let $E_n, n = 1,2,3\ldots$ be a ...
- 59
-1
votes
1
answer
82
views
How to solve this equation for x?
$$\frac{d}{dx}\int_{0}^{1}|log_2(1+t)-(t+x)|\,dt=0$$
Is this solvable at all?
- 3
2
votes
0
answers
115
views
Showing that for measurable $\Omega \subseteq \mathbb{R}^n$, $L^1(\Omega; C_0(\mathbb{R}^n))$ is separable
Here we're integrating "Banach-valued" functions $u: \Omega \rightarrow C_0(\mathbb{R}^n))$ , and by $u \in L^1(\Omega; C_0(\mathbb{R}^n))$ I mean that
$$\int_{x \in \Omega} \| u(x) \|_{\...
- 21
0
votes
0
answers
95
views
When does a potential function with given partial derivatives exist
I am looking for the answer to the following question:
Consider an integrable function $f:X\rightarrow X$ with $X$ being a compact subset of $\mathbb{R}^n$. What are the conditions on $f$ so that a ...
- 37
0
votes
1
answer
210
views
Integral estimate (inequality) with a Schwartz function
$\DeclareMathOperator\supp{supp}\newcommand\abs[1]{\lvert#1\rvert}\newcommand\Bigabs[1]{\Bigl\lvert#1\Bigr\rvert}$Given a Schwartz function $f \in \mathcal{S}(\mathbb{R})$ with $\supp(f) \subseteq [-A,...
user168186
4
votes
2
answers
404
views
Integration in a finite dimensional vector space
Let $V$ be a finite dimensional complex vector space. Let $G$ be a compact group with normalized Haar measure $\mu$. In the representation theory of compact groups, I encounter
$$\int_G f(g) \mu(dg)$$
...
user167952
4
votes
0
answers
73
views
Saddle point approximation for multiple contour integrals
General Question: Is there a reference where the saddle point approximation is applied to multiple contour integrals?
In particular, say we have the integral
$$ I_N = \frac{1}{(2\pi i)^N} \oint \left[\...
- 241
0
votes
0
answers
82
views
Integral equality involving fractional laplacian
Let $s\in(0,1)$, let $u\in H^s(\mathbb{R}^n)$. For all $\psi\in\mathcal{S}(\mathbb{R}^n)$, let:
$$ (-\Delta)^s\psi(x)=c(n,s)\lim_{\epsilon\to0^+}
\int_{\mathbb{R}^n\setminus B_\epsilon(0)}\frac{\psi(...
- 339
0
votes
0
answers
82
views
A question about Fourier transform of a function defined by an integral
I have the function:
$$ G_k(x)_=\frac{1}{(4\pi)^{k/2}\Gamma(k/2)}\int_0^\infty e^{-\pi|x|^2/\delta}e^{-\delta/4\pi}\delta^{-(n-k)/2}\,\frac{d\delta}{\delta}, $$
for all $x\in\mathbb{R}^n$ and $k>0$....
- 339
4
votes
1
answer
2k
views
Exchanging series and integrals
I know that I can use Lebesgue or monotone convergence theorem to exchange limit of partial sums and a Lebesgue integral, given a power series or a generic function series. But in general given a ...
- 91
14
votes
1
answer
919
views
Was Cantor aware of Lebesgue theory of integration?
Georg Cantor died in 1919, more than ten years after appearance of the Lebesgue theory of measure and integration at the beginning of the twentieth century. Lebesgue theory has a deep connection with ...
- 747
2
votes
1
answer
168
views
Integral inequality for Schwartz function
Let $s\in(0,1)$, $u\in\mathcal{S}({\mathbb{R}^n})$, $x\in\mathbb{R^n}$ with: $|x|\geq1$, i have to prove that:
$$ \int_{B_{|x|/2}(0)} \frac{|u(x+y)+u(x-y)-2u(x)|}{|y|^{n+2s}}\,dy\leq c|x|^{-n-2s}, $$
...
- 339
1
vote
0
answers
121
views
Haar measure decomposition using orbital integrals
Let $G$ be a unimodular locally compact group, $N,A \le G$ be unimodular closed subgroups. Suppose that $A$ normalizes $N$. Let $N_0 \le N$ be a compact open subgroup. Suppose that a function $f : N \...
- 730
2
votes
1
answer
291
views
An inequality involving fractional Laplacian
I have to prove that for $s\in(0,1)$, $u\in\mathcal{S}(\mathbb{R}^n)$, (i.e. $u$ is a Schwartz function):
$$ |(-\Delta)^su(x)|\leq c_{n,s}|x|^{-n-2s},\quad\forall x\in\mathbb{R}^n\setminus B_1(0), $$
...
- 339