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Results tagged with integration
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user 21907
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, ...
6
votes
About the definition of Borel and Radon measures
Let $(X,\mathcal M, \mu)$ be a measure space, where $\mu$ is a positive measure and $X$ is topological space. Let $\mathcal B$ the Borel $\sigma$-algebra on $X$.
The measure $\mu$ is called a Borel m …
- 16.2k
4
votes
Accepted
One-sided Cauchy principal value
Let me give an example: you want to define a distribution on $\mathbb R$ which coincides with $1/t$ on $(0,+\infty)$ and vanishes on $(-\infty,0)$. Let us take
$$
T=\frac{d}{dt}(H(t)\ln t),\quad H=1_{ …
- 16.2k
4
votes
0
answers
244
views
English language and Mathematics
I have a question maybe more relevant to an English language section of StackExchange, but I doubt that anybody but a Mathematician could properly answer my question.
Let $\mathcal M$ be a smooth man …
- 16.2k
4
votes
0
answers
189
views
The Poincaré Lemma
Let me consider an $L^1(\mathbb R^N)$ function $f$ such that $$
\int_{\mathbb R^N} f(x) dx =0.
$$
Then I claim that the $N$-form $f(x) dx_1\wedge\dots\wedge dx_N$ is closed, i.e. there exists a vector …
- 16.2k
3
votes
Accepted
Stationary phase in spherical integral
You have
$
I(\lambda, x)=x\cdot\int_{\mathbb S^{n-1}} ye^{i \lambda x\cdot y} d\sigma(y)=x\cdot J(x,\lambda)
$
and you claim that for $\vert x\vert \lambda \ge 1$, you have
$$
J(x,\lambda)=O((\vert …
- 16.2k
3
votes
0
answers
335
views
Norm of a singular integral operator
Let $H$ be the characteristic function of $(0,+\infty)$ and let us define for $(x,y)\in \mathbb R^2$, $x\not=y$
$$
k(x,y)=\frac{H(x+y)}{iπ(x-y)}.
$$
For $u\in C^1_c(\mathbb R)$, we define for $x\in \m …
- 16.2k
2
votes
Exterior derivative independence from coordinate systems
A remark, too long for a comment. To check that the exterior derivative is a geometric operation, coordinate-free, it seems better to define first the Lie derivative of a form $\omega$ with respect to …
- 16.2k
1
vote
($n$-dimensional) Inverse Fourier transform of $\frac{1}{\| \mathbf{\omega} \|^{2\alpha}}$
Let's start with a simple change of notation. Let me consider first on $\mathbb R^n_x$ the function $f_{\beta}(x)=\Vert x\Vert^{\beta-n}$ for $0<\beta< n$, which is locally integrable and homogeneous …
- 16.2k
1
vote
Integration of a particular rational expression
Too long for a comment. You should discuss on the denominator, which has 0 as a triple root if $f\not=0$. Assuming this, you have three non-zero roots for $X^3+eX+f$ and you have explicit formulas to …
- 16.2k
0
votes
Topological properties of complex valued Riemann sum limit curve and a particular integral i...
More a comment than an answer, but too long anyway for a comment. There is nothing weird or mysterious about your first equalities: with $R>a>0$, we have from the residue formula,
$$
\int_{[-R,R]}\fra …
- 16.2k
-1
votes
Convergence of an oscillatory integral
We set $h=1/t$ so that $h\rightarrow0_+$. We have with $I_f(t)=J_f(h)$
$$
J_f(h)=\int e^{ih\vert x\vert^2} f(x) g(h,\vert x\vert)dx,
$$
with
$
g(h,y)=\int_0^{+\infty} e^{-s+ih s^2+2ihsy}ds.
$
The func …
- 16.2k