All Questions
Tagged with integral or integration
1,507 questions
0
votes
1
answer
215
views
Generalised limits via derivatives of integrals?
Assuming that $f$ is a continuous function, we have that
$$f(x) = \frac{d}{dx}\int f(t)\,dt.$$
Assuming instead that $f$ has a removable singularity at $x=a$, and is otherwise continuous, we have ...
- 4,943
2
votes
2
answers
399
views
Asymptotic decay rate of an oscillatory integral
Consider the following oscillatory integral
$$
I(n):=\int_{-\pi}^\pi\int_{-\pi}^\pi e^{i n(x+y)}\frac
{(1 - \cos(2x)) (1 - \cos(2y))}
{2k - (\cos x + \cos y)}\ \mathrm{d}x\,\mathrm{d}y.
$$
where $...
- 2,712
2
votes
0
answers
110
views
The mean along the eccentric anomaly of an ellipse log distance to a point within the ellipse
Conjecture.
Let
$$ f(r,\alpha,p, \theta) = \ln\left(\left(r\sin\alpha-\sin\theta\right)^{2}\left(1-p\right)^{2}+\left(r\cos\alpha-\cos\theta\right)^{2}\left(1+p\right)^{2}\right). $$
Then for any ...
0
votes
0
answers
105
views
Is this integral finite and how does it decay to zero?
I would like to know if the following is convergent/finite (it represents a bound from a truncated Legendre series approximation)
\begin{equation}
\varepsilon_n \leq \int_{-1}^1 \left(\int_{n\gg 1}^\...
- 201
1
vote
0
answers
78
views
Convolve a 4D Gaussian function along a plane?
There is a 4D Gaussian function $G(u,s)=G(x|c,\mu,\Sigma )$ where $x=\begin{bmatrix}u\\ s\end{bmatrix}$,$u$ and $s$ is all 2D vector.
Now I want to blur (convolve) it along with $u$ by another 2D ...
- 11
1
vote
0
answers
103
views
Spectrum of large random asymmetric matrices with correlation
Background:
In their paper, Sommers Crisanti Sompolinsky and Stein derive the spectral distribution of large random matrices $\mathbf{J}$ by studying the following integral:
\begin{equation}
I=\left[\...
- 117
2
votes
1
answer
843
views
Integral involving associated Laguerre polynomial and Bessel function
In a quantum mechanics problem I encountered the following integral
\begin{equation*}
\int_0^\infty t^{\nu+1}J_\nu(\beta t)L_{\mu-\nu}^{2\nu}(t)e^{-t/2}dt\,,
\end{equation*}
where $L$ denotes the ...
- 23
3
votes
0
answers
464
views
Accuracy of Richardson's error estimate in the presence of rounding errors
Richardson extrapolation is a well known technique in scientific computing. It is used to compute error estimates when our approximations can be viewed as a function of a parameter $h$. Common ...
- 221
0
votes
1
answer
129
views
Evaluation of $\int \prod_{j=1}^u \frac{x+j}{j-x}~dx$
Let $I_{n,k} = \frac{(n+k)!}{k!(k-1)!(n-k)!}$. This is a sort of generalization of the Apéry's numbers, with $I_{n,n} =$ the $n$-th Apéry number. I am studying integrals of the form:
$$f_u(x)=\int \...
6
votes
1
answer
339
views
Theory of integration for functions from $\mathbb{Z}_{p}$ to $\mathbb{Z}_{q}$ for distinct primes $p,q$
Let $p$ and $q$ be prime numbers. When $p=q$, Mahler's Theorem gives a complete description of $C\left(\mathbb{Z}_{p};\mathbb{Z}_{p}\right)$, the space of continuous functions from $\mathbb{Z}_{p}$ to ...
- 1,284
0
votes
1
answer
546
views
Orlicz–Sobolev spaces
Let $A$ be an N-function and suppose that
$$\int^{+\infty}_1\frac{A^{-1}(\tau)}{\tau^{1+\frac{1}{n}}}d\tau=+\infty. $$
We denote by $\widehat{A}$ an N-function equal to $A$ near infinity and $\widehat{...
0
votes
1
answer
102
views
Sign of expectation value
Consider a multivariate Gaussian-type measure $$d\lambda(x):=\nu_{\mu,\Sigma} e^{-\langle (x-\mu), \Sigma^{-1}(x-\mu) \rangle - \vert x \vert^2} $$
with vector $\mu \in \mathbb R^n$ and $\Sigma$ ...
- 536
3
votes
1
answer
156
views
$L_p(I,Y)^\perp=L_q(I,Y^\perp)$?
Let $X$ be a Banach space and $Y$ be a closed subspace of $X$. For $1<p<\infty$ consider the $p$-th power Bochner Integrable functions which takes values in $X$ and defined on the unit interval $...
- 521
45
votes
1
answer
2k
views
Existence and uniqueness of Haar measure on compacta; a cohomological approach
I am trying to use a modification of group cohomology to prove the existence and uniqueness of Haar measure on a compact Hausdorff group.
I think the best way of introducing the idea I am pursuing is ...
user30211
2
votes
0
answers
200
views
The collection of mean value abscissas in the Mean value theorem
The integral mean value theorem for continuous f on [0,b] and finite positive continuous measure $\mu$ we have
$$\frac{1}{\mu[a,b]}\int_{a}^{b}f(x)d\mu(x)=f(c)(*)$$
for at least one $c\in [a,b]$. We ...
- 5,474
6
votes
1
answer
1k
views
Fubini's theorem on arbitrary foliations
In what follows $ \mathbb{R}^{n+m} = \{(x,y): x \in \mathbb{R}^n, \ y \in \mathbb{R}^m \} \ .$
Suppose $G: U \to V $ is a $C^1$-diffeomorphism from an open subset of a manifold to an open subset of $...
- 622
5
votes
1
answer
432
views
Functions that are Khinchin integrable but not Henstock-Kurzweil integrable
I posed this question on Mathematics SE recently, though by the total lack of attention it has gotten, I do not anticipate an answer and bring it here.
What are some Khinchin integrable $f$ which ...
0
votes
0
answers
146
views
Why is the divergence theorem used in the Eells-Sampson paper slightly different from that in a textbook?
I am reading Harmonic Mappings of Riemannian Manifolds by Eells and Sampson. In chapter 2, the author(s) used the divergence theorem, which does not look like the usual divergence theorem for ...
- 283
4
votes
0
answers
194
views
The Poincaré Lemma
Let me consider an $L^1(\mathbb R^N)$ function $f$ such that $$
\int_{\mathbb R^N} f(x) dx =0.
$$
Then I claim that the $N$-form $f(x) dx_1\wedge\dots\wedge dx_N$ is closed, i.e. there exists a vector ...
- 16.2k
1
vote
1
answer
571
views
How can we do a Gaussian integral over matrix elements?
I am integrating the following Gaussian over all possible matrix elements $J_{ij}$:
$$ I=\int \exp{\left\{-a\sum_{ij}J_{ij}^2+b\sum_{ij}J_{ij}+c\sum_{ij}J_{ij}J_{ji} \right\}} \left (\prod_{ij}\mathrm{...
- 117
0
votes
1
answer
107
views
Integration of a particular rational expression [closed]
I am trying to solve the following integration, where $a,b,c,d,e$ and $f$ are constants:
$$I=\int\frac{x^4+ax^3+bx^2+cx+d}{x^3(x^3+ex+f)}dx$$
I tried to solve the integral using the following two ...
- 51
5
votes
3
answers
265
views
Is there a good approximation for this Gaussian-like integration?
Is there an analytic solution or approximation for the following Gaussian-like integration? $\frac{1}{\eta^{2n}} \frac{1}{\sqrt{2 \pi}} \int_{-\eta}^{+\eta} e^{-x^2/2} x^{2n} dx$? The numerical plot ...
- 51
5
votes
0
answers
266
views
Hadamard lemma without integration
Let $I$ be the ideal of smooth germs vanishing at zero. Let $I^{k+1}$ be the ideal generated by $(k+1)$-fold product of such germs. Write $F_k$ for the ideal of $k$-flat germs at zero.
By the product ...
- 10.5k
1
vote
0
answers
374
views
Adaptive Simpsons Quadrature Algorithm for Double Integrals? [closed]
I'm currently using Numerical Analysis 10th edition by Richard L Burden as a reference for approximate Integration techniques. In there it describes the Adaptive Simpsons Quadrature rule that inputs ...
- 11
0
votes
1
answer
366
views
Integral $ g(a)= \int_{0}^{\frac{\pi}{2}} \frac{\arctan(a \tan x)}{\tan x}dx $ [closed]
I am having trouble calculating this integral:
$$ g(a) = \int_{0}^{\frac{\pi}{2}} \frac{\arctan(a \tan x)}{\tan x}dx $$
I tried calculating $g'(a)$ but then I get stuck.
- 1
1
vote
0
answers
96
views
Is harmonic mean of linear functions a Bernstein function?
According to some experiments I've been running, for any $n$ and non-negative $a_1, a_2, \ldots a_n$, the following function:
$f(t) = \frac{n}{\sum_{i=1}^n 1/(a_i+t)}$
is a Bernstein function, ...
- 93
1
vote
0
answers
100
views
Expressing 1-e^{-z} as a Fourier integral
According to the theory of screw functions and screw lines by John Von Neumann and Issai Schoenberg (see here), any function $F:\mathbb{R} \rightarrow \mathbb{R}$ such that $F(|x_i - x_j|) = \|f(x_i)-...
- 93
1
vote
1
answer
179
views
Integration of a particular quartic form
I would like to solve the following integral:
\begin{equation}
\int \prod_i d x_i e^{a x_i^2 + b x_i^4 + c x_i^2 x^2_{i+1}}
\end{equation}
This integral can be for sure lead back to a common ...
- 11
0
votes
1
answer
135
views
Bounds on expectation of $X/(X^2 + c)$ with $X$ ~ Gaussian and $c > 0$
I'm trying to compute expectation of $X / (X^2 + c)$ when $X$ is normally distributed with mean $\mu$ and variance $\sigma^2$, and $c$ is some positive constant. I think this cannot be solved ...
- 11
8
votes
1
answer
1k
views
On the closed-form of $\int_0^1\int_0^1\int_0^1\frac{dxdydz}{1-\frac{z}{3}(x+\sqrt{xy}+y)}$
I would like to know if it is possible to calculate in closed-form, or well what work can be done about it, the definite integral $$\int_0^1\int_0^1\int_0^1\frac{3dxdydz}{3-z(x+\sqrt{xy}+y)},\tag{1}$$
...
5
votes
2
answers
2k
views
Elementary calculus estimate or not?
Does there exist a constant $C>0$ such that for all $f \in H^3(\mathbb R)$
$$\int_{\mathbb R} \vert x f''(x) \vert^2 \ dx \le C \int_{\mathbb R} \vert f'''(x) \vert^2 + \vert x^3f(x) \vert^2 + \...
user69109
14
votes
1
answer
655
views
Almost all non-negative real numbers have only finitely many multiples lying in a measurable set with finite measure
Let $A$ be Lebesgue measurable subset of $[0,\infty)$ such that Lebesgue measure of $A$ is positive i.e. $0<\lambda(A)<\infty$. Let $S$ be the set defined as follows:
$$S:=\{t\in [0,\infty):nt\...
- 632
2
votes
0
answers
55
views
Solving the inverse of a matrix under a uniform distribution
I am looking to solve the following equation:
$$\left(\begin{array}{cc}{ g_{11}} & { g_{12}} \\ { g_{21}} & { g_{22}}\end{array}\right)=\int_{a}^{b} \frac{1}{b-a}\left(\begin{array}{cc}{- g_{...
- 117
0
votes
1
answer
234
views
Logarithm of an integral involving generalized real binomial coefficients
I could not find a closed form for this integral although I think it should have been studied.
What is a good approximation to $I$ in
$$I=\ln\Bigg(\int_{0}^y\binom{2m}{m(1+x)}dx\Bigg)$$ where $0\...
- 1,826
0
votes
0
answers
299
views
Derivative of a convolution integral of the following type?
I'm looking to find the derivative of a convolution integral of the following form:
\begin{equation}
\frac{d}{dr}((G(r,t)*f(t)) = \frac{d}{dr} (\int_{-\infty}^{\infty} G(r,t-\tau)f(\tau) d\tau)
\end{...
1
vote
0
answers
422
views
Integral of matrix determinant with respect to Lebesgue measure
$\newcommand\norm[1]{\lVert#1\rVert}
\newcommand\opnorm[1]{\norm{#1}_{\text{op}}}
\newcommand\Frnorm[1]{\norm{#1}_{\text F}}$Define
\begin{align*}
S_t=\{
(A,B)\in\mathbb{R}^{n\times n}\times\...
- 1,495
1
vote
0
answers
171
views
Prove the following property about natural integral
Natural integral is the distinguished antiderivative of a function that can be understood as an analytic continuation of consecutive derivatives of a function towards $-1$th order. It is defined as
$...
- 10.1k
3
votes
2
answers
168
views
A definite integral related to sample variances of bivariate Gaussians
This integral is needed to obtain the joint
distribution of the sample variances of a random sample from a bivariate
Gaussian distribution. For details on the joint distribution of the sample
means, ...
- 984
3
votes
0
answers
116
views
Comparing an integral to zero, by slicing and stacking
Let $f \colon [0,1] \rightarrow {\mathbb R}$ be a nice function -- real-analytic, or maybe definable in some o-minimal structure, let's say. Let $0 < \alpha_1 < \cdots < \alpha_n < 1$ be ...
- 13.3k
2
votes
3
answers
806
views
A Curved/Warped Version of Fubini's Theorem
I will think of $ \mathbb{R}^{n+m}$ as $\mathbb{R}^n \times \mathbb{R}^m$.
Let $ V \subset \mathbb{R}^{n+m}$ be open and $g:V \to U \subset \mathbb{R}^{n+m} $ be a $C^1$ diffeomorphism. For a fixed ...
- 622
1
vote
0
answers
119
views
|Evaluating integral on $ \mathbb S^{d-1}$
I am trying to evaluate the following integral:
$$ \int_{\mathbb S^{d-1}} \exp \bigg(-\frac{(1+x\cdot y)^2}{\|x+y\|^2} \bigg) \ dx $$
for $x,y \in \mathbb R^d$. Does anyone know a solution or an ...
user148767
1
vote
1
answer
548
views
Integration on sphere $\mathbb{S}^{d-1}$ for $d$ large -- Change of variables
I'm trying to integrate a function over two vectors which lie on the surface of the unit sphere in D dimensions. The function depends only on the difference between the two vectors, and their dot ...
user148767
4
votes
3
answers
600
views
Meaning of divergent integrals
In quantum field theory, one usually encounters divergent integral when one calculates loop functions, such as the integral $\int_{0}^{\infty}dkk^{3}\frac{1}{(k^{2}-m^{2})^{2}}$ which is divergent. ...
2
votes
1
answer
340
views
An integral of $|\sin(x)\cos(nx)^{-2/n}|$ from $-\pi$ to $\pi$
For an integer $n \geq 3$, define
$$A_n = \int\limits_{-\pi}^\pi\frac{|\sin(x)|}{|\cos(nx)|^{2/n}}dx.$$
It is a fact that $A_n$ is finite for all such $n$. I am interested in the behaviour of a ...
- 1,625
1
vote
1
answer
415
views
What are the consequences if we could express tangent via logarithm in an algebraic system? [closed]
Working on an algebra of divergent integrals I came to the following relation:
If $\tau=\int_0^\infty dx$ then
$$\ln (\tau+a)=\int_{0}^\infty \psi'(x+1/2+a)dx$$
and this directly gives the following ...
- 10.1k
1
vote
0
answers
158
views
Solving an equation of function
How to solve, or at least how to proceed to solve, the following equation for $g(u)$
$$\int_0^{\infty} \{1-\cos(2\pi uh)\} g(u)du = (1+h^{\alpha})^{\beta/\alpha} -1?$$
Here $0<\alpha\leq2$ and $-\...
- 133
7
votes
1
answer
625
views
Possible application of divergence Theorem?
suppose that $f \in C^1 (\mathbb{R}^{N+1},\mathbb{R})$. It's well known that if all his points are regular points i.e.
$$\nabla f (x) \neq 0 \; \; \; \forall x \in \mathbb{R}^{N+1}$$
then, for every ...
- 51
52
votes
4
answers
6k
views
A historical mystery : Poincaré’s silence on Lebesgue integral and measure theory?
Lebesgue published his celebrated integral in 1901-1902. Poincaré passed away in 1912, at full mathematical power.
Of course, Lebesgue and Poincaré knew each other, they even met on several occasions ...
- 1,026
0
votes
0
answers
91
views
Does $L^1$ convergence preserve the regularity of this sequence of functions?
Let $f_n$ be a sequence of $L^1(]0,1[)$ functions such that $f_n$ is non-decreasing, at least left-continuous, $f_n(0^+) <0$, $f_n(1^-) >0$, for all $n \in \mathbb N$. This sequence converges
$...
- 143
4
votes
2
answers
274
views
Is it true that the quantile function of an $L^1$ random variable is $L^2(]0,1[)$?
Let $(\Omega, \mathcal A, P)$ be a probability space. Let $X:\Omega \rightarrow \mathbb R$ be an $L^1(\Omega, \mathcal A, P)$ random variable.
We define the distribution function of $X$ by
$$F(x) = ...
- 143