All Questions
Tagged with integral or integration
1,506 questions
0
votes
1
answer
58
views
Getting a finite value for curve Dissidence
If I've got $f: x \to f(x)$, one may define the Arithmetic Dissidence $\delta_A[f(x)]$ as the real value of the difference between the length following the curve of $f$ and the length of the $x$ axis (...
- 111
10
votes
1
answer
431
views
An integral identity involving cotangents and Bessel functions
Numerical experiments suggest that the following integral identity holds for Bessel functions of the first kind,
$$J_2(t) = 12 \int_0^{1/2}\mathrm{d}x\,\cot \pi x \int_0^x \mathrm{d}y\, \cot \pi y \, ...
- 3,927
1
vote
1
answer
532
views
Convolution of an Airy function with a Gaussian
I wonder if the convolution
\begin{equation}
f(y)=\int_{-\infty}^{+\infty} \mathrm{Airy}(a\cdot x)\cdot e^{-b(y-x)^2} dx
\end{equation}
can be solved analytically. Or in case not, if there is an ...
- 113
5
votes
2
answers
466
views
About the integral $\int_{0}^{1}\frac{\log(x)}{\sqrt{1+x^{4}}}dx$ and elliptic functions
NOTE: I post this question on math.stackexchange but nobody answered, so I try here.
For a work we need to evaluate the following integral $$\int_{0}^{1}\frac{\log\left(x\right)}{\sqrt{1+x^{4}}}dx=\,-...
- 319
4
votes
1
answer
191
views
Scaling of double convolution
I am interested in the scaling of
$$F(x_1,x_4)=\int_{\mathbb R^2} e^{-\vert x_1 -x_2 \vert -\varepsilon \vert x_2 -x_3 \vert- \vert x_3 -x_4 \vert } \ dx_2 dx_3 $$
In particular, I suspect that
$$F(...
- 192
2
votes
0
answers
65
views
Reference request for type of specific integral equation in two variable:
Consider the following integral equation:
$$\int_0^\infty K(t,y)\phi(t,x)dt=0$$
Here, $K(t,y)$ is a trigonometric kernel and
$\phi(t,x)$ is monotonic wrt $x$ ( for fixed $t$).
I want to find the ...
- 41
0
votes
1
answer
541
views
Interchange of integration and supremum
Let $u \in C^0(-T,T; L^2(B_R))$ be a measurable function, then is the following true?
$$
\int_0^R \sup_{-T<t<T} \int_{S_r} |u(\sigma ,t)|^2 \ d \sigma \ dr = \sup_{-T<t<T}\int_0^R \int_{...
- 455
2
votes
1
answer
339
views
$\int\limits_{-\infty}^\infty \left(f(\frac{x-\mu}{1+\Psi/2})-f(\frac{x+\mu}{1-\Psi/2})\right)\frac{x\gamma}{(x-x_{0})^{2}+a}dx$
EDIT: I realized from numerical implementation that the step from
\begin{align}
\mathcal{I}_2=&\frac{\gamma}{2}\int\limits_{-\infty}^\infty \left(f_T(\frac{x-\mu}{1+\Psi/2})-f_T(\frac{x+\mu}{1-\...
4
votes
1
answer
96
views
Estimate of $\frac{\int x^{2p}\,e^{-x^{2n}\,+\,\omega(x,y)}\;dx}{\int e^{-x^{2n}\,+\,\omega(x,y)}\;dx}$
For every $x,y\in\mathbb R$ let
$$ V(x,y) \,\equiv\, a\,x^{2n} + b\,y^{2m} - \omega(x,y)\,$$
where $a,b>0$, $n,m\in\mathbb N$, $n\geq m\geq1$, and $\omega$ is such that $\omega(x,y)/(x^{2n}+y^{2m})...
- 311
2
votes
0
answers
132
views
Green's identity with a different norm
Let $\Omega \subset \mathbb{R}^n$ be a domain with a smooth boundary $\Gamma$. Suppose that $f, g \colon \mathbb{R}^n \to \mathbb{R}$ are of class $C^\infty( \overline{\Omega})$. Then Green's first ...
- 820
2
votes
1
answer
369
views
Ratios of Gaussian integrals with a positive semidefinite matrix
Cross-post from MSE
https://math.stackexchange.com/questions/4118128/ratios-of-gaussian-integrals-with-a-positive-semidefinite-matrix
Generally speaking, I’m wondering what the usual identities for ...
- 1,026
3
votes
1
answer
183
views
On integral representation of Whittaker $W$ functions
According to NIST, the integral representation of Whittaker $W$ functions
$$
W_{\kappa,\mu}\left(z\right)=\frac{z^{\mu+\frac{1}{2}}2^{-2\mu}}{\Gamma\left(%
\frac{1}{2}+\mu-\kappa\right)}\int_{1}^{\...
- 373
0
votes
2
answers
682
views
On integral relating logarithm of absolute value of Zeta function
Sorry for such a direct question:
Consider the following integral:
$$I(t)=\int_{1/2}^{1} {\log|\zeta(a+it)|}da.$$
How to find the nature of $I(t)$ as $t\rightarrow\infty$?
- 790
2
votes
0
answers
44
views
Derivatives of $G_h(u):=\int_0^{2\pi} h(\cos t)h(\cos(t - \arccos(u)))dt$ when $h$ is positive-homogeneous
Let $h:\mathbb R \to \mathbb R$ be a continuous which is positive-homogeneous of order $p \ge 1$, and define $G_h:[-1,1] \to \mathbb R$ by
$$
G_h(u):=\int_0^{2\pi} h(\cos t)h(\cos(t - \arccos(u)))dt.
$...
- 6,853
1
vote
2
answers
390
views
On the Bochner spaces $L^\infty(a,b;L^p(c,d))$ and $L^p(c,d;L^\infty(a,b))$, or: Interchange of supremum and integral
I am asking whether the Bochner spaces $L^\infty(a,b;L^p(c,d))$ and $L^p(c,d;L^\infty(a,b))$ are the same. Or, whether one is included/embedded in the other.
We have the norms
$$\|u\|_{L^\infty L^p}=\...
- 51
0
votes
1
answer
195
views
Sufficient conditions for finite mean of a non-negative random variable
Consider a continuous random variable that takes only non-negative values. Let the cumulative distribution function be $F(\cdot)$. Consider the following condition:
$$\lim_{x\rightarrow\infty} x(1-F(x)...
- 79
0
votes
1
answer
86
views
Is integration against an indicator Wasserstein-Continuous
Let $\mathcal{P}_p(X)$ denote the Wasserstein space over a compact metric space $X$, and $1\leq p<\infty$. Fix a non-empty closed subset $C\subseteq X$. Then is the map:
$$
\mathbb{P} \mapsto \...
- 5,405
1
vote
0
answers
240
views
Riemann-Stieltjes integral of a distribution function
I recently learned the basics of Riemann-Stieltjes integral, and based on the sources I found, we can define the expectation of random variables quite naturally with the R-S integrals: if $X$ is a ...
- 21
1
vote
1
answer
95
views
Decide the order of of an integration involving the $\log$ function
Let $$A_n=\int_{n^{-\frac{1}{2}}}^{1}\frac{\log(nx)}{nx(\log\log(nx)-\log\log(1+x))}dx.$$
I want to discribe the order of $A_n$, by geting a progressive formula or a good lower bound for it. The order ...
- 622
2
votes
1
answer
194
views
Generalized Selberg integral
I am interested in expressing the following generalization of the Selberg integral in terms of Gamma functions
$$ \int_0^1 \ldots \int_0^1 \prod_{i=1}^d u_i^{\frac{k_i-1}{2}} \prod_{m=1}^d (1-u_m)^{\...
- 83
2
votes
0
answers
126
views
How does the area affect the integral?
Let $\Omega\subset\mathbb{R}^n$ a open bounded set. For any $r>0$ consider the integral:
$$J_\Omega(r)=\int_{\Omega}\frac{|x^s|dx}{r^c+\sum_{i=1}^m|x^{p_i}|r^{d_i}},$$
where $s,p_i\in\mathbb{N}^n$ ...
- 561
0
votes
0
answers
83
views
The loss of double periodicity (ellipticity)
Consider a meromorphic function $f(\mathfrak{a}_1, \mathfrak{a}_2)$, such that
$$
\begin{align}
f(\mathfrak{a}_1, \mathfrak{a}_2) = f (\mathfrak{a}_1 + 1, \mathfrak{a}_2) = f(\mathfrak{a}_1 + \tau, \...
- 857
3
votes
4
answers
366
views
Integrals involving fractions of exponentials
I am trying to calculate the average degree of a complex network, which requires me to solve for the following integral:
$$\int \mathrm{d} x \frac{\exp{\left[-x -\frac{1}{2}\left(\frac{x-\mu}{\sigma}\...
- 33
1
vote
1
answer
174
views
Faster than Euler's substitution. How to derive this formula?
I wish someone could help me derive this expression. ($K$ is a constant coefficient. $P_n(x)$ is a polynomial function of degree n.)
$$
\int\frac{P_n(x)\mathrm{d}x}{\sqrt{ax^2+bx+c}} \equiv P_{n-1}(x) ...
- 11
5
votes
1
answer
155
views
Which averages of products of a function give a norm?
Let $f: [0,1] \rightarrow \mathbb{R}$ be a bounded measurable function. For some real non-negative numbers $a_1, a_2, b_1, b_2$ with $a_1+b_1=a_2+b_2=1$ consider the quantity
$$N(f)=\int_{[0,1]} \int_{...
- 2,288
8
votes
0
answers
362
views
The many theories of integration
Diclaimer: In what follows, I will be loose in the usage of terminology since the very nature of the question is of a similar flavour.
In the mathematics literature, one can find a zoo of theories of ...
- 183
3
votes
0
answers
269
views
definite integral with incomplete gamma function and exponential
While working with electron density computations in quantum chemistry, I encountered the following improper integral:
$$
I(k, n) = \int\limits_0^\infty \Gamma\left(\frac{3}{n},\ k r^n\right) r \exp(-k ...
- 31
2
votes
2
answers
296
views
Laplace transform calculation
Please can someone help me? I have tried to find the Laplace transform of the form:
$$\int_{0}^{+\infty} (v+1)^{\nu}(2v+1)^{k}v^{\alpha} \exp(-pv), \mbox{ where }\alpha,\nu, k \mbox{ are integers }$$
...
- 159
3
votes
1
answer
384
views
A possible error in Villani's monograph "Hypocoercivity"
I think there is an (possible) error in Villani's monograph titled "Hypocoercivity". To be specific, in page 48, he wrote "For the second term in (6.9), we use the identity $$\nabla\...
- 730
3
votes
1
answer
702
views
van Cittert deconvolution method
In the early 1930s, van Cittert published a deconvolution method. Although his method was not perfect but it is the forefather of many improved spectral deconvolution methods. The basic idea is that ...
- 879
2
votes
3
answers
425
views
Help with a limit involving incomplete beta integral
In trying to prove that the limit of a certain function approaches 1 as the positive integer parameter $n$ approaches infinity, I have ended up with the following intermediate expressions:
$$f(n)=2^{...
- 826
6
votes
4
answers
628
views
Generalizing contour integration to quaternions and bicomplex numbers
I am interested in the possibility of generalizing the notion of contour integration to the quaternions or bicomplex numbers. I am aware that the Frobenius theorem prevents the construction of a true ...
- 547
5
votes
1
answer
2k
views
Question on an exercise from Terry Tao's blog
I've been reading Tao's An introduction to measure theory, a draft can be found here. An exercise from it is
Exercise 30 (Rising sun inequality) Let ${f: {\bf R} \rightarrow {\bf R}}$ be an absolutely ...
- 95
1
vote
0
answers
129
views
Sources for multiple Stieltjes integral
My research involves multiple Stieltjes integral or multiple Lebesgue-Stieltjes integral. But after searching online, I can not find what I need. So I ask this question on which sources (books or ...
- 663
9
votes
2
answers
1k
views
A tricky integral to evaluate
I came across this integral in some work. So, I would like to ask:
QUESTION. Can you evaluate this integral with proofs?
$$\int_0^1\frac{\log x\cdot\log(x+2)}{x+1}\,dx.$$
- 43.2k
0
votes
0
answers
72
views
Integration of fractional function over Rice distribution
Let $a>2$ be a real variable. My objective is to find an approximation of the integral defined as
\begin{equation}
\int_0^{\infty } {\frac{1}{{1 + {x^a}}}} f\left( {x|y} \right)\, dx
\end{equation}...
- 377
0
votes
1
answer
128
views
integral of fractional function
Let $a>2$ be a real variable. My objective is to find an approximation of the integral defined as
\begin{equation}
\int_b^{ + \infty } {\frac{x}{{1 + {x^a}}}}.
\end{equation}
Here $b$ is a ...
- 377
8
votes
3
answers
629
views
Expected distance between two uniform points in distinct rectangles
Are there any good approximations (especially upper bounds) for the quantity $E(\|X_1-X_2\|$), where each $X_i$ is uniformly distributed in a rectangle $[a_i,b_i]\times[c_i,d_i]$? It does not appear ...
- 4,049
1
vote
0
answers
106
views
Length of isoline $x(1-x)y(1-y)=c$
For the integral appearing in this answer, it may be beneficial to derive the length $L(c)$ of the isoline:
$$x(1-x)y(1-y) = c,$$
where $x,y$ are ranging in $[0,1]$, and constant $c\in [0,\frac1{16}]$....
- 34.3k
17
votes
3
answers
1k
views
Decoupling a double integral
I came across this question while making some calculations.
QUESTION. Can you find some transformation to "decouple" the double integral as follows?
$$\int_0^{\frac{\pi}2}\int_0^{\frac{\pi}...
- 43.2k
3
votes
0
answers
65
views
How to find or approximate (e.g. using method of steepest descent ) integral?
Can you give any advice on how to find or approximate the following integral
$$
F(t,y) = \int_{0}^{y}\frac{i e^{-\frac{3 t^2 \left(x^2+1\right)}{2 \left(9 x^2+1\right)}-i \frac{4 t^2 x}{9 x^2+1}}}{\...
- 31
0
votes
1
answer
223
views
Distribution of $\sqrt{X^2+Y^2}$ when $X$ and $Y$ are Cauchy
Suppose that $X$ and $Y$ are Cauchy-distributed with $\gamma=1$, i.e., with PDF $\frac 1 \pi \frac 1 {1+x^2}$. I tried to find the distribution of $R = \sqrt{X^2+Y^2}$. The PDF of $R$ should be given ...
- 3,738
2
votes
2
answers
321
views
Asymptotic of an improper integral
I found myself stuck with an "elementary" claim in some article. A simplified version of the problem is:
Let $p : [0,1]\to [0,1]$ be a continuous and non decreasing function such that $p(0)=...
- 43
24
votes
2
answers
1k
views
Is $\iiint_{[0, 1]^3} \lvert f(x)+f(y)+f(z)\rvert\, dx\, dy\, dz \ge \int_0^1 \lvert f(x)\rvert\, dx$?
$\newcommand\abs[1]{\lvert#1\rvert}\newcommand\Abs[1]{\left\lvert#1\right\rvert}$Question: Let $f(x)$ be a continuous real-valued function defined on the interval $[0, 1]$. Does the following ...
- 4,280
1
vote
0
answers
77
views
How does one interpret the wetting area?
This may be a simple question, but I decided to post it here (not just on MSE) because it is very related to a research topic: capillary surfaces.
Let $(M^3,g)$ be a Riemannian $3$-manifold with ...
- 2,095
18
votes
0
answers
571
views
Fundamental Theorem of Algebra via multiple integrals
Consider the product of complex linear monic polynomials times polynomials of degree less than $n$, that is $\big( (z-\lambda), p(z)\big)\mapsto (z-\lambda)p(z)$. If we represent a polynomial by its ...
- 60.5k
2
votes
1
answer
239
views
Injectivity of an integral transform
For a bounded function $F: \mathbb R_{\ge 0} \to \mathbb R$ (not necessarily non-negative), is it true that
$$\int_0^\infty \frac{x^ks}{(s^2+x^2)^{(k+3)/2}} F(x) dx = 0 \text{ for all $s >0$} \iff ...
- 303
2
votes
1
answer
592
views
Estimate on an integral involving the Japanese bracket
I'm currently reading the paper Well-posedness for the Zakharov system with the periodic boundary condition by Takaoka. In the proof of Lemma 2.3 about the integral $I_1$ one needs to establish the ...
- 469
2
votes
4
answers
1k
views
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 ...
- 93
2
votes
0
answers
259
views
Bochner integral in a Fréchet space
I have a Fréchet space $V$ whose topology is (if it helps) induced by a family $\mathcal{P}$ of norms - not just seminorms - and on this space I have a Borel probability measure $\nu$. Now, I would ...
- 651