All Questions
Tagged with integral or integration
1,506 questions
1
vote
1
answer
51
views
An integral representation of a two-variable function
Hellow.
I don't understand why the following formula is valid.
Can you please tell me the proof?
Let $f(x,y)$ be a function on star-shaped domain of $\mathbb{R}^2$, and let $c(x,y):=\int^1_0tf(tx,ty)...
- 83
28
votes
2
answers
3k
views
Importance of integral equations
Differential equations are at the heart of applied mathematics - they are used to great success in fields from physics to economics. Certainly, they are very useful in modelling a wide range of ...
- 3,738
0
votes
1
answer
390
views
expectation of log(1-x^a) if x is a beta random variable
How can I compute $\mathbb{E}_{q}\Big[\log (1-x^a)\Big]$ when the distribution of $q$ is given as $q(x)\sim\mathrm{Beta}(\alpha,\beta)$?
- 37
1
vote
3
answers
215
views
Solution to $\int_{0}^{y} x^{-a} \exp \left[- \frac{(b - cx^{-d})^2}{2} \right] dx$
Is there a solution to this integral?
$$\int_{0}^{y} x^{-a} \exp \left[- \frac{(b - cx^{-d})^2}{2} \right] dx,$$
where $a > 0$ and $d > 0$.
1
vote
0
answers
132
views
Numerical calculation of a double integral from the slowly-decaying oscillating function
Let us consider the following integral
$$
I = \int\limits_{0}^{+\infty}dx\int\limits_{-\infty}^{+\infty}dy \left[f(x,y) + g(x,y) \right].
$$
We know several properties of these functions.
There are ...
- 335
1
vote
0
answers
128
views
Relation between sum and integral [closed]
The Euler-Maclaurin formula helps to relate sums and integrals. I am particularly interested in one case of equation and want to get it clarified.
$$\lim_{x\to\infty} \left(\sum_{n=0}^{x}f(n)-\int_{0}^...
- 111
0
votes
2
answers
355
views
Integration of the product of sin and exponential with power [closed]
$$\ \int_0^{\pi} \bigl(\sin(x)\bigr)^{2n-2k+1} e^{a\cos(x)} dx , \qquad a,n,k\in\mathbb Z.$$
I tried to solve this integral by parts, but I didn't get any result. I look forward to your experience.
- 41
4
votes
0
answers
113
views
Minimal regularity for domains in Green's formula
The Green formula is well-known for smooth bounded domains of $\mathbb R^d$. My question is:
What is the minimal regularity known for domains where Green's formula still holds?
- 395
2
votes
0
answers
57
views
Is the lattice of bounded Henstock Kurzweil integrable functions countably complete?
The set of HK integrable functions with an integrable upper bound $f$ forms a lattice, and satisfies the MCT and DCT. Does this mean that the lattice is countably complete?
Indexing any countable set, ...
- 1,947
1
vote
1
answer
124
views
Integral of $I = \int_{a}^{\infty}dx \frac{x^s}{(1+x)^{n}}$ [closed]
I have been trying to evaluate the following integral:
$$I = \int_{a}^{\infty}dx \frac{x^s}{(1+x)^{n}}$$
If $a=0$, then this is the Mellin transform of $\frac{1}{(1+x)^n}$. However, suppose $a \neq 0$....
- 11
1
vote
2
answers
630
views
Action of Bochner integral of operator-valued functions on vectors
Consider a separable Hilbert space $\mathcal H$ and the bounded linear operators $B(\mathcal H)$.
Consider a function $T: [0, \infty) \to B(\mathcal H)$, under what assumptions on $T(t)$ is it true ...
- 151
2
votes
1
answer
119
views
Exact formula or non-trivial upper bound on p-norm of $f(x)=\|x\|_2$ in $[0,1)^d$
I wonder whether one can exactly calculate the following integral in terms of $d$ and $p\geq 1$ or not, or a better bound(than the trivial one I am going to give) in terms of $d,p$:
$$\left(\int_{[0,1)...
- 715
0
votes
1
answer
202
views
When does the measure integral of the form $\int_{\log(S)} f d \mu$ exist?
When does the measure integral of the form $\int_{\log(S)} f d \mu$ exist ?
Here $\mu$ can be any measure (Lebesgue, Borel, Haar etc), $f$ is a measurable function, $S$ is any measurable set with ...
- 930
1
vote
0
answers
45
views
Asymptotic solution of wrinkle amplitude
I have encountered an equation determining amplitude of sinusoidal wrinkle
$\int_{0}^{\lambda_0} \sqrt{1+[(A_0\sin{\frac{2\pi x}{\lambda_0}})']^2} \,dx = \int_{0}^{\lambda_0(1+\epsilon_a)} \sqrt{1+[(A\...
- 11
0
votes
0
answers
79
views
Question on evaluating double integral
I am trying to prove the following inequality:
$\int_{\tau}^{B} \int_{b}^{A} \frac{a(a-b)}{4a-b} dadb + \int_{\tau}^{B} \int_{\tau}^{b} \frac{b(b-a)}{a-4b} dadb \geq \int_{\tau}^{B} \int_{\tau}^{A} \...
- 1
1
vote
1
answer
522
views
Stochastic Integral + conditional expectation
Let $\overline{\widehat{Z}_i} = \frac{E_i\left[ \int_{t_i}^{t_{i+1}}\widehat{Z}_sds\right] }{\Delta t_i}$ with $\widehat{Z}$ a square integrable process, $\Delta t_i := t_{i+1} - t_i$, and $E_i$ ...
- 43
4
votes
0
answers
826
views
Showing that $\int_0^\pi\frac{x\ln(1-\sin x)}{\sin x}dx=3\int_0^\frac{\pi}{2}\frac{x\ln(1-\sin x)}{\sin x}dx$
Prove, without evaluating the integrals that:
$$\int_0^\pi\frac{x\ln(1-\sin x)}{\sin x}dx=3\int_0^\frac{\pi}{2}\frac{x\ln(1-\sin x)}{\sin x}dx$$
Originally I posted this here on MSE, however it's ...
- 215
22
votes
1
answer
1k
views
A multiple integral that seems related to the $\zeta$ function at even integers
I came across this integral that seems related to the Riemann zeta function $\zeta(2n)$ evaluated at even integers $2n \in 2\mathbb{Z}$. Letting $n$ be an even integer, define the multiple integral ...
- 545
2
votes
1
answer
670
views
Integral on level sets
Let $g_\epsilon : K \subset \mathbb{R}^d \rightarrow \mathbb{R}$ (more regularity can be assumed if necessary) be defined on a compact set (with regular boundary) $K \subset \mathbb{R}^d$, and the ...
user140442
4
votes
2
answers
316
views
Average value of $\frac{x'A^2x}{x'A^3x}$ over surface of $n$-dimensional sphere
Suppose $A$ is a diagonal matrix with eigenvalues $1,\frac{1}{2},\frac{1}{3},\ldots,\frac{1}{n}$ and $x$ is drawn from standard Gaussian in $n$ dimensions. Define $z_n$ as follows
$$z_n=E_{x\sim \...
- 2,252
4
votes
1
answer
357
views
Haar integral of rational function of unitaries
I'm trying to compute the following Haar integral over the unitary group:
$$
\int\limits_{\mathbb{U}(d)}\dfrac{1}{\sum_{k,l=1}^d u_{ik}\overline{u_{il}}c_{kl}}dU.
$$ Is there anything known about the ...
0
votes
1
answer
314
views
How to find $\int_{0}^{\infty} \log(1+x)(x^a)\exp(-bx) dx$ step by step?
How can I find $\int_{0}^{\infty} \log(1+x)x^a\exp(-bx) dx$ step by step?
I've got a Mathematica solution, but I'd like to know and understand the steps taken to reach the solution.
On page 469 of &...
4
votes
1
answer
322
views
Integrals involving $1/|\zeta(1+i t)|^2$: closed expressions?
Is there by any chance anything resembling a closed expression for, say, the integral
$$I = \frac{1}{2 \pi} \int_{-\infty}^\infty \frac{dt}{|\zeta(1+i t)|^2 t^2} ?$$
It is easy to show (by Plancherel) ...
- 20.2k
0
votes
1
answer
171
views
How to compute $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{[-1,1]^n}\exp[2\pi i(\theta_1 v_1.x+\theta_2v_2.x)]d^nx d\theta_1d\theta_2$
Let $\mathbf{v}_1, \mathbf{v}_2$ be two vectors in $\mathbb{R}^n$. I would like to compute the following singular integral:
$$\int_{-\infty}^{ \infty} \int_{-\infty}^{\infty}
\int_{[-1,1]^n}
e(\...
- 3,625
1
vote
0
answers
87
views
Estimating the impact of replacing a negative exponential by a truncation of its Taylor series in an integral
Let $f(x)$ be a smooth function that takes both positive and negative values and suppose there exists an increasing sequence of positive numbers $R_i$ diverging to $\infty$ such that
$$\lim_{i \...
- 81
2
votes
1
answer
132
views
A "uniform continuity" type condition on a Hammerstein integral equation
I asked the following question on MathStackExchange, but I have not received the answer that I'm looking for. Although it may not be a research-level question, I thought I could ask it here.
I'm ...
- 291
7
votes
0
answers
332
views
Integration à la Mirzakhani
Let $$
\gamma = \sum_i c_i \gamma_i
$$
be a multi-curve on a hyperbolic surface $S$. For any $f: \mathbb{R}^+ \to \mathbb{R}^+$ one can define $$
f_\gamma (X) = \sum_{\alpha \in \mathrm{Mod} . \gamma} ...
- 191
2
votes
0
answers
97
views
Finite version of Mehlers formula?
This is a crosspost from Math Stack Exchange, please let me know if this is not an appropriate use of crossposting, and I will delete.
Mehler's formula is the following identity for Hermite ...
- 233
4
votes
0
answers
75
views
Marginalization of Wishart distribution
Consider the following Wishart distribution
$$
f({\bf W}) = \frac{ |{\bf W}|^{(n-p-1)/2} \exp\big[-\frac{1}{2}\text{tr}({\bf V}^{-1}{\bf W} ) \big] }{2^{np/2} |{\bf V}| \Gamma_p(\frac{n}{2})} \tag{1}
$...
3
votes
1
answer
456
views
Fast computation of convolution integral of a gaussian function
Given a convolution integral
$$
g(y) =\int_a^b\varphi(y-x)f(x)dx=\int_{-\infty}^{+\infty}\varphi(y-x)f(x)\mathbb{I}_{[a,b]}(x)dx
$$
where
$\varphi(x)= \frac{1}{\sqrt{2\pi}}\exp{\left(-\frac{x^2}{2}\...
- 250
2
votes
1
answer
178
views
Given the integral. What's the relation between $I_{n+1}(t)$ and $I_n(t)$?
$$I_n(t)=\int_0^t\frac{1}{\left(x^5+1\right)^n}dx.$$
What is the relation between $I_{n+1}(t)$ and $I_n(t)$?
Can it be done with integration by parts?
- 33
2
votes
0
answers
162
views
Integral rewritten in terms of a modified Bessel function
I am reading this paper by Kunz and Shapiro: they state that the integral (Eqs. 3.17-3.19)
$$\int_{-\infty}^\infty\frac{dy}{2\pi}e^{ib(y-i\delta)}\left[\exp\left(-\frac{ia}{y-i\delta}\right)-1\right]\...
- 21
5
votes
1
answer
561
views
Upper bound an integral with exponential function
I am working on my research about approximation a function. I come up with the following integral. I run some simulations and saw that the integral would converge to zero as n goes to infinty. Here is ...
- 87
2
votes
1
answer
102
views
Approximation of $\Phi (p)$
I am trying to find the asymptotic behavior (with respect to N) of the integral $$ \frac{2}{\sqrt{\pi}}\int_0^\infty \varPhi^{N-2}(p)e^{-p^2}\ dp. $$ In Rényi and Sulanke's paper Uber die konvexe ...
- 23
2
votes
0
answers
108
views
Are a.e. derivatives of continuous $VBG_*$ functions Denjoy–Perron integrable?
I would like to ask a question pertaining to the Denjoy–Perron (Henstock–Kurzweil) theory of integration. It is simple enough that I have entertained the idea that perhaps an answer is known, but I ...
4
votes
0
answers
311
views
Approximation of integral of gaussian function over a parallelepiped
Remark: I posted this question in math stackexchange here and computer science stackexchange https://cs.stackexchange.com/ few weeks ago but obtain no answer.
Given a multi-dimensional gaussian ...
- 250
0
votes
1
answer
198
views
Prove that $f(0+)=f(0)$ if $f \in R(\beta_1)$ [closed]
Let $\beta_1$ be a function defined by $$\beta_1(x)= \begin{cases} 0 & x \le 0\\ 1 & x >0 \end{cases} $$
Now we define $f(x)$ which is a bounded function on $[-1,1]$.
We need to how that $ ...
- 117
1
vote
1
answer
343
views
Simple example of Hammerstein integral equation
I'm currently reading this paper (and working on a similar one). The main goal is to study the Hammerstein integral equation (in $\mathcal{C}(I,E))$:
$$x(t) = \int_{0}^{t} K(t,s)f\big(s,x(s)\big)ds,\...
- 291
1
vote
1
answer
441
views
A question about eigenvalue equation of Hankel transform
When we think about the Fourier transform in two dimensional polar coordinates, the Hankel transform is the transformation with respect to the polar diameter. Now I have a question, why is the ...
- 11
1
vote
0
answers
100
views
$ \lim _{n \rightarrow \infty} \int_{E} \frac{f_{n}^{2}(x)}{1+f_{n}^{2}(x)} \mathrm{d} m=0 $ associated with convergence in measure [closed]
For $m E<+\infty$, why the sufficient and necessary condition of $\left\{f_{n}(x)\right\}$ converge in measure to $0$ is
$$
\lim _{n \rightarrow \infty} \int_{E} \frac{f_{n}^{2}(x)}{1+f_{n}^{2}(x)}...
- 11
1
vote
0
answers
245
views
A characterization of the integral
Let $I(f)$ be an endomorphism of the smooth functions with zero value in zero such that:
$$\ln[1+I(f)]=I\left(\frac{f}{1+I(f)}\right).
$$
Then, does it exist $g$ smooth such that:
$$I(f)(x)=\int_0^x f(...
- 377
6
votes
3
answers
714
views
Expected absolute value of the average of two points from the disc
Looking at Average distance of the mean of n random complex numbers in a unit disc, I tried to figure out
what is the expected absolute value $|\frac{z_1 + z_2}{2}|$ of two numbers $z_1, z_2\in\...
- 10.7k
1
vote
1
answer
118
views
Asymptotics of the integral of an oscillating function
I would like to know the asymptotics of the following sequences of integrals:
$$ I_n = \displaystyle { \int _0 ^{+ \infty}
\dfrac{t^n}{(t + i)^{n + 1}}
...
- 97
3
votes
0
answers
129
views
Inverse Laplace transform through contour integration
How can I prove that in formal way, this function doesn't have inverse Laplace transform.
$$
F(s)=\frac{\sin(s)}{\sqrt{s}}
$$
Strictly it should be in Bromwich contour method.
Could you please tell ...
- 31
1
vote
1
answer
136
views
Prove the integral of multi-variable rational fraction is convergent
I have posted this problem in MSE long ago:
https://math.stackexchange.com/questions/3782868/multi-variable-rational-fraction-integral. But it hasn't been solved yet so I post it here. Maybe this ...
- 561
2
votes
0
answers
303
views
Infinite sum of iterated integrals of matrix products
Originally asked over at Stackexchange (https://math.stackexchange.com/questions/4169812/infinite-sum-of-iterated-integrals-of-matrix-products), but this forum was deemed more appropriate.
The problem:...
0
votes
0
answers
50
views
Integral of Airy function containing a second order polynom
I wonder if there is an analytic expression for
$ \int_{-\infty}^{\infty} \mathrm{Ai}(a x^2 + b x + c) dx$
As a Bonus:
$ \int_{-\infty}^{\infty} e^{- d x^2} \mathrm{Ai}(a x^2 + b x + c) dx$
where $a,...
- 113
7
votes
1
answer
307
views
Reference for proof of an integral from the "Tables of Integral Transforms" involving a Gaussian and a Laguerre polynomial
I am looking for a proof of one of the integrals presented in Harry Bateman's Tables of Integral Transforms. The specific integral in question is presented on page 42 in chapter 8.9 as equation (3):
$$...
- 73
2
votes
0
answers
162
views
$\int_{\mathbb{R}^{N}\setminus\Omega}\vert x-z\vert^{-N-\alpha} dz = c \ \forall x\in\partial U$ implies $dist(x,\partial\Omega)=c, x \in \partial U$?
Let $\alpha \in \mathbb R_+$, $\Omega \subset \mathbb R^N$ and $U \subset \Omega$. Is it true that if
$$\int_{\mathbb R^N \setminus \Omega} |x - z|^{-N-\alpha} dz = \text{constant} \quad \text{for all ...
- 61
3
votes
2
answers
135
views
Asymptotics of a sequences of integrals
I would like to know the asymptotics of the following sequences of integrals:
$$ I_n = \int _0
^{+ \infty}
e^{-t} \left ( \dfrac{t}{1 + t} \right )^n
\...
- 97