Questions tagged [integration]

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, ...

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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} ...
EtienneBfx's user avatar
2 votes
0 answers
84 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 ...
fewfew4's user avatar
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4 votes
0 answers
74 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} $...
RenatoRenatoRenato's user avatar
3 votes
1 answer
313 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}\...
NN2's user avatar
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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?
Dutu Mircea's user avatar
2 votes
0 answers
144 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]\...
baderi's user avatar
  • 21
5 votes
1 answer
466 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 ...
Quicky2357's user avatar
2 votes
1 answer
101 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 ...
user311932's user avatar
2 votes
0 answers
100 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 ...
David Manolis's user avatar
4 votes
0 answers
273 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 ...
NN2's user avatar
  • 250
0 votes
1 answer
175 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 $ ...
ThirstForMaths's user avatar
1 vote
1 answer
263 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,\...
Motaka's user avatar
  • 291
1 vote
1 answer
344 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 ...
Jiang's user avatar
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1 vote
0 answers
95 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)}...
Ad_M's user avatar
  • 11
1 vote
0 answers
243 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(...
Antoine Balan's user avatar
6 votes
3 answers
703 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\...
Moritz Firsching's user avatar
1 vote
1 answer
116 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}} ...
MathTolliob's user avatar
3 votes
0 answers
114 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 ...
meli0das's user avatar
1 vote
1 answer
114 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 ...
Houa's user avatar
  • 561
2 votes
0 answers
294 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:...
genus_3_amoeba's user avatar
0 votes
0 answers
47 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,...
Luke's user avatar
  • 113
7 votes
1 answer
255 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): $$...
schade96's user avatar
2 votes
0 answers
159 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 ...
user175203's user avatar
3 votes
2 answers
133 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 \...
MathTolliob's user avatar
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 (...
Aileann D. PRET's user avatar
10 votes
1 answer
421 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 \, ...
Timothy Budd's user avatar
  • 3,545
1 vote
1 answer
447 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 ...
Luke's user avatar
  • 113
4 votes
2 answers
428 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=\,-...
Marco Cantarini's user avatar
4 votes
1 answer
189 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(...
Kung Yao's user avatar
  • 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 ...
GSA_1's user avatar
  • 41
0 votes
1 answer
428 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_{...
Adi's user avatar
  • 483
2 votes
1 answer
311 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-\...
Bhagwan rajneesh's user avatar
4 votes
1 answer
95 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})...
tituf's user avatar
  • 311
2 votes
0 answers
131 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 ...
Kacper Kurowski's user avatar
2 votes
1 answer
331 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 ...
Fabrice Pautot's user avatar
3 votes
1 answer
170 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}^{\...
Y.Okuyama's user avatar
  • 373
0 votes
2 answers
479 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 $t\rightarrow\infty$?
TPC's user avatar
  • 690
2 votes
0 answers
42 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. $...
dohmatob's user avatar
  • 6,716
1 vote
2 answers
306 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}=\...
Cahn's user avatar
  • 51
0 votes
1 answer
184 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)...
liuchun deng's user avatar
0 votes
1 answer
83 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 \...
ABIM's user avatar
  • 5,019
1 vote
0 answers
216 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 ...
gouhaha's user avatar
  • 21
1 vote
1 answer
94 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 ...
ZZP's user avatar
  • 404
2 votes
1 answer
172 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)^{\...
esner1994's user avatar
2 votes
0 answers
124 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 $s,p_i\in\mathbb{N}^n$ ...
Houa's user avatar
  • 561
0 votes
0 answers
79 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, \...
Lelouch's user avatar
  • 857
3 votes
4 answers
330 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}\...
dancer's user avatar
  • 33
1 vote
1 answer
171 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) ...
RengarJG's user avatar
5 votes
1 answer
154 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_{...
TOM's user avatar
  • 2,218
8 votes
0 answers
325 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 ...
genfuntranslate's user avatar

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