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-1
votes
1answer
90 views

Suppose I know $\int h(t) dt = H(t)$, is there a way to find $\int h(t)^N dt$?

I am trying to find the -1 moments of sum of N geometric random variable, i.e. $E[\frac{1}{\sum_{i=1}^N X_i}]$ Suppose the probability mass function is $f_X(x) = (1 - p)^{x - 1} p$ The moment ...
-5
votes
1answer
75 views

computing integral of dz/(z+1) on unit circle [on hold]

I guess it must be a simple matter in complex analysis, but I would like to compute the the following integral: $$ \oint_C\frac{dz}{z+1} $$ and $$ \int_C\frac{dz}{z+1} $$ where $C=\{z\in \mathbb{C}: ...
-1
votes
0answers
56 views

Rising Sun Inequality (Dunford-Schwartz maximal inequality) [migrated]

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be an absolutely integrable function, and let $f^*:\mathbb{R} \rightarrow \mathbb{R}$ be the one-sided signed Hardy-Littlewood maximal function $$f^*(x) := ...
6
votes
1answer
144 views
+100

On the convexity of certain integrals involving Bessel functions

Let $n\geq 0$ be an integer and let $J_n=J_n(r)$ denote the usual Bessel function (of the first kind) of order $n$ i.e. one of the solutions to Bessel's differential equation ...
0
votes
1answer
95 views

Is any derivative of $f_1^x f_0^{1-x}$ w.r.t. $x$ integrable?

For $f_0$ and $f_1$ two continuos probability density functions on $\mathbb{R}$, by Hölder, I know that $f_1^x f_0^{1-x}$ is integrable on $\mathbb{R}$, where $0 \leq x \leq 1$. Let $l=f_1/f_0$, then ...
3
votes
1answer
75 views

Convergence of the Double Integral of a Polynomial Reciprocal

Let $f \in \mathbb{R}[x,y]$ be a polynomial satisfying the following conditions: (i) $f(\mathbb{R}^2) \subset [a,\infty)$ where $a>0$; (ii) $f$ is non-degenerate, in the sense that there isn't a ...
-1
votes
0answers
7 views

How to compute $\int_0^1 [ln(1/x)]^5 dx$ [migrated]

Why is $\int_0^1 [ln(1/x)]^5 dx=120$? More generally, is $\int_0^1 [ln(1/x)]^n dx$ equal to n!? If so why?
2
votes
2answers
288 views

How to integrate an exponential function of an exponential function?

Does any one know how to calculate the following integration? $$ \int_{\mathbb{R}} \left(\exp(z \: e^{-y^2})-1\right)^2 dy=?,\quad z>0. $$ This post is related to my previous question here , ...
4
votes
2answers
391 views

I don't understand behavior of this integral, help!

In an answer to a question I needed the following integral: $$ f(z):=\int\limits_0^\infty t\coth(zt)e^{-t^2}dt; $$ it represented deviation from modularity of some other function. However I noticed ...
1
vote
0answers
64 views

Is there an example where the error of Gauss-Laguerre quadrature does not vanish?

The $n$th Gauss-Laguerre quadrature aims to approximate integral $$\int_{\mathbb{R}_+} f(x) \exp(-x)$$ by the sum $$\sum_{i=1}^n f(x_i) w_i$$ where $x_1,...,x_n$ are the roots of the $n$th Laguerre ...
5
votes
1answer
107 views

Approximations of the identity on Lie groups and homogenous spaces

I'm looking for a nice (and preferably classic or book) reference for the following type of result: Consider a transitive action of a compact Lie group $G$ on a compact manifold $M$ and a continuous ...
1
vote
0answers
109 views

Bound for a certain integral expression

I am working to establish an estimate in $X^{s,b}$ spaces to prove local well-posedness of a certain equation, and I need to consider some sub-cases. In particular, I wish to show that the following ...
2
votes
0answers
174 views

How is the deconvolution of a fat gaussian from a polynomial derived?

We have a 2D order-2 polynomial, a Gaussian and a 'box' indicator function. Let: $\begin{eqnarray} p(x,y) &=& c_0+c_1x+c_2y+c_3xy+c_4x^2+c_5y^2+c_6xy^2 \\ G(x,y) &=& ...
1
vote
2answers
80 views

Numerical calculation of Fourier transform with a nice error bound

I'd like to have an algorithm for a numerical calculation of Fourier transform with a nice error bound. To be precise, if $f$ is a function from $L_1(R)$, $F[f]$ is it's exact Fourier transform and ...
1
vote
1answer
181 views

Request for help with two integrals

It would be great if someone can help me do these integrals - using numerical integration on Mathematica it seems that these converge - in what follows $a \in \mathbb{R}$ and $q \in \mathbb{N}$ and $n ...
3
votes
2answers
232 views

Riesz's representation theorem for non-locally compact spaces

Every version of Riesz's representation theorem (the one expressing linear functionals as integrals) that I have found so far assumes that the underlying topological space is locally-compact. (For ...
3
votes
1answer
296 views

Does there exist a function such that $\int_{\mathbb{R}_+^{\star} } t^nf(t)dt=0$? [closed]

Let $f\in C([a,b],\mathbb{R})$ such that $\displaystyle\int_{a}^{b} t^nf(t)dt=0$ for all integer n. We know that $f\equiv 0$. It's call Hausdorff theorem. This theorem is wrong on $\mathbb{R^+}$, a ...
2
votes
0answers
71 views

The Haar integral on uniform spaces

Does anybody know what the current research status is on the topic of Haar measures on uniform spaces? I'm specifically interested in compact uniform spaces and its corresponding Haar probability. As ...
4
votes
2answers
88 views

Reference for the fact that the images of the narrow and wide Denjoy integrals are respectively $ACG_\ast$ and $ACG$?

The narrow Denjoy integral (which also goes by the names Henstock-Kurzweil integral, Perron integral, and Lusin integral) is a transfinite integration process defined by Denjoy in the early 20th ...
4
votes
0answers
189 views

Inverse of matrix-valued function

Given $c>0$. Let $\gamma_c:{\cal M}_{k \times k}^+\mapsto {\cal M}_{k \times k}^+$ is a function defined by \begin{equation} ...
2
votes
0answers
104 views

How to show the identity $\int_0^T \int_{\Gamma(t)}f(s,t)\;dsdt = \int_S f(\sigma)(1+(\mathbf w \cdot \mathbf n)^2)^{-\frac{1}{2}}\;d\sigma$?

I am reading this paper. Let $\Gamma(t)$ be a smooth closed connected oriented hypersurface for each $t \in [0,T]$. Define the set $$S = \bigcup_{t \in (0,T)}\Gamma(t) \times \{t\}.$$ On page 5 of ...
0
votes
0answers
58 views

What function is “$U_{\nu}(\cdot, \cdot)$”?

I was searching in the Prudnikov (vol. 2) how to solve an integral and I finally found it. However, I didn't recognized a function that appears in the answer. Integral 1.8.2.4: $$ \int_0^x x^{\nu+1} ...
1
vote
1answer
127 views

Characterization of a particular integrable function

Let $f$ be a strictly positive function such that $\int_{0}^{\infty}f(x)dx=\int_{0}^{\infty}xf(x)=1$ (i.e., a probability density function with expectation one). Let also $g$ be a nonnegative ...
2
votes
1answer
238 views

Integral wrt probability measure

Let $\Theta\subseteq\mathbb{R}^d$ is open set and $(\cal X, \cal A)$ be a measurable space . For every $\theta\in\Theta$, suppose that $P_\theta$ is a probability measure on $(\cal X, \cal A)$. ...
0
votes
2answers
133 views

Indefinite integration of multiplication of two Bessel function

I am trying to calculate this integral. I know it has an analytic expression when $a = 0$. But, is there any analytic expression for this case? $$\int_{a}^{\infty}J_2(bx)J_1(cx)\,dx$$ Thanks in ...
4
votes
2answers
209 views

Summary of Lie-Algebra integration tactics

If this is in the scope of MO, I would like to gather here the known tactics of Lie algebra integration, since it appear surprisingly hard to find such a compendium, library or any other kind of ...
0
votes
0answers
147 views

Contour integral (inverse Laplace transform) with arctan

I have what I think is a relatively simple contour integral involving arctan, but it is giving me difficulty. I would really appreciate any help. The integral itself is, with $\tau$, $\lambda$, and ...
52
votes
1answer
2k views

A hard integral identity on MATH.SE

The following identity on MATH.SE $$\int_0^{1}\arctan\left(\frac{\mathrm{arctanh}\ x-\arctan{x}}{\pi+\mathrm{arctanh}\ x-\arctan{x}}\right)\frac{dx}{x}=\frac{\pi}{8}\log\frac{\pi^2}{8}$$ seems to be ...
1
vote
2answers
131 views

How to show this integral on boundary of Lipschitz domain is finite?

Sorry for asking a basic question but this did not get answered on M.SE. Let $\Omega \subset \mathbb{R}^n$ be a Lipschitz domain. How do I show rigorously that $$\int_{\partial\Omega} ...
6
votes
2answers
247 views

$\mathrm{Bessel}^3$ Integral

I'm trying to calculate the following integral: $\int_0^\infty \mathrm{BesselJ}[l_0,k_0r] \cdot \mathrm{BesselJ}[l_1,k_1r] \cdot \mathrm{BesselJ}[l_0-l_1,kr] \cdot r\,dr$ ($\mathrm{BesselJ}[n,x]$ is ...
0
votes
2answers
116 views

Defining surface integral on boundary of $C^1$-domain

Let $\Omega$ be a bounded $C^1$ domain with bounded boundary $\partial\Omega$. Can someone point me to a reference where the surface integral of a measurable function $f\colon \partial\Omega \to ...
1
vote
1answer
319 views

Prove or disprove $ \int_{0}^{\infty} \int_{-x}^{0} f(x)f(y)dydx > \int_{0}^{\infty} \int_{-\infty}^{-x} f(x)f(y)dydx. $

Consider a symmetric, unimodal distribution $f(x)$ such that $\int_{0}^{\infty} f(x) > 1/2$. I want to prove or disprove the following: $$ \int_{0}^{\infty} \int_{-x}^{0} f(x)f(y)dydx > ...
2
votes
2answers
142 views

dense lattices in high dimensions

I want a collection of points $\{ x_1, \dots, x_m\}$ to sample a unit cube $[0,1]^n$ with $n >>1 $ in high dimensions so that summing over these points is approximate the integral over that ...
5
votes
2answers
182 views

Integration on Compact Semirings

I want to know if integration of functions $f:X\rightarrow G$ where $G$ is a compact semiring has been defined and if it is possible to ensure that all continuous functions are integrable. This is ...
0
votes
1answer
82 views

Estimating a quantity from an estimate in its integral

I am reading a paper in which the following argument is made. We have two positive real valued functions $f(x)$ and $g(x)$. We know that $$\int_0^x \int_0^y f(z) \ dz \ dy \leq g(x).$$ It is then ...
1
vote
1answer
212 views

From Lebesgue Integral to Stieltjes Integral, and integration by parts

Let $X$ be a real random variable with c.d.f function $F$. Let $g$ be an increasing measurable real function and assume that $\mathbb{E}\left[g(X)\right]$ exists (and is finite). What additional ...
2
votes
0answers
125 views

How to perform this matrix integral?

Edit: some backgrouds added. In quiver matrix model which is reviewed DV or CKR, the path integral reduce to the matrix integral $$Z \sim \int \prod_{i=1}^r d\Phi_i \prod_{<a,b>} dQ_{ab} ...
1
vote
1answer
75 views

Is sequential completeness of LCS strictly stronger that Riemann integrability of curves?

$\def\RiemInt{\,\text{-}\,\lower.5mm\hbox{$^{^{\rm Riem}}$}\kern-1mm\int}$ This is essentially a reformulation of this MSE question which has not received any answers for about three weeks. To ...
1
vote
0answers
115 views

Are the two functions equal? If not, what relation are they?

We have $$A=\frac{1}{2\pi}\int_{-\infty}^{\infty}\prod_{i=1}^{\infty}\frac{\sin(v2^{-k})}{v2^{-k}}e^{ivx}dx$$ $$B=\frac{1}{2}\sum_{k=1}^{n}\cos (\pi kx)\prod_{i=1}^{m}\frac{\sin(\pi k2^{-i})}{\pi ...
1
vote
2answers
190 views

Positivity of the Coulomb energy in two dimensions

In dimensions $d\geq 3$ the Coulomb energy is always non-negative (since the Fourier transform of $\frac{1}{|\cdot|^{d-2}}$ is non-negative). What can one say about positivity properties of the ...
13
votes
2answers
708 views

Can we simplify $\int_{0}^{\infty}\frac{{\sin}^px}{x^q}dx$?

We know the followings : $$\int_{0}^{\infty}\frac{{\sin}x}{x}dx=\int_{0}^{\infty}\frac{{\sin}^2x}{x^2}dx=\frac{\pi}{2},\int_{0}^{\infty}\frac{{\sin}^3x}{x^3}dx=\frac{3\pi}{8}.$$ Also, we can get ...
3
votes
3answers
194 views

How to extract the divergent part from the singular integral

How to extract the divergent part of the following integral simply as $u \rightarrow \infty$ $$g(u) = \frac{\sqrt{2u}}{\pi} \int^1_{\frac{1}{u}} dz \frac{\sqrt{z-1}}{\sqrt{z^2-u^{-2}}} $$
3
votes
1answer
184 views

Integral of a product of Laguerre polynomials

In order to estimate the non linear term in a particular PDE, I have to decompose $L_k^\alpha(x)^3\cdot x^{-\delta}$ (with $0<\delta<\alpha+1$) into a basis consisting of Laguerre polynomials ...
2
votes
0answers
202 views

An integral inequality

Let $g:\mathbb{R}\rightarrow\mathbb{R}$ be bounded with derivative $g'$. I have shown that the following inequality holds for all $w\in\mathbb{R}$, ...
1
vote
3answers
196 views

limit of a singular integral

Denote $f_{\gamma}(x) =\frac { (1+\gamma)}{2} |x|^{\gamma}$. We consider: $$I(\gamma) = \int_{-1}^1\int_{-1}^1 \ln (|x-y|) f_{\gamma}(x) f_{\gamma}(y) dx dy$$ I would like to know the limit of ...
4
votes
0answers
122 views

About arithmetic-geometric mean

It's well known that if we set $a_0=x \geq 0, \ g_0=y \geq 0$, and $$ a_{n+1}=\dfrac{1}{2}(a_n+g_n), \ g_{n+1}=\sqrt{a_n g_n} ,$$ then both $\{a_n\}$ and $\{g_n\}$ will converge to $AGM(x,y)$. ...
0
votes
3answers
136 views

Definite integral of a function containing an exponential

I have to calculate analytically this integral: $$ {\rm J}\left(q\right) = \int_{0}^{\infty}{{\rm d}x \over x^{q}\left({\rm e}^{kx}-1\right)} $$ where $-1\le q\le N$ with: $N\in\mathbb{N}$ and ...
1
vote
1answer
129 views

Lebesgue's integrability condition in several variables

The well known Lebesgue's condition of Riemann integrability says that a bounded function in one variable $f\colon [a,b] \to \mathbb{R}$ is Riemann integrable if and only if it is continuous almost ...
1
vote
0answers
223 views

exponential integral $e^{x/t^{3}}$ and floor function

Is it possible to give a closed form for the integral $$ \int_{0}^{\infty} \frac{dt}{t^{3}}\rho (t)e^{-\frac{x}{n^{2}t^{2}}}$$ where $ \rho (t) = t- \left\lfloor t\right\rfloor $ is the fractional ...
2
votes
0answers
78 views

Distribute Monte Carlo samples among dimensions

Simplified problem: Given a $d$-times nested convolution of an input function $g(x):\mathbb{R}\mapsto \mathbb{R}$ with the same band-limited smooth function $f(x):\mathbb{R}\mapsto \mathbb{R}$. I am ...