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-2
votes
0answers
107 views

Show that $\int_0^\infty \sin\left(x^2\right)dx$ converges, but that $\int_0^\infty \sqrt{\sin^2\left(x^2\right)}dx$ does not. [on hold]

Show that $\int_0^\infty \sin\left(x^2\right)dx$ converges, but that $\int_0^\infty \sqrt{\sin^2\left(x^2\right)}dx$ does not. The first part I think I proved using triangles, but I could not prove ...
6
votes
1answer
189 views

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 ...
-2
votes
0answers
26 views

Area between 2 curves with absolute value [on hold]

I have the two equations: y = 1 - 2x^2 y = abs(x) I solved the two equations using the absolute value for: abs(x) and ...
-1
votes
1answer
96 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 ...
-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) := ...
0
votes
1answer
97 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
80 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 ...
3
votes
1answer
189 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
1answer
239 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 $(\cal X, \cal A)$. ...
-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
297 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
397 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 ...
5
votes
2answers
183 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 ...
2
votes
3answers
345 views

Defining the integral of a function using the product measure

Imagine that we're trying to define the expression $$\int_U f(x)dx$$ in a rigorous way. Assume that $f:X \rightarrow \mathbb{R}^{\geq 0}$ where $(X,\mu)$ is a measure space, and suppose that $U$ is a ...
1
vote
0answers
65 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 ...
12
votes
5answers
1k views

An algebra of “integrals”

When discussing divergent integrals with people, I got curious about the following: Is there an $\mathbb{R}$-algebra $A$ together with a map (could be defined on just a subspace) $$\int_0^{\infty}: ...
5
votes
1answer
108 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
110 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
175 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) &=& ...
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 ...
14
votes
2answers
437 views

Evaluation of an $n$-dimensional integral

I asked the same question on math.se but got no answer there. Since it pertains to my current research, I decided to ask here: Let $n\in 2\mathbb{N}$ be an even number. I want to evaluate $$I_n := ...
1
vote
1answer
182 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 ...
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 ...
3
votes
2answers
235 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 ...
4
votes
0answers
191 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} ...
11
votes
3answers
772 views

Applications of visual calculus

Mamikon's visual calculus (see Mamikon, Tom Apostol, Wikipedia) is a very beautiful and surprisingly efficient tool. The basis is Mamikon's theorem. The area of a tangent sweep is equal to the ...
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 ...
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 ...
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 ...
7
votes
5answers
705 views

$\int^{\infty}_{0}x^{r +s- 1}(1 + x)^{-s}(1 + x^2)^{-\frac{rm}{2}}dx$

I'm trying to solve the integral $\int^{\infty}_{0}x^{r +s- 1}(1 + x)^{-s}(1 + x^2)^{-\frac{rm}{2}}dx$, where $s$, $r$ and $m$>1 are positive integers. My question is whether a closed form ...
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} ...
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 ...
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 ...
0
votes
2answers
137 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
210 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
82 views

An integral over Clifford torus

This is part of my research project and I hope somebody can help with it. Let $F_{ij}$ be the antisymmetric function $$\frac{1}{4}L^4 \left(\text{cos}\left[\frac{f_{2j}}{2}-\frac{g_{2j}}{2}\right] ...
3
votes
2answers
331 views

Integration in several variables and elementary applications

This fall I'm teaching the "second half" of the standard entry-level undergraduate multivariable calculus course: the focus is on double and triple integrals, path integrals, Green's theorem, Stokes' ...
0
votes
0answers
160 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 ...
3
votes
3answers
196 views

How to extract the divergent part from the singular integral

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

evaluating an integral related to the volume of Hessenberg orthogonal matrices

Consider the following integral, $$ {1 \over 4\pi^{2}}\int_{0}^{2\pi}\int_{0}^{2\pi} \sqrt{\, 9 -\sin^{2}\left(\theta_{1} \over 2\right) \sin^{2}\left(\theta_{2} \over 2\right)\,} \,{\rm ...
6
votes
2answers
700 views

Contour integration problem from probability

Can integrals of the form $$ \int_{-\infty}^{\infty}{\exp\left(-\left[x - c\right]^{2}\right) \over 1 + x^{2}}\, {\rm d}x $$ be computed in closed form using contour integration (or any other ...
1
vote
1answer
238 views

Integrate kˆ(n-1) / prod_{i=1…n} (kˆ(2)+x_iˆ{2}) dk between 0 and infinity, with x_i constants and n>=1? [closed]

[some formatting tweaked, and the question copied from the title to the main body, by YC] Hi, I've been struggling a lot to calculate this integral. $$ \int_0^\infty \frac{k^{n-1}}{\prod_{i=1}^n ...
3
votes
3answers
281 views

Asymptotic behaviour/upper bound for $\int_0^{\infty} \exp(-c x^a+K x^b)dx$ for $a>b>0$ as $K\rightarrow \infty$?

What is the asymptotic behaviour or an upper bound for $\int_0^{\infty} \exp(-c x^a+K x^b) \, dx$, for $a>b>0,$ as $K\rightarrow \infty$? Or any good reference for tools to tackle this question? ...
8
votes
3answers
2k views

Is $x \, \tan(x)$ integrable in elementary functions?

I'm teaching Calculus and my students asked me to calculate the integral of $x \, \tan(x)$. I spent quite a lot of effort to do this, but I'm now even not sure if the integral could be presented in ...
0
votes
1answer
311 views

integrate of functions involving floor

Is there any exact formula or at least exact inequalities for the following intehral $$ \int_{2}^{x}{{\rm d}t \over \left\lfloor\vphantom{\large h}% \log(x)/\log(t)\right\rfloor \log\left(t\right)} ...
1
vote
2answers
79 views

Finding Kuramoto Model coupling strength with limits?

The following is an equation describing the coupled phases of N oscillators according to the Kuramoto model: $$ 1 = K \int_{-\pi/2}^{\pi/2}\cos^{2}\left(\theta\right)\, {\rm ...
2
votes
2answers
400 views

Sum involving binomial coefficients

I have the following sum $\sum_{j=1}^K {K \choose j} (-1)^{j+1}/j$. Now I can write this as the integral $\int_{-1}^0 \frac{(1+x)^K - 1}{x} dx$. However, I wonder whether there is a closed form ...
1
vote
3answers
197 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 ...
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 ...
3
votes
1answer
515 views

Action Integral

In the theory of action-angle variables, you wind up having to solve integrals with a characteristic square-root behavior near the endpoints to express the action in terms of the orbital quantities. ...