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21 votes
0 answers
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A multiple integral

Let us consider the multiple integral $$I_{n}=\int_{-\infty }^{\infty }ds_{1}\int_{-\infty}^{s_{1}}ds_{2}\cdots \int_{-\infty }^{s_{2n-1}}ds_{2n}\;\cos {(s_{1}^{2}-s_{2}^{2})}\;\cdots \cos {(s_{2n-1}...
Zurab Silagadze's user avatar
11 votes
0 answers
137 views

Assymptotics of a Selberg type integral

Does any one know some references/ ideas on how to study the assymptotics as $N$ goes to $\infty$ of the following Selberg type integral $$\int _{\mathbb R^N} e^{-|x|^2}\ \prod_{1\le i<j\le N} \...
Hatem's user avatar
  • 111
9 votes
0 answers
1k views

How complicated can an elementary antiderivative get?

I asked this question on MSE here. I recently learned that there are many very large numbers that have been defined, such as $\operatorname{TREE}(3)$ and many others that are too big to be written ...
pie's user avatar
  • 541
8 votes
0 answers
296 views

Is there a real-analytic approach to evaluate a definite integral (with an elementary integrand) whose value involves Lambert $W$?

I have never seen a real-analytic approach to evaluate integrals of the form below $$\int_a^b\text{elementary function}(x)\,dx=\text{constant non-trivially involving}\,W(\cdot)\tag1$$ The elementary ...
TheSimpliFire's user avatar
6 votes
0 answers
431 views

How to prove these identities for $\log(2)$ based on $_3F_2$ integrals?

In this MO post I have placed 4 Ramanujan-type hypergeometric series found using the LLL algorithm for fast computing of some logarithms. I could prove 3 of them by means of classical methods based on ...
Jorge Zuniga's user avatar
  • 2,836
6 votes
0 answers
2k views

Interchange of integral and infimum

Can anyone please suggest how to justify widely used formula for interchange of integral and infimum: $ \inf_{u(t)\in U}\int_{t_0}^{t_1}g(t,u(t))dt=\int_{t_0}^{t_1}\inf_{u\in U}g(t,u)dt, $ where $ U\...
Shadowman's user avatar
5 votes
0 answers
652 views

Nature of function as $x\rightarrow\infty$

I'm studying the limits and applicability of Abel Plana summation for different test functions (class of functions). In doing so this just pops out and couldn't handle the said integral so asked here (...
TPC's user avatar
  • 790
5 votes
0 answers
266 views

Hadamard lemma without integration

Let $I$ be the ideal of smooth germs vanishing at zero. Let $I^{k+1}$ be the ideal generated by $(k+1)$-fold product of such germs. Write $F_k$ for the ideal of $k$-flat germs at zero. By the product ...
Arrow's user avatar
  • 10.5k
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 ...
Zacky's user avatar
  • 215
4 votes
0 answers
261 views

Is the following integral positive or not?

Let $n$ be a given even positive integer. We have the following integral \begin{eqnarray} &&\int_0^1\cdots\int_0^1\prod\limits_{i=1}^n\prod\limits_{j=1}^n(x_i-y_j)dx_1\cdots dx_ndy_1\cdots ...
user173856's user avatar
  • 1,997
4 votes
0 answers
349 views

Fractional integral inequality (Hardy-Littlewood-Sobolev)

I am investigating the following integral \begin{equation} I^*(x) = \int_{\mathbb{R}} \frac{f(y) \ln |y-x| }{|y - x|^{\mu}} \, dy \end{equation} where $f \in L_p(\mathbb{R})$, $ 1 < p < q <...
Narek Margaryan's user avatar
4 votes
0 answers
672 views

Proofs of the second fundamental theorem of calculus

I am referring to the following version of the theorem, in the setting of the Lebesgue integral. Theorem Let $f: [a,b] \rightarrow \bf R$ be an everywhere differentiable function whose derivative is ...
coudy's user avatar
  • 18.7k
4 votes
0 answers
684 views

A difficult integral which the Risch algorithm shows is not elementary

For reasons which aren't conceptually related to the problem a few of my colleagues and I are in need of finding an expression for the following integral in terms of $a$ and $\delta$: $$\int_{\delta}^...
Samuel Reid's user avatar
  • 1,431
4 votes
0 answers
428 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} \gamma_c(\Omega)=\frac1{\sqrt{(2\pi)^{k}|\Omega|}}\int_{\mathbb{R}^k}\{(-...
Jlamprong's user avatar
  • 133
3 votes
0 answers
95 views

Is it true that p-integrable function can be written as a convolution of an integrable function and p-integrable function?

We know that convolution of an integrable function with an $p$-integrable is an $p$-integrable function. This follows from Young's inequality. My question: Is it true that $L^p(\mathbb{R}^n)\subseteq ...
user531870's user avatar
3 votes
0 answers
94 views

Question on an integral inequality

I am reading van de Vaart and Weller, Weak Convergence and Empirical Processes With Applications to Statistics. And I am stuck in the proof of Theorem 2.6.7 on page 141. For simplicity I restae the ...
newbie's user avatar
  • 53
3 votes
0 answers
646 views

On properties on a certain functional

Consider the following function: $$F(z) = \omega(z)\sin^2\left(\frac{c\Gamma(z)}{z}\right)$$ Here, $\omega(z)$ is a weight we have to construct and $c$ is a constant. The following three conditions ...
bambi's user avatar
  • 375
3 votes
0 answers
238 views

Dominated convergence Theorem

I am struggling to understand the proof in the paper, Learning Temporal Evolution of Spatial Dependence with Generalized Spatiotemporal Gaussian Process Models. Theorem 2.1 in the page 33 uses ...
ChangYong Oh's user avatar
3 votes
0 answers
132 views

A new characterization of Riemann-Integrability

Question : Given two bounded functions $\,f:[a,b]→\mathbb{R}\,$ and $\;θ:(0,b−a]→[0,1]$. Suppose $\,P:a=x_0<x_1<⋯<x_n=b\;$ is a partition of $\,[a,b]$. Let $\,Δx_k=x_k−x_{k−1}\,$ and $\,\...
Mather1's user avatar
  • 31
3 votes
0 answers
160 views

integral with simple approximation. But why?

I have the following integral $$g(x_0) = \int_{-\infty}^{\infty} \frac{1}{(1+x^2)^{3/4}}\frac{1}{(1+(x+x_0)^2)^{3/4}}\exp\left(-\frac{2\pi i}{\lambda}\left[\sqrt{1+x^2}-\sqrt{1+(x+x_0)^2} \right] \...
Karen Schmidt's user avatar
3 votes
0 answers
187 views

An upper bound for a average of a function in $L_{p}([0,1))$

Suppose that $ f $ is $ 1 $-periodic and that $ f \in {L^{p}}([0,1) $, where $ p > 1 $. Let $$ (D_{n})_{n \in \mathbb{N}_{0}} = \left( \left\{ I^{n}_{j},~ 1\leq j \leq 2^{n} \} \right\} \right)_{n ...
Alex's user avatar
  • 103
2 votes
0 answers
101 views

An inequality related to Problem 10210 AMM 1992 No. 3

Problem. Let $A$ be a $N \times N$ real matrix whose $(i,j)$ entry is $a_{ij} \ge 0, \forall i, j$. Let $1$ denote $N\times 1$ all-ones vector. Prove that $$N^2 1^\top A^\top A A^\top 1 \ge (1^\top A ...
River Li's user avatar
  • 1,053
2 votes
0 answers
99 views

Closed form for $\int_0^{+\infty} \ln^p(t) \frac{\sin^q(t)}{t^r}dt$

Do you know if there exists a closed form for the integral : $$I_{p,q,r} = \int_0^{+ \infty} \ln^p(t) \frac{ \sin^q (t)}{t^r} dt$$ where $p$, $q$, $r$ are natural integers such as this integral ...
Azoth's user avatar
  • 69
2 votes
0 answers
946 views

On a deceptively tricky calculus problem

Motivation for this question: If the operators $B_i'$ satisfy an inequality, prove that $B_1'+\dots B_n'$ also satisfies the same inequality Let $A$ be a non-constant operator acting on $C^...
matilda's user avatar
  • 90
2 votes
0 answers
303 views

an upper bound for $L^1$ norm of the mollifier function

The standard mollifier function is defined as follows $$f(x)=\begin{cases} 0 & \text{if } |x| \ge 1\\ \exp \left(-\cfrac{1}{1-x^2}\right) & \text{if } |x|<1.\end{cases}$$ It is well known ...
Johnny T.'s user avatar
  • 3,625
2 votes
0 answers
63 views

Evaluation of a certain area

I asked a version of this question on Math Stack Exchange 6 days ago, but without any responses: The area of a certain region I am interested in evaluating the area of the region defined by $$A_{L_1, ...
Stanley Yao Xiao's user avatar
2 votes
0 answers
252 views

Power series of the modified Bessel function of the second kind

I am looking for a power series representation of $$ \frac{1}{K_{\nu}(x)}, $$ where $K_{\nu}$ denotes the modified Bessel function of the second kind and $\nu>-1/2$ is not an integer. I know that ...
esner1994's user avatar
2 votes
0 answers
136 views

Multiple integral with diagonal constraint (short-range)

I am looking for an upper bound on the following integral: $$\int_{X_{\delta}}\prod_{j\neq i=1}^{n}\left ( \frac{\delta}{\min (\max(\epsilon, |a_i-a_j|),\delta)}\right )^{b} \prod_{i=1}^{n} da_{i},$$ ...
Thomas Kojar's user avatar
  • 5,474
2 votes
0 answers
155 views

Second differential of total variation

I am trying to give meaning to the notion of second differential of total variation. For sufficiently regular $u:\Omega \subset \mathbb{R}^2 \to \mathbb{R}$ let the total variation be given by $$TV(u)=...
Marko Rajkovic's user avatar
2 votes
0 answers
104 views

Coercivity of an integral operator in control theory

Let us consider the integral operator $T:\mathbb{R}^{n\times d}\to [0,\infty)$ such that for all $K\in \mathbb{R}^{n\times d}$, $$ T(K)=\int_0^1 \operatorname{tr}(KK^\top \Sigma_t) \,d t, $$ where $\...
John's user avatar
  • 503
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, ...
saolof's user avatar
  • 1,947
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 ...
David Manolis's user avatar
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
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 ...
Kacper Kurowski's user avatar
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. $...
dohmatob's user avatar
  • 6,853
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 $s,p_i\in\mathbb{N}^n$ ...
Houa's user avatar
  • 561
2 votes
0 answers
84 views

A problem of uniqueness

Let $f\in\mathcal{S}(\mathbb{R}^n)$, $a\in(-1,1)$, how i can prove that the following problem: $$\text{div}(t^a\nabla u)=0,\quad\text{in }\mathbb{R}^n\times(0,\infty),$$ $$ u(x,0)=f(x),\quad\forall x\...
inoc's user avatar
  • 339
2 votes
0 answers
115 views

Showing that for measurable $\Omega \subseteq \mathbb{R}^n$, $L^1(\Omega; C_0(\mathbb{R}^n))$ is separable

Here we're integrating "Banach-valued" functions $u: \Omega \rightarrow C_0(\mathbb{R}^n))$ , and by $u \in L^1(\Omega; C_0(\mathbb{R}^n))$ I mean that $$\int_{x \in \Omega} \| u(x) \|_{\...
brighton's user avatar
2 votes
0 answers
84 views

How to define Lebesgue Integrability of functions assuming values in an arbitrary topological vector space over an arbitrary topological field?

I have already asked this question in this MSE thread, but some people suggested me to ask to the MO community also. Preliminaries An algebra of sets in a set $X$ is an $\mathcal{X}\subseteq\mathcal{P}...
Daniel Kawai's user avatar
2 votes
0 answers
200 views

The collection of mean value abscissas in the Mean value theorem

The integral mean value theorem for continuous f on [0,b] and finite positive continuous measure $\mu$ we have $$\frac{1}{\mu[a,b]}\int_{a}^{b}f(x)d\mu(x)=f(c)(*)$$ for at least one $c\in [a,b]$. We ...
Thomas Kojar's user avatar
  • 5,474
2 votes
0 answers
249 views

Calculating $\int_1^{\infty}\frac{\operatorname{ali}(x)}{x^3}dx$, where $\operatorname{ali}(x)$ is the inverse function of the logarithmic integral

It is well-known that we can compute the closed-form of the integrals $$\int_1^{\infty}\frac{\log x}{x^2}dx$$ and $$\int_1^{\infty}\frac{\operatorname{li} (x)}{x^3}dx,$$ where $\operatorname{li} (x)$ ...
user142929's user avatar
2 votes
0 answers
197 views

Orthogonality relation in $L^2$ implying periodicity

Let $\theta(t)$ and $\phi(t)$ be two real $C^1$ functions $[0,2\pi]\rightarrow \mathbb{R}$. Let us assume $\theta$ has the properties $$ \int_0^{2\pi} e^{i\theta(t)} dt=0. $$ Geometrically this means ...
Leonardo's user avatar
  • 405
2 votes
0 answers
114 views

Is there an explicit version of Morse Lemma used in stationary phase method?

In the proof of the stationary phase method (at least the one I have seen) Morse lemma shows up, which states: Let $g:\mathbb R^n\to \mathbb R$ be a function of class $C^\infty$ for which $0$ is a ...
Johnny T.'s user avatar
  • 3,625
2 votes
0 answers
60 views

Finding a function in contour integration involving Riemann mapping

Let $T$ be a rectifiable Jordan curve in $\mathbb{C},$ $G$ be the interior of $T,$ and $\Phi$ be a conformal map of the unit disk $\mathbb{D}$ onto $G.$ Let $\mathcal{P}_{n}$ be the space of algebraic ...
Suracha Bosunoi's user avatar
2 votes
0 answers
571 views

Integrating a product of integrals involving Bessel functions

I have asked similar questions on Math Stack Exchange, but not been able to receive many helpful responses. Therefore, I am posting this problem here, and any input would be extremely valuable. I ...
user363087's user avatar
2 votes
0 answers
254 views

Prove this function is increasing

I'm stuck in showing that the following function is increasing over the domain $\left[0,\hat{b}\right]$: \begin{eqnarray} \Pi\left(z\right) & = & \int_{0}^{\phi\left(z\right)}\int_{x}^{\bar{x}...
Emmanuel's user avatar
2 votes
0 answers
79 views

Compute Mixed Volume with Respect to Some Regular Sets

Let $( \mathbb{R}^n, \mathcal{B}, \gamma)$ be a measure space where $\mathcal{B}$ is the Borel sigma algebra and $\gamma$ is a continuous measure. For $A, B\in \mathcal{B}$ that are convex, the mixed ...
Steve's user avatar
  • 1,127
2 votes
0 answers
259 views

How to analytically evaluate this n-dimensional iterated integral?

I would very much appreciate any suggestions and/or pointers to references relevant for the analytic evaluation of the following n-dimensional iterated integral $$\int_{-\infty}^{+\infty}dx_1\int_{-\...
Andrea Becker's user avatar
2 votes
0 answers
181 views

Is the implication ($f$ is Riemann integrable over $D_1$ and $D_2$) $\Rightarrow $ ($f$ is Riemann integrable over $D=D_1\cup D_2$) true?

Let $D_1,D_2$ be a bounded subset of $\mathbb{R}^n$ and $\partial D_1,\partial D_2$ are both of Lebesgue measure zero (that is to say: $D_1,D_2$ are Jordan measurable). Also, let $f:D_1\cup D_2=D\...
Elliot's user avatar
  • 121
1 vote
0 answers
146 views

integral over the unit sphere of $\Bbb C^n$

Please, is there a way to calculate this integral $$\int_{S_{2n-1}} \frac{e^{a \langle z, \zeta \rangle}}{|z - \zeta|^{\beta}} \, d\sigma(\zeta)$$ where $ z $ is a fixed point in the complex unit ball ...
zoran  Vicovic's user avatar