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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1answer
216 views

Computing the integral $\int_{-1}^1 dx \, |x| J_0(\alpha \sqrt{1 - x^2}) P_\ell(x)$

I would like to compute the following integral: $$ I_\ell(\alpha) := \int_{-1}^1 dx \, |x| J_0(\alpha \sqrt{1 - x^2}) P_\ell(x) \tag{1} \label{1} $$ where $\alpha \geq 0$, $J_0$ is the zeroth-order ...
2
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0answers
39 views

Removing integral from norm by inequality

My first question on Math Overflow. For my Mathematics Bachelor thesis I am looking at a paper called "Deep Limits of Residual Neural networks" by Matthew Thorpe and Yves van Gennip. (arxiv....
6
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2answers
224 views

An abstract characterization of line integrals

Let $M$ be a smooth manifold (endowed with a Riemann structure, if useful). If $\omega \in \Omega^1 (M)$ is a smooth $1$-form and $c : [0,1] \to M$ is a smooth curve, one defines the line integral of $...
-2
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1answer
190 views

Is it possible to express $\int_{0+\epsilon}^{1-\epsilon}\left(\sqrt{1-x^2}^{\sqrt{1-x^2}^{\cdots}}\right) dx$ in elementary functions?

let $\epsilon >0$, I tried to evaluate $\int_{0+\epsilon}^{1-\epsilon}\left(\sqrt{1-x^2}^{\sqrt{1-x^2}^{\cdots}}\right) dx$ , using the fact $x= \cos t$ yield to have integrand using $\sin $ ...
5
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2answers
418 views

Compute $ \int_{0}^{+\infty} \left( \frac{\ln(x)}{e^x}\right)^2 dx $ [closed]

How can I compute this integral? $$ \int_{0}^{+\infty} \left( \frac{\ln(x)}{e^x}\right)^2 dx $$
-1
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1answer
199 views

What is the integral of $r \frac{2^{r-1} \log (2) e^{-\frac{\sqrt{2^r-1}}{b}} \left(2^r-1\right)^{\frac{d}{2}-1}}{b^d \Gamma (d)}$?

I have been trying to solve a research problem for a while now and in doing so, I stumbled upon the following integral: $$\int_0^{\infty } r \frac{2^{r-1} \log (2) e^{-\frac{\sqrt{2^r-1}}{b}} \left(2^...
0
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1answer
47 views

Looking for a family of random variables such that only the second clause is fulfilled [closed]

Working with the epsilon-delta-criterium, a family $(X_i)_{i \in I}$ on $(\Omega,A,P)$ is uniformly integrable if i) $sup_{i \in I} E(X_i) <\infty$ ii) $\forall \epsilon>0$ ex. $\delta>0$ s.t....
1
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0answers
85 views

A Fredholm equation with non-separable kernel [closed]

I'm trying to solve this form of Fredholm equation: $$ g(v)=f_1(v)+\int\limits_{0}^{v_\mathrm{th}} g(v_s)\frac{e^{-\tfrac{\big[(v-v_\mathrm{init})-(1-v_\mathrm{leak})(v_s-v_\mathrm{init})\big]^2}{2v_\...
5
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1answer
216 views

Three integral expressions for integer values of $\zeta(s)$. Could these be further reduced to known integrals?

In this MSE-question I've asked about three, similarly shaped, integrals for integer vales of $\zeta(s)$ that I found numerically: $$\zeta \left( 3 \right) =\frac12{\int_{0}^{1} \frac{1}{x}\big(\zeta(...
4
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2answers
304 views

From Zurab's integral representation for the Apéry's constant to almost impossible integrals

I would like to know if the following integrals are known, or in case that aren't in the literature we can calculate these in closed-form (in terms of elementary and standard functions). I wondered ...
0
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0answers
92 views

Solving integral equation with an unknown probability distribution

Considering this system of integral equations, where $\gamma \in \mathbb{R} $ and $\alpha\in \mathbb{C}$ are the unknown to solve : $$ 1=\int_{-\infty}^{\infty} p(u) \frac{ -1}{\gamma-\left(u-z^{*}+\...
7
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0answers
147 views

A function with unexpectedly simple Legendre transformation

Let $I(x) = \frac{1}{2\pi} \int_{-2}^2 \sqrt{4-y^2}\ln|x-y|dy$. Then $I(x)$ is a concave function and \begin{equation} I(x)= \begin{cases} \frac{1}{4}x^2-\frac{1}{2}, &\text{if } |x|\leq2 \\ \...
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0answers
53 views

Differentiable dependence on the initial condition of the solution of a SDE

Let $b,\sigma:\mathbb R\to\mathbb R$ be differentiable and Lipschitz continuous $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge0}$ be a complete and right-...
4
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1answer
292 views

Two definitions of $L^p$ spaces that are not always equivalent

There are two definitions of $L^p(S, \Sigma,\mu)$ in the literature. (Here $S$ is a set, $\Sigma$ is a $\sigma$-algebra of subsets of $S$ and $\mu$ is a positive measure.) The two definitions are ...
2
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0answers
67 views

Constructing a small Radon-Nikodym derivative

Let $u:\mathbb{R}^n\to\mathbb{R}^n$ be a $C^1$ function. Is it possible to (explicitly construction) a function $h$ such that: $0<h(x)$. $\int_{x \in \mathbb{R}^n} |h(x)|<\infty$, $\sup_{x \in ...
0
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0answers
39 views

Bounding the absolute value of a complex integral with itself

I already asked a similar question on this topic, but after a small discussion, I noted that I did must boil down the problem such that the solution space so to say to maybe have a concrete answer. I ...
0
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0answers
54 views

Bounding the absolute value of a complex integral

I'm working on some problems involving Fourier transforms and convolution problems and there is one problem I cannot solve. In my situation we have a complex valued function $f(ix)$, with $x\in\mathbb{...
0
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1answer
54 views

Variant of modified Bessel functions

Consider the integral \begin{align*} g_f(x)=\int_{\phi=0}^{2\pi} f(\phi) ~e^{x cos(\phi)}~\mathrm{d}\phi, \end{align*} where $f(\phi)$ is a probability density functions defined over $[0,2\pi]$, \...
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0answers
43 views

Oscillatory integral independent of a parameter

Let $h: \mathbb{R} \mapsto \mathbb{C}$ be a positive definite function, continuous at the origin. (In fact, $h$ is the Fourier transform of a finite measure). Define the oscillatory integral $$Q(t) :=...
2
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1answer
61 views

$ \int_{E}^{*}{\psi (t) d\mu(t)}=\int_{E}{\phi (t) d\mu(t)} $

Let $(T, \mathcal{A}, \mu)$ be an arbitrary measure space. The outer integral over $(T, \mathcal{A}, \mu)$ of a (possibly nonmeasurable) function $\psi: T\to (-\infty, +\infty]$ is defined by: $$ \...
3
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1answer
222 views

Integral convergence implies pointwise

This is a simplified version of a question I am looking at, but embarrassingly I can't do this one anyway. Let's assume $f(x)$ is a decreasing positive function, $f(0)$ is infinite and, moreover, $f(...
1
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1answer
107 views

Asymptotic development of Integral of $e^xx^r$

Let $\alpha \in (0,1)$ and $\delta \in (0,1/2)$ be fixed, and consider the following integrals for each integer $j \geq 0$: $$I_j(u):= \frac{e^u}{u^{j+\alpha}} \int_{-u\delta}^0 e^t t^{j-1+\alpha}\...
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1answer
64 views

Limit of the convolution of derivative of Gaussian heat kernel

I'm looking for the following limit: $$\lim_{\varepsilon\to 0^+}\int_{-\sqrt{\varepsilon}}^{\sqrt{\varepsilon}}\frac{1}{\sqrt{2\pi}\varepsilon^{3/2}}\left(-1+\frac{x^2}{\varepsilon}\right)e^{-\frac{x^...
0
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0answers
53 views

Integration of exponential of square root of quadratic polynomial

I know that $\int_0^\infty \exp(-\sqrt{x^2 + b^2}) \mathrm{d}x = b K_1(b)$, where $K_1$ denotes the modified Bessel function of second kind and $\mathrm{Re } b > 0$ (Gradshteyn-Ryzhik 7th ed., 3....
23
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1answer
418 views

A characterization of constant functions

In How to recognize constant functions. Connections with Sobolev spaces (Russian Math Surveys 57 (2002); MSN), H. Brezis recalls the following fact: Let $\Omega\subset{\mathbb R}^N$ be connected ...
0
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0answers
74 views

How to integrate an exponential function of a rational function?

Can anyone help me to calculate the following integral? \begin{align} \int\limits_0^t {{{(x - t)}^2}} x\,{e^{ - \left(x + \frac{a}{{bx + 1}}\right)}}\mathrm{d}x \end{align} where $a$ and $b$ are ...
2
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1answer
115 views

An interpolation inequality

I am interested in the following statement. Let $q>p$. Then there are positive numbers $\alpha$ and $\beta$ so that for all $f\in C^1(\mathbb{R}^n)$, one has $$ \left(\int|\nabla f|^p dx\right)^\...
0
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0answers
41 views

Interpret certain expressions in terms of classical quadratic surfaces

I have two primary constraints, a linear one, \begin{equation} \label{C1} C_1=Q_1>0\land Q_2>0\land Q_3>0\land Q_1+3 Q_2+2 Q_3<1, \end{equation} and a quadratic one (incorporating $C_1$), \...
3
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0answers
56 views

References on integration on non-compact manifolds

I am looking for references on integration on non-compact Riemannian manifolds, specially on the change of variables theorem. In particular I have non-compact manifold $M$ and I have an integral (in ...
2
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0answers
33 views

First moments of uniform distribution on a curve from (0,0) to (1,1) in two-space

The curve $\Gamma$ in $\mathbb{R}^2$ is defined by a continuous and monotonically increasing function $f(x)\in\text{C}[0,1]$, where $f(0)=0$, $f(1)=1$. Let $(X,Y)$ is jointly and uniformly ...
-2
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1answer
491 views

Prove or disprove this integral of a function, defined on a countable set with infinite limit points, converges to zero [closed]

Edit: I got rid of my old definitions. Everything should be clear now Since no one has answered my question on MSE, I’m hoping to get an answer here. I apologize if you dislike my writing. I am an ...
6
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1answer
130 views

When is the Radon-Nikodym derivative locally essentially bounded

Let $\mu\lll\nu$ be $\sigma$-finite Borel measures, which are not finite, on a topological space $X$. Under what conditions is $0<\operatorname{ess-supp}(\frac{d\mu}{d\nu}I_K)<\infty$ for every ...
0
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0answers
47 views

Integral of an expression including a fraction having modified Bessel functions of the first kind on both numerator and denominator

I am looking for an analytic result of the following integral $$\iint_0^\infty {{\rm{d}}x{\rm{d}}y{x^2}{y^2}\exp \left\{ { - {\alpha _1}{x^2} - {\alpha _2}{y^2}} \right\}\frac{{{\rm{I}}_1^2\left[ {\...
2
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0answers
61 views

Comparison of inductive limit topology with very-rapidly decaying non-convex $L^{!/2}$-space topology

This is related to these posts and here. Let $L^1([n,n+1])$ denote the subspace of $L^p$-functions on $[0,\infty)$ essentially supported on $[-n,n]$. Denote the accelerated $\ell^1$-direct sum ...
2
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1answer
128 views

Integral with 4 Bessel functions and an exponential

I would like to solve the following integral $$ \int_0^\infty e^{-a k^2} J_{3/2}(b k) J_{3/2}(c k) J_{3/2}(f k) J_{1/2}(r k) k^{-3} dk, $$ where $a,b,c,f,r > 0$, and $J_\nu(x)$ is the Bessel ...
6
votes
1answer
189 views

Is there a known closed form expression for this integral?

I am interested in the following integral: $$f(x,y) = \int_{\mathbb{S}^d} \max(0,x^Tw)\cdot\max(0,y^Tw) \, dw, \qquad x,y\in\mathbb{S}^d,$$ where $\mathbb{S}^d\subset\mathbb{R}^{d+1}$ is the $d$-...
3
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1answer
108 views

Euler–Maclaurin formula in $\mathbb{Z}^d$

I was wondering whether there is a Euler–Maclaurin formula of sorts for expressions such as $$ \sum_{x \in [a,b]^d\cap \mathbb{Z}^d} f(x) - \int_{[a,b]^d}f(x) $$ where $d\ge 2$ is an integer, $a,b \...
0
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1answer
77 views

Integral rising from difference of chi-squared random variables

Let $X,Y$ be independent random variables such that $X\sim\chi_{n-1}^{2}, Y\sim\chi_{1}^{2}$ are chi-squared distributed (where $n\geq2$ is a natural number). I am trying to evaluate $\mathbb{P}[X\leq ...
2
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0answers
27 views

An exponential integral over a closed convex polytope

For any $T\geq 2$, let us define the polyhedron $S$ given by \begin{align*} S:=\{\underline{t}:=(t_0,t_1,t_{2},t_{3},t_{4},t_{5},t_{6},t_{7})\in [0,+\infty)^{8}:A\underline{t}\leq (\log T)\textbf{1}\} ...
7
votes
2answers
366 views

Why is $-\int_{-\infty}^\infty \log\left[1+2f'(x)(1-\cos\phi)\right]\,dx$ equal to $\phi^2$?

I came across this integral involving the derivative $f'(x)$ of the Fermi function $f(x)=(1+e^x)^{-1}$: $$I(\phi)=-\int_{-\infty}^\infty \log\left[1+2f'(x)(1-\cos\phi)\right]\,dx.$$ I'm pretty certain ...
2
votes
1answer
78 views

fractional moments of multivariate normal distributions

is there an analytic formula for fractional moments of multivariate normal distribution? $E(\prod_{i=1}^k x_i^{\nu_i})=?$ where $X=(x_1,\ldots,x_k) \sim N_k(\mu, \Sigma)$, $\nu_i\in \mathcal{R}$ and $\...
0
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0answers
103 views

Finding a square integrable dominating function for function class

problem statement For any $x \in R^d$, consider the function $$\phi(x,a) = \min_{1\leq j \leq k} \|x-a_j\|^2,$$ where $a = (a_1, \dots, a_k) \in R^{kd}$ and $\| \cdot \|$ is the $L_p$ norm for any $p ...
0
votes
1answer
82 views

$ \{X_n\mathbb{1}_{X_n\in[-n,n]}\}$ is uniformly integrable

Let $(\Omega,\mathcal{A},\mathbb{P})$ be a probability space. Suppose $\{X_n\}$ is a sequence of random variables satisfying : $$ \sup_{n}{\mathbb{E}(|X_n|)} <\infty $$ Suppose that $$ \dfrac{M_j}{...
2
votes
1answer
89 views

Duality form of $L^q$ norm, without assumption that $\int fg$ defined?

The following theorem is found, for example, in the Real Analysis books by Folland, by Yeh, and (in a slightly different form) by Royden. Theorem. Let $(X,\mathcal{A},\mu)$ be a measure space. Let ...
3
votes
2answers
134 views

Meinardus theorem at use: problems with conditions

I am working on an enumerative problem related to knot theory, and I have found the following generating function $$F(z)=\prod_{n\geq 1} \frac{1}{(1-z^{2n+1})^2}.$$ I am interested on getting ...
1
vote
1answer
156 views

A Bessel-like integral

I encounter the following integral when trying to find the inverse Fourier transform of the characteristic function of a certain sum of random variables. Here, $0\le\lambda\le1$, $p\ge0$, $q\ge0$ are ...
1
vote
0answers
31 views

How do I find the portion of a cell/voxel lying within a defined surface?

We have a 3-dimensional grid of voxels (or cells), with individual voxels being of volume $dx\,dy\,dz$ where $dx=dy=dz=1$. A cone-like surface is defined by some function, $z = f(x, y)$, which in ...
7
votes
1answer
561 views

Newman's proof of the prime number theorem

I am teaching a graduate course in Complex Analysis and I am covering Newman's proof of the prime number theorem. I have been using the simplified version in the papers of Zagier and Korevaar. However,...
1
vote
1answer
61 views

Integral average near a point of dispersion

Let $\Omega\subset\subset\mathbb R^{n}$ be a bounded domain and let $E\subset \Omega$ be a Lebesgue measurable set. Let $f\in L^{1}(\Omega)$ and let $x\in \Omega$ be a point of dispersion of $E$, that ...
0
votes
1answer
195 views

What can this $\int_{0}^{t} (\pi(x)-Li(x)) dx$ tell us about primes distribution?

Many papers I have read which are related to primes distribution only it discussed sign and refinement Bounds of $\pi(x)-Li (x)$ with $\pi(x)$ is a prime counting function and $Li (x)$ is the ...

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