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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2
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1answer
50 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
votes
1answer
206 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(...
-3
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0answers
28 views

Prove the following path integral in the complex plane [closed]

enter image description here The image contains an equation of a path integral.
-3
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0answers
107 views

Integration modulo integers

$f(x,\theta)=\frac{g(x,\theta)}{h(x,\theta)}$ be a function parametrized by $x\in\mathbb N$ such that its integral with $\theta\in[a,b]$ for fixed $a,b\in\mathbb R$ is always an integer. The ...
4
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0answers
144 views
+50

Batalin-Vilkovisky integral is invariant under infinitesimal deformation

This is from page 90 and 93 of Mnev's paper BV formalism and applications. Let $\mathcal L_{t} \subset \Pi T^{*}M$ be a smooth family of Lagrangians with $t \in [0,1]$ a parameter, s.t. $\...
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0answers
35 views

Find the volume of the solid obtained by rotating the region [closed]

Let R be the region in the first quadrant bounded by the curves y = f(x) = 2x+ 1 and y = g(x) = 2x^2 − 8x + 9 Find the volume of the solid obtained by rotating the region R about y-axis using two dy-...
0
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1answer
76 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}\...
0
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1answer
45 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^...
-2
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0answers
43 views

Range of i.i.d random variables [closed]

The density funtion of the range of n identically distributed random variables taking values in $(a,b)$ with the common cumulative distribution function $F(x)$ and pdf $f(x)$ is known to be given by $$...
0
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0answers
37 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....
22
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1answer
387 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
63 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
votes
1answer
102 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)^\...
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0answers
262 views

Analyzing my definition of Average which uses a variation of the Lebesgue Integral and Measure [closed]

Consider $f:A\to[a,b]$ where $A\subseteq[a,b]$ and $S\subseteq A$. As noted in previous questions, I want to define an average using a new measure and integral since I found certain aspects of the ...
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0answers
38 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 $C_1$), \...
1
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0answers
119 views

A question about rationality, irrationality or transcendence of definite integral [closed]

Forgive me for the following fundamental question. But I think I require the accuracy of an expert. Consider an integral of the form: $$\int_a^b f(x)dx,$$ where $f(x)$ is analytic and real valued for ...
3
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0answers
53 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 ...
-5
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2answers
418 views

Coming up with a equivalent (or close) definition for an average which is easier to compute? [closed]

Continuing from my last question, I understand that my definition is unclear so I have modified it. Since no one has answered my question on math stack exchange, I decided to ask here. Definition ...
2
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0answers
31 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 ...
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1answer
422 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
108 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
40 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
votes
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 ...
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0answers
50 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 ...
4
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0answers
73 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
votes
1answer
98 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
75 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
104 views

How do I evaluate the following double integral?

I would like to evaluate the following double integral: $$ \int_{-1}^1d\zeta\int_{-1}^1 d\bar{\zeta} (\zeta+\bar{\zeta})^{d-2}[(1+\zeta\bar{\zeta})(\zeta-\bar{\zeta})]^J \,\times [(1-\zeta)(1+\bar{\...
2
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0answers
26 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}\} ...
6
votes
2answers
301 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
50 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
99 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
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1answer
78 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
84 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
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2answers
113 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
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0answers
44 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
30 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
541 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
59 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
187 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 ...
6
votes
1answer
79 views

Stationary phase in spherical integral

I'm trying to find asymptotics for an oscillatory integral on $\mathbb{S}^{n-1}$, for which my advisor said I should use stationary phase arguments. The particular, he claims that: If $\lambda\gg 1$...
0
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0answers
73 views

Existence of the inverse Fourier transform, Carr Madan

I have a function $C_T(k)$ that is not $L_1$, because its limit in negative infinity is a constant. So I dampened it by $ e^{\alpha k} $. Let's call the transformed function (of the dampened function) ...
3
votes
0answers
51 views

Standard definition: vector-valued essential support

Let $f \in L^p(\mathbb{R}^n,\mathbb{R}^m)$. If $m=1$ then the essential support of $f$ is a mainstream definition; see here for example. However, when $m>1$ is the following definition used? $$ \...
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0answers
75 views

Closed form for a double integral over the first quadrant of the $L^p$ disk

Is there, by any chance, a closed form for the following integral $$ I_p=\iint_{Q_p}(x+y)\log(x+y)dxdy, $$ where $Q_p=\{(x,y)\in\mathbb{R}^2, x>0,y>0,x^p+y^p\leq1\}$, $0<p\leq\infty$ ? ...
0
votes
0answers
35 views

Weak formulation of PDE with weighted inner product

In Boyd's book on spectral methods (available here: https://depts.washington.edu/ph506/Boyd.pdf), I stumbled in section 3.5 "Weak & Strong Forms of Differential Equations: the Use-fulness of ...
2
votes
0answers
159 views

Asymptotic of a functional as $x\rightarrow \infty$

Consider the following functional : $$ I(x,s) =\int_0^\infty\mathrm dy \frac{F(x + \mathrm iy, s) − F(x −\mathrm iy, s)}{\mathrm e^{2πy}-1}, $$ where $ F(z, s) = \dfrac{\sinh(\sin^2[π\Gamma(z)/(2z)])...
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0answers
82 views

A conjecture on integrals of infinite products

The problem I would like to discuss in this post is about a conjecture on the following integrals, \begin{align} \int_0^\infty \prod_{n=1}^\infty \cos(x/n)\,dx \stackrel{?}= \pi/4 \tag{1}\\ \int_0^\...
1
vote
2answers
136 views

Connection between Volkenborn integral and Haar measure on $\mathbb{Q}_p$

This may be a rather elementary question, but I haven't been able to figure it out on my own, and the literature appears to be eerily silent on the topic. Since $\mathbb{Q}_p$ is a locally compact ...
-1
votes
1answer
97 views

Inequality involving Gaussian integral [closed]

I'm looking to prove the following inequality: $$ \left| \int_0^1 e^{-x^2} \sin(x) \, dx \right| \leq \frac{1}{2} \left(1- \frac{1}{e}\right) $$ So far I have no idea on how to prove it. Anybody?
1
vote
0answers
75 views

L2 norm of the diagonal entries of a random rotation of a fixed matrix?

Let $X\in\mathbb{R}^{d\times d}$ be the diagonal matrix with $d/2$ entries equal to $1$ and $d/2$ entries equal to $-1$. Let $F_U \triangleq \frac{1}{d}\|\operatorname{diag}(U^{\dagger}XU)\|^2_F$ ...

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