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, ...

Filter by
Sorted by
Tagged with
0
votes
0answers
15 views

Polya-Laguerre Class and inversion formula

Is it possible to write the following statement: If (i) $E(s)=\frac{1}{\int_{0}^{+\infty}K(x) x^{s-1} dx}$ (ii)$E(s) \in E_{0}$ (i.e $E(s)= e^{bs} \prod_{k=1}^{\infty}(1-\frac{s}{a_{k}}) e^{s/a_{k}} $...
0
votes
1answer
50 views

Interchange of integration and supremum

Let $u \in C^0(-T,T; L^2(B_R))$ be a measurable function, then is the following true? $$ \int_0^R \sup_{-T<t<T} \int_{S_r} |u(\sigma ,t)|^2 \ d \sigma \ dr = \sup_{-T<t<T}\int_0^R \int_{...
2
votes
0answers
153 views

$\int\limits_{-\infty}^\infty \left(f(\frac{x-\mu}{1+\Psi/2})-f(\frac{x+\mu}{1-\Psi/2})\right)\frac{x\gamma}{(x-x_{0})^{2}+a}dx$

EDIT: I realized from numerical implementation that the step from \begin{align} \mathcal{I}_2=&\frac{\gamma}{2}\int\limits_{-\infty}^\infty \left(f_T(\frac{x-\mu}{1+\Psi/2})-f_T(\frac{x+\mu}{1-\...
4
votes
1answer
79 views

Estimate of $\frac{\int x^{2p}\,e^{-x^{2n}\,+\,\omega(x,y)}\;dx}{\int e^{-x^{2n}\,+\,\omega(x,y)}\;dx}$

For every $x,y\in\mathbb R$ let $$ V(x,y) \,\equiv\, a\,x^{2n} + b\,y^{2m} - \omega(x,y)\,$$ where $a,b>0$, $n,m\in\mathbb N$, $n\geq m\geq1$, and $\omega$ is such that $\omega(x,y)/(x^{2n}+y^{2m})...
2
votes
0answers
95 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 ...
0
votes
1answer
138 views

Ratios of Gaussian integrals with a positive semidefinite matrix

Cross-post from MSE https://math.stackexchange.com/questions/4118128/ratios-of-gaussian-integrals-with-a-positive-semidefinite-matrix Generally speaking, I’m wondering what the usual identities for ...
3
votes
1answer
74 views

On integral representation of Whittaker $W$ functions

According to NIST, the integral representation of Whittaker $W$ functions $$ W_{\kappa,\mu}\left(z\right)=\frac{z^{\mu+\frac{1}{2}}2^{-2\mu}}{\Gamma\left(% \frac{1}{2}+\mu-\kappa\right)}\int_{1}^{\...
0
votes
2answers
106 views

On integral relating logarithmic of absolute value of Zeta function:

Sorry for such a direct question: Consider the following integral: $$I(t)=\int_{1/2}^{1} {\log|\zeta(a+it)|}da$$ How to find the nature of $I(t)$ as $t\rightarrow\infty$?
2
votes
0answers
31 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. $...
1
vote
2answers
44 views

On the Bochner spaces $L^\infty(a,b;L^p(c,d))$ and $L^p(c,d;L^\infty(a,b))$, or: Interchange of supremum and integral

I am asking whether the Bochner spaces $L^\infty(a,b;L^p(c,d))$ and $L^p(c,d;L^\infty(a,b))$ are the same. Or, whether one is included/embedded in the other. We have the norms $$\|u\|_{L^\infty L^p}=\...
0
votes
1answer
55 views

Sufficient conditions for finite mean of a non-negative random variable

Consider a continuous random variable that takes only non-negative values. Let the cumulative distribution function be $F(\cdot)$. Consider the following condition: $$\lim_{x\rightarrow\infty} x(1-F(x)...
0
votes
1answer
40 views

Is integration against an indicator Wasserstein-Continuous

Let $\mathcal{P}_p(X)$ denote the Wasserstein space over a compact metric space $X$, and $1\leq p<\infty$. Fix a non-empty closed subset $C\subseteq X$. Then is the map: $$ \mathbb{P} \mapsto \...
1
vote
0answers
51 views

Riemann-Stieltjes integral of a distribution function

I recently learned the basics of Riemann-Stieltjes integral, and based on the sources I found, we can define the expectation of random variables quite naturally with the R-S integrals: if $X$ is a ...
1
vote
1answer
69 views

Decide the order of of an integration involving the $\log$ function

Let $$A_n=\int_{n^{-\frac{1}{2}}}^{1}\frac{\log(nx)}{nx(\log\log(nx)-\log\log(1+x))}dx.$$ I want to discribe the order of $A_n$, by geting a progressive formula or a good lower bound for it. The order ...
2
votes
1answer
83 views

Generalized Selberg integral

I am interested in expressing the following generalization of the Selberg integral in terms of Gamma functions $$ \int_0^1 \ldots \int_0^1 \prod_{i=1}^d u_i^{\frac{k_i-1}{2}} \prod_{m=1}^d (1-u_m)^{\...
2
votes
0answers
114 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 $s,p_i\in\mathbb{N}^n$ ...
0
votes
0answers
12 views

Which units describe the area under a curve? [migrated]

A car traveling in a straight line from its starting position at a speed of 65 miles per hour for 3 hours is represented by the function v(t) where t is the number of hours traveled. What units ...
0
votes
0answers
68 views

The loss of double periodicity (ellipticity)

Consider a meromorphic function $f(\mathfrak{a}_1, \mathfrak{a}_2)$, such that $$ \begin{align} f(\mathfrak{a}_1, \mathfrak{a}_2) = f (\mathfrak{a}_1 + 1, \mathfrak{a}_2) = f(\mathfrak{a}_1 + \tau, \...
3
votes
4answers
215 views

Integrals involving fractions of exponentials

I am trying to calculate the average degree of a complex network, which requires me to solve for the following integral: $$\int \mathrm{d} x \frac{\exp{\left[-x -\frac{1}{2}\left(\frac{x-\mu}{\sigma}\...
1
vote
0answers
57 views

'Partial Integral' Equation

Given the following 'partial integral' equation of a measurable function $u$: $$\int_\ell u(x,y) \, d\mathcal{H}^1=F(\bar{\ell})$$ where $\ell = \{(x,y)\mid x \cos \theta + y \sin \theta = p,p \ge 0\}$...
1
vote
1answer
138 views

Faster than Euler's substitution. How to derive this formula?

I wish someone could help me derive this expression. ($K$ is a constant coefficient. $P_n(x)$ is a polynomial function of degree n.) $$ \int\frac{P_n(x)\mathrm{d}x}{\sqrt{ax^2+bx+c}} \equiv P_{n-1}(x) ...
5
votes
1answer
141 views

Which averages of products of a function give a norm?

Let $f: [0,1] \rightarrow \mathbb{R}$ be a bounded measurable function. For some real non-negative numbers $a_1, a_2, b_1, b_2$ with $a_1+b_1=a_2+b_2=1$ consider the quantity $$N(f)=\int_{[0,1]} \int_{...
8
votes
0answers
208 views

The many theories of integration

Diclaimer: In what follows, I will be loose in the usage of terminology since the very nature of the question is of a similar flavour. In the mathematics literature, one can find a zoo of theories of ...
3
votes
0answers
130 views

definite integral with incomplete gamma function and exponential

While working with electron density computations in quantum chemistry, I encountered the following improper integral: $$ I(k, n) = \int\limits_0^\infty \Gamma\left(\frac{3}{n},\ k r^n\right) r \exp(-k ...
2
votes
2answers
198 views

Laplace transform calculation

Please can someone help me? I have tried to find the Laplace transform of the form: $$\int_{0}^{+\infty} (v+1)^{\nu}(2v+1)^{k}v^{\alpha} \exp(-pv), \mbox{ where }\alpha,\nu, k \mbox{ are integers }$$ ...
2
votes
1answer
283 views

A possible error in Villani's monograph “Hypocoercivity”

I think there is an (possible) error in Villani's monograph titled "Hypocoercivity". To be specific, in page 48, he wrote "For the second term in (6.9), we use the identity $$\nabla\...
3
votes
1answer
59 views

van Cittert deconvolution method

In the early 1930s, van Cittert published a deconvolution method. Although his method was not perfect but it is the forefather of many improved spectral deconvolution methods. The basic idea is that ...
2
votes
2answers
142 views

Help with a limit involving incomplete beta integral

In trying to prove that the limit of a certain function approaches 1 as the positive integer parameter $n$ approaches infinity, I have ended up with the following intermediate expressions: $$f(n)=2^{...
5
votes
4answers
264 views

Generalizing contour integration to quaternions and bicomplex numbers

I am interested in the possibility of generalizing the notion of contour integration to the quaternions or bicomplex numbers. I am aware that the Frobenius theorem prevents the construction of a true ...
7
votes
1answer
991 views

Question on an exercise from Terry Tao's blog

I've been reading Tao's An introduction to measure theory,a draft can be found here.An exercise from it is Exercise 30 (Rising sun inequality) Let ${f: {\bf R} \rightarrow {\bf R}}$ be an absolutely ...
1
vote
0answers
48 views

Sources for multiple Stieltjes integral

I'd like to know which sources (books or papers) provide a detailed discussion of multiple Stieltjes integral or multiple Lebesgue-Stieltjes integral.
8
votes
2answers
1k views

A tricky integral to evaluate

I came across this integral in some work. So, I would like to ask: QUESTION. Can you evaluate this integral with proofs? $$\int_0^1\frac{\log x\cdot\log(x+2)}{x+1}\,dx.$$
0
votes
0answers
50 views

Integration of fractional function over Rice distribution

Let $a>2$ be a real variable. My objective is to find an approximation of the integral defined as \begin{equation} \int_0^{\infty } {\frac{1}{{1 + {x^a}}}} f\left( {x|y} \right)\, dx \end{equation}...
0
votes
1answer
107 views

integral of fractional function

Let $a>2$ be a real variable. My objective is to find an approximation of the integral defined as \begin{equation} \int_b^{ + \infty } {\frac{x}{{1 + {x^a}}}}. \end{equation} Here $b$ is a ...
0
votes
0answers
597 views

Trying to evaluate an integral relating to $\zeta (3)$

So similarly to my search for $\zeta (3)$ over at the mathematics stack exchange, I have continued to attempt to work towards a closed-form for it. The following integral is related to a search of ...
8
votes
3answers
319 views

Expected distance between two uniform points in distinct rectangles

Are there any good approximations (especially upper bounds) for the quantity $E(\|X_1-X_2\|$), where each $X_i$ is uniformly distributed in a rectangle $[a_i,b_i]\times[c_i,d_i]$? It does not appear ...
1
vote
0answers
75 views

Length of isoline $x(1-x)y(1-y)=c$

For the integral appearing in this answer, it may be beneficial to derive the length $L(c)$ of the isoline: $$x(1-x)y(1-y) = c,$$ where $x,y$ are ranging in $[0,1]$, and constant $c\in [0,\frac1{16}]$....
16
votes
3answers
750 views

Decoupling a double integral

I came across this question while making some calculations. QUESTION. Can you find some transformation to "decouple" the double integral as follows? $$\int_0^{\frac{\pi}2}\int_0^{\frac{\pi}...
3
votes
0answers
43 views

How to find or approximate (e.g. using method of steepest descent ) integral?

Can you give any advice on how to find or approximate the following integral $$ F(t,y) = \int_{0}^{y}\frac{i e^{-\frac{3 t^2 \left(x^2+1\right)}{2 \left(9 x^2+1\right)}-i \frac{4 t^2 x}{9 x^2+1}}}{\...
0
votes
0answers
25 views

Integrating multinomial likelihood over intersection of two simplexes

I am trying to calculate this integral, coming from a multinomial likelihood with an extra condition ($\sum_{i=1}^n p_i w_i = 1$). Essentially it can be seen as integrating a Dirichlet pdf over a ...
0
votes
1answer
82 views

Distribution of $\sqrt{X^2+Y^2}$ when $X$ and $Y$ are Cauchy

Suppose that $X$ and $Y$ are Cauchy-distributed with $\gamma=1$, i.e., with PDF $\frac 1 \pi \frac 1 {1+x^2}$. I tried to find the distribution of $R = \sqrt{X^2+Y^2}$. The PDF of $R$ should be given ...
2
votes
2answers
227 views

Asymptotic of an improper integral

I found myself stuck with an "elementary" claim in some article. A simplified version of the problem is: Let $p : [0,1]\to [0,1]$ be a continuous and non decreasing function such that $p(0)=...
24
votes
2answers
905 views

Is $\iiint_{[0, 1]^3} \lvert f(x)+f(y)+f(z)\rvert\, dx\, dy\, dz \ge \int_0^1 \lvert f(x)\rvert\, dx$?

$\newcommand\abs[1]{\lvert#1\rvert}\newcommand\Abs[1]{\left\lvert#1\right\rvert}$Question: Let $f(x)$ be a continuous real-valued function defined on the interval $[0, 1]$. Does the following ...
1
vote
0answers
65 views

How does one interpret the wetting area?

This may be a simple question, but I decided to post it here (not just on MSE) because it is very related to a research topic: capillary surfaces. Let $(M^3,g)$ be a Riemannian $3$-manifold with ...
5
votes
0answers
238 views

Wiener-Hopf Factorization: How are these contour integrals done?

$$\Phi_+(\alpha) = \frac{1}{2\pi i} \int_{C_1}\ln\left(2 e^{-\frac{c | z| }{2}} \cosh \left(\frac{c z}{2}\right) \right) \frac{dz}{z-\alpha}$$ and $$\Phi_-(\alpha) = - \frac{1}{2\pi i} \int_{C_2} \ln\...
18
votes
0answers
478 views

Fundamental Theorem of Algebra via multiple integrals

Consider the product of complex linear monic polynomials times polynomials of degree less than $n$, that is $\big( (z-\lambda), p(z)\big)\mapsto (z-\lambda)p(z)$. If we represent a polynomial by its ...
0
votes
0answers
95 views

Multivariate gaussian integral

I am looking for the solution of a multivariate gaussian integral over a vector x with an arbitrary vector a as upper limit and minus infinity as lower limit. The dimension of the vectors x and a are ...
1
vote
0answers
48 views

Integration by parts with fractional Laplacian and divergence

Let $\eta \in C^\infty_c$ and $\xi \in C^2$ Is it true that the integration by parts formula $$ \int_{\mathbb R^N} \nabla \eta \cdot \nabla((-\Delta)^{(\alpha-2)/2} \xi) = \int_{\mathbb{R}^N} \nabla \...
2
votes
1answer
112 views

Injectivity of an integral transform

For a bounded function $F: \mathbb R_{\ge 0} \to \mathbb R$ (not necessarily non-negative), is it true that $$\int_0^\infty \frac{x^ks}{(s^2+x^2)^{(k+3)/2}} F(x) dx = 0 \text{ for all $s >0$} \iff ...
0
votes
0answers
33 views

Expectation of ratios of probability density functions

I'm trying to solve/simplify the expression $$\mathbb{E_{x \sim b(x)}} B\ [\log\left(1 - \frac{A\ a(x)}{2\ c(x)}\right)], $$ or $$B \int_{x}b(x)\log\left(1 - \frac{A\ a(x)}{2\ c(x)}\right)dx,$$ where ...

1
2 3 4 5
22