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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Integral with inequality

Let $p(u,x):=(4 \pi u)^{-1/2}e^{-\frac{x^2}{4u}},u>0,x \in \mathbb{R}.$ Let $\mathcal{E}:=\{\phi \in C_c^\infty (\mathbb{R}),\operatorname{supp}(\phi) \subset B(0,1),\|\phi\|_\infty \leq 1\}.$ ...
mathex's user avatar
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2 answers
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Perhaps an application of Hardy's inequality

Let $f \in H_{0}^{1}(0,1)$ and $\lambda >0$ big enough. Consider $0 <\alpha < 1$ and some $k > 0$. I would like to show the following inequality $$ \int_{\lambda^{-k}}^{1}|f(x)|^{2}dx \leq ...
user253963's user avatar
3 votes
0 answers
140 views

Monotonicity of a function defined by an integral

The question below is motivated by the related question Integral of a function changing sign and the associated answer: Can we study the monotonicity of the following function on $(0,1)$? $$\small f(x)...
Migalobe's user avatar
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Closed formula for iterated Fourier series

I'm trying to obtain a closed formula for the following integral. \begin{align} I_n = {} & \int_0^h \Bigr[\sum_{r_1=1}^\infty a_{1,r} \cos\left(\frac{2\pi}{h} r_1t_1\right) \\[6pt] & {}+ b_{1,...
Marco's user avatar
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How to perform this integral to get a closed/ semi closed form

I want to get a closed/ semi-closed form of the integral given below. $$ \int_{-\infty}^{+\infty} \exp{\left(-\frac{(x - \mu)^2}{2\sigma^2}\right)} \text{erf}\left(\alpha \frac{x-\mu}{\sqrt{2}\sigma}\...
CfourPiO's user avatar
  • 159
2 votes
0 answers
126 views

Multiple Wiener integral as Witt polynomial of Brownian motion

I know that if i have a Brownian motion $W_t$ the multiple Wiener integral $\int_0^t \int_0^{\xi_1}...\int_0^{\xi_n} dW_{\xi_1}...dW_{\xi_n}$ can be expressed as $H_n(\int_0^t dW_s)$ where $H_n$ is ...
Marco's user avatar
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-2 votes
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Simple closed form for $\int \lfloor x \rfloor dx$? [closed]

Wolfram Alpha claims there is no closed form in terms of standard funcions for $\int \lfloor x \rfloor dx$ but we believe we found simple closed form agreeing with experimental data. Define $i_1(x)=x -...
joro's user avatar
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2 votes
0 answers
184 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
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198 views

Distribution of "occupation times" of Brownian Motion

Let $B_t(\omega)$ be a standard Brownian motion and let $A\in\mathcal{B}(\mathbb R)$ be a Borel set. I would like to find the distribution of $$Y_A(\omega):=\lambda(\{t\in[0,1]:B_t(\omega)\in A\})=\...
Andrea Aveni's user avatar
7 votes
1 answer
201 views

Average of polynomials over the real sphere

In quantum information, much can be done with the averaging formula $$ \int_{\mathbb{C}P^{n-1}} (zz^*)^{\otimes t} dz = {n + t -1 \choose t}^{-1} \operatorname{\Pi}_{\mathrm{Sym}^t}$$ Here the ...
eepperly16's user avatar
3 votes
1 answer
311 views

Integral of a function changing sign

By some numerical tests, we can see that the following function is negative on $(0,1)$: $$\small f(x)=\int_0^\infty\frac{s^{x-1} e^{-2 s} (\pi \cos(\pi x) (s^{2 x}+(0.1)^2)-\sin(\pi x) \ln(s) (s^{2 x}-...
Migalobe's user avatar
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4 answers
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Some Log integrals related to Gamma value

Two years ago I evaluated some integrals related to $\Gamma(1/4)$. First example: $$(1)\hspace{.2cm}\int_{0}^{1}\frac{\sqrt{x}\log{(1+\sqrt{1+x})}}{\sqrt{1-x^2}} dx=\pi-\frac{\sqrt {2}\pi^{5/2}+4\sqrt{...
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Fourier transform of an exponential function with radical argument divided by a radical

I have $f(t)=\dfrac{e^{-i\sqrt{(t-t_0)^2+A^2}}}{\sqrt{(t-t_0)^2+A^2}}$ where $t_0$ and $A$ are constant. I need to take the Fourier transform of $f(t)$. I made few substitutions to take it to a form ...
Ft insat's user avatar
28 votes
4 answers
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If $X$ and $Y$ independent and identically distributed, then $E(|X-Y|)\leq E(|X+Y|)$. Are other proofs of this known?

I know a proof of the theorem that if $X$ and $Y$ independent and identically distributed, then $E(|X-Y|)\leq E(|X+Y|)$. The proof uses an integral representation of the absolute value, $$\int_0^\...
janis's user avatar
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5 votes
2 answers
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Integral of a product between two normal distributions and a monomial

The integral of the product of two normal distribution densities can be exactly solved, as shown here for example. I'm interested in compute the following integral (for a generic $n \in \mathbb{N}$): $...
user1172131's user avatar
3 votes
1 answer
128 views

Convolution between normal distribution and the maximum over $m$ Gaussian draws

$\DeclareMathOperator\erf{erf}$ Let's consider the Gaussian distribution $P_X(x)= \frac{1}{\sqrt{2 \pi \sigma^2}} e^{- \frac{x^2}{2 \sigma^2}}$. Now consider the random variable $W \equiv \max \{ X_1, ...
user1172131's user avatar
2 votes
1 answer
244 views

Proof of covariant convolution for a kernel function that is rotation symmetric in Fourier space

Problem Statement Let $g:\mathbb R^{d}\to \mathbb R,d\in\{2,3\}$ be an integrable function (assumption I1). Suppose $\mathcal T$ is a rotation, and $f:\mathbb R^d\to\mathbb C$ (assumption C) is an ...
Jacob Helwig's user avatar
3 votes
0 answers
151 views

What is the meaning of big-O of a random variable?

I encountered this problem in a book "Pattern Recognition and Machine Learning" by Christopher M. Bishop. I excerpt it below: screenshot of the book In the excerpt, the big-O notation $O(\xi^...
zzzhhh's user avatar
  • 31
12 votes
2 answers
1k views

Counterexamples to differentiation under integral sign, revisited

Let $f\colon\mathbb R^2\to\mathbb R$ be a measurable function such that \begin{equation*} F(t):=\int_{\mathbb R}dx\,f(t,x) \end{equation*} exists and is finite for all real $t$. Suppose that \...
Iosif Pinelis's user avatar
1 vote
1 answer
187 views

Probability of multivariant gaussian random variables in different areas

$\newcommand{\sgn}{\operatorname{sgn}}$Let $X_i$ is a gaussian random variable correlated with others. we want to find the probability of each possible case to find the expectation of following ...
A. R.'s user avatar
  • 25
9 votes
2 answers
573 views

Integral inequality: Prove $\int_0^1 f\int_0^1 1/f \leq 1$ for a certain function $f$

Let $g$ be a piecewise smooth, zero average, function over $[0,1]$ such that $\min g^2>0$. I would like to show that $$ \int_0^1 g\sqrt{1-r/g^2}\int_0^1 \frac{1}{g\sqrt{1-r/g^2}} \leq 1 $$ for all $...
Hussein's user avatar
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1 answer
708 views

The derivation of Reynolds-averaged Navier-Stokes equations

The following procedure is used to derive the Reynolds-averaged Navier-Stokes equations (Wikipedia: RANS equations) When we talk about turbulent flows we can represent the velocity of the fluid as: $$ ...
Maman's user avatar
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1 answer
191 views

Calculation of solid angle for rectangle in 6DOF [closed]

I am an undergrad trying to understand and use solid angle calculations: I have a point source in R3 space (x_source, y_source, z_source) and a rectangle with given center (x_center, y_center, ...
Vojtooo's user avatar
1 vote
1 answer
82 views

Representing an $L^2$-functional by a non-$L^2$-function on a dense subspace - Part II

This is a follow-up to this previous question, but under stronger assumptions. Let $(X, \mu)$ be a (say, $\sigma$-finite) measure space, let $g \in L^2$ (say, over the real scalar field). Let $\tilde ...
Jochen Glueck's user avatar
7 votes
2 answers
379 views

Representing an $L^2$-functional by a non-$L^2$-function on a dense subspace

Let $(X, \mu)$ be your favourite measure space (finite or $\sigma$-finite if you like), let $g \in L^2$ (say, the scalar field of $L^2$ is $\mathbb{R}$, though this probably doesn't matter). Let $\...
Jochen Glueck's user avatar
4 votes
1 answer
132 views

Decreasing tail integrals for nonnegative random variable $X$

Let $X$ be a nonnegative random variable with density function $f(x)$, distribution function $F(x)$, survival function $S(x)=1-F(x)$ and finite first and second moments. Let also $$\ell(x):=\frac{1}{...
Jimmy R.'s user avatar
2 votes
2 answers
196 views

Assigning values to divergent oscillating integrals

I have recently run into a number of divergent oscillating integrals in various contexts. Thus, I have been led to desire general methods for assigning values to divergent oscillating integrals. All ...
Caleb Briggs's user avatar
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2 votes
0 answers
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How to perform this Gaussian matrix integral?

I'm reading this book The supersymmetric method in random matrix theory and applications to QCD. In page 302-303, the author calculate the following integral $$ Z=\int d\psi \, dH \, P(H)\exp\left(-\...
Guoqing's user avatar
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1 vote
1 answer
206 views

On a property of complex exponentials

Does there exist a simple smooth closed curve $\gamma:S^1\to \mathbb C$ such that $$ \int_{0}^{2\pi} e^{\gamma(e^{it})} \, |\gamma'(e^{it} )|\,dt =0?$$
Ali's user avatar
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2 votes
1 answer
96 views

Power series expansion of the order parameter in the Kuramoto model

In this review of the Kuramoto model, Eq. 14 is obtained by expanding the following integral in powers of $K r$, $$ r = K r \int_{-\pi/2}^{\pi/2}\cos^2(\theta) g(K r \sin{\theta}) \mathrm{d}\theta $$ ...
apg's user avatar
  • 612
0 votes
1 answer
72 views

A solution satisfying an integral inequality is bounded [closed]

Let $y$ be a positive function and $c>0$. If $y$ satisfies the following integral inequality \begin{equation} y(t)+\int_{0}^{t}y(s)ds\leq c_{1}\int_{0}^{t} y^{\frac{3}{2}}(s)ds+c_{2} \end{equation} ...
Mee Na's user avatar
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1 vote
1 answer
175 views

Finding $W^{1,\infty}$ solutions to an integral equation by fixed point

Let $f \in L^\infty((0,1)\times(0,1))$ and consider the integral equation \begin{align*} u(x, y)= \int_0^y \int_0^{x-y} u(x-y+\tau, s+\tau) f(s+\tau, \tau) d s d \tau -\int_0^y f(x-y+s, s) d s. \end{...
user avatar
2 votes
0 answers
82 views

A closed expression for definite integral of a rational function

Suppose $F(x) = P(x)/Q(x)$ is an integrable rational function on $\mathbb R$, that is, $\deg Q \geq \deg P + 2$, and $Q$ has no real roots. Does there exist an expression for the definite integral $...
Troshkin Michael's user avatar
16 votes
1 answer
2k views

Equivalence between Lebesgue integrable and Riemann integrable functions

As the title says, for every Lebesgue integrable function $f:\mathbb{R}\to\mathbb{R}$ is there a Riemann integrable function $g:\mathbb{R}\to\mathbb{R}$ such that $f=g$ almost everywhere? For example, ...
SilverBladeII's user avatar
5 votes
0 answers
194 views

If $\lim_{t\to +\infty} \int_{0}^{\pi} f(x)\exp(e^{xt}) \, dx=0$ then $f=0$ a.e?

Question, Let $f \in L^1(\alpha, \beta)$ , $\beta>0$ and $$ F(x)= \int_{\alpha}^{\beta} f(t)\exp(e^{xt}) \, dt $$ such that $\displaystyle \lim_{x\to +\infty}F(x)=0$. Does this imply that $f$ is ...
Pascal's user avatar
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1 vote
0 answers
167 views

Integration with respect to Haar measures normalised over a subspace

Coming from physics I have come across the following integral over a haar measure (for $U$ unitary as an example) for something I am trying to determine for my work $\int_{\mathcal{U}(d)} \frac{\...
N A McMahon's user avatar
0 votes
1 answer
144 views

Inhomogeneous Markov chains and the product-integral as a solution to the Kolmogorov forward equation

We have a inhomogeneous continous $K$-State Markov chain $X(t)$ with transition intensity matrix $Q(t)$. Therefore its entries are: $$q_{ij}(t)= \lim_{\delta \to 0} \frac{1}{\delta} \mathbb{P}(X(t+\...
Bloble's user avatar
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2 votes
0 answers
93 views

Computing $\int_0^{2\pi}\frac{e^{ikt}}{|e^{it}-e^{it_0}|^m}~\text{d}t$, where $k\in\mathbb{Z}$, $t_0\in\mathbb{C}$, and $m=1,3,5,\dots$

I am working on a project on accurate numerical quadrature where I need to compute the following integral in order to find my quadrature weights, $$ \int_{0}^{2\pi}\frac{{\rm e}^{{\rm i}kt}}{\,\left\...
David's user avatar
  • 21
2 votes
1 answer
298 views

Bounded operator on $L^2(\Bbb R^2)$

Let $\lambda\in \{z\in\Bbb C\mid \text{Re}(z)>0\quad \text{and}\quad \text{Im}(z)>0 \}$. Consider the operator $T_\lambda: L^2(\Bbb C)\to L^2(\Bbb C)$: $$ f\mapsto T_\lambda(f)(z)=\int_{\Bbb C}f(...
zoran  Vicovic's user avatar
0 votes
0 answers
126 views

Bounded sums involving primes

I'm trying to generalize the Theorem 2.7.1 in [1] where they prove: $$\sum_{p \leq x} f(p) = \int_{2}^{x} \frac{f(t)}{\log{t}} dt + \epsilon(x)f(x) - \int_{2}^{x} \epsilon(t) f^{'}(t) dt $$ where $\...
Pierluigi's user avatar
  • 109
2 votes
1 answer
275 views

Volume of submanifold as integral of delta-function

Let $M$ be an $n-m$ dimensional sub-manifold of $\mathbf R^n$ defined by the following set of equations: \begin{equation} f_1(\vec x)=0, \\ \vdots \\ f_m(\vec x)=0, \end{equation} (where $\vec x$ are ...
dennis's user avatar
  • 423
0 votes
0 answers
120 views

How to prove an equality involving Laguerre polynomials

Assume $\mu<0$. Let $L^\alpha_n(x)$ be Laguerre polynomials of type $n$ and $f\in L^2(\Bbb C)$. How to prove that $$\sum^\infty_{k=0}\int_{\Bbb C} f(z)\frac{1}{2(2k+1)-\mu}L^0_k(\lvert z\rvert^2)...
zoran  Vicovic's user avatar
4 votes
1 answer
164 views

Estimate an improper integral

Suppose that $f$ satisfies $a$-Hölder condition on $[0,1]$ ($0<a<1$). Fix $0<b<a$. For any $x\in [0,1]$ and sufficiently small $\delta>0$, is the following inequality established? $$ \...
Watheophy's user avatar
  • 419
8 votes
1 answer
2k views

How badly can the Lebesgue differentiation theorem fail?

Suppose $f:\mathbb{R}^n\to\mathbb{R}$ is integrable. Is it true that $$ \lim_{r\to 0}\frac{\displaystyle\int_{B_r(0)}f(y)~\mathrm dy}{r^{n-1}}=0 \quad ? $$ This is obvious if $0$ is a Lebesgue point ...
No-one's user avatar
  • 1,037
1 vote
1 answer
195 views

Integration by parts for indicator of a sphere to indicator of a ball

Broadly speaking, I have a radial distribution on $\mathbb R^n$, i.e., the pdf only depends on the $\ell_2$-norm of the argument. I would like to obtain an expression for the pdf in the form $\int_{w=...
Nicolas Resch's user avatar
3 votes
0 answers
146 views

Is there a general relationship between definite integrals over functions involving the complete elliptic integral of the first kind and zeta values?

Background Let $\textbf{K}(k)$ be the complete elliptic integral of the first kind, where $k$ is its elliptic modulus [1]. Moreover, define $k' := \sqrt{1-k^{2}} $ as its complementary modulus. I've ...
Max Muller's user avatar
  • 4,485
6 votes
3 answers
1k views

Integral of product of gaussian CDF and PDF

Looking for an analytic solution to the integral below: $$ \int_{-\infty}^\infty \Phi\left(\frac{x - a}{\tau}\right) \phi\left(\frac{x - b}{\sigma}\right)dx $$ where $\Phi(\cdot)$ and $\phi(\cdot)$ ...
user489812's user avatar
3 votes
0 answers
44 views

Evaluate $\int_\phi e^{tr(RM)} dR$ where $\phi$ is a set of all real orthonormal matrices of a certain size

I am trying to evaluate $\int_\phi e^{tr(RM)} dR$ where $\phi$ is a set of all real orthogonal matrices of a certain size. $M$ is an arbitrary real matrix (of a certain size). This is equivalent to $$\...
CWC's user avatar
  • 359
8 votes
1 answer
321 views

Malmsten-like integrals for $\zeta(n) $

Context Let us define a Malmsten integral when it is of the form $$M_{P,Q} := \int_{0}^{1} \frac{P(x)}{Q(x)} \ln \ln \left( \frac{1}{x} \right) dx. \tag{1} $$ In a paper by Blagouchine [1], the author ...
Max Muller's user avatar
  • 4,485
1 vote
2 answers
161 views

Asymptotic estimation of an integral

I have an integral of the form $$ I = \int\limits^{1}_{0} \exp\left(\dfrac{vt}{(v+1)^2 + v^2} - vt\right) dv $$ and I want to prove that $I\leq c t^{-1}$ for the sufficiently large $t$, where $c$ is a ...
minxin jia's user avatar

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