The integration tag has no wiki summary.

**16**

votes

**2**answers

890 views

### Can we simplify $\int_{0}^{\infty}\frac{{\sin}^px}{x^q}dx$?

We know the followings :
$$\int_{0}^{\infty}\frac{{\sin}x}{x}dx=\int_{0}^{\infty}\frac{{\sin}^2x}{x^2}dx=\frac{\pi}{2},\int_{0}^{\infty}\frac{{\sin}^3x}{x^3}dx=\frac{3\pi}{8}.$$
Also, we can get
...

**3**

votes

**3**answers

231 views

### How to extract the divergent part from the singular integral

How to extract the divergent part of the following integral simply as $u \rightarrow \infty$
$$g(u) = \frac{\sqrt{2u}}{\pi} \int^1_{\frac{1}{u}} dz \frac{\sqrt{z-1}}{\sqrt{z^2-u^{-2}}} $$

**4**

votes

**1**answer

356 views

### Integral of a product of Laguerre polynomials

In order to estimate the non linear term in a particular PDE, I have to decompose $L_k^\alpha(x)^3\cdot x^{-\delta}$ (with $0<\delta<\alpha+1$) into a basis consisting of Laguerre polynomials ...

**2**

votes

**0**answers

210 views

### An integral inequality

Let $g:\mathbb{R}\rightarrow\mathbb{R}$ be bounded with derivative $g'$. I have shown that the following inequality holds for all $w\in\mathbb{R}$,
...

**1**

vote

**3**answers

219 views

### limit of a singular integral

Denote $f_{\gamma}(x) =\frac { (1+\gamma)}{2} |x|^{\gamma}$. We consider:
$$I(\gamma) = \int_{-1}^1\int_{-1}^1 \ln (|x-y|) f_{\gamma}(x) f_{\gamma}(y) dx dy$$
I would like to know the limit of ...

**4**

votes

**0**answers

129 views

### About arithmetic-geometric mean

It's well known that if we set $a_0=x \geq 0, \ g_0=y \geq 0$, and
$$ a_{n+1}=\dfrac{1}{2}(a_n+g_n), \ g_{n+1}=\sqrt{a_n g_n} ,$$
then both $\{a_n\}$ and $\{g_n\}$ will converge to $AGM(x,y)$. ...

**0**

votes

**3**answers

164 views

### Definite integral of a function containing an exponential

I have to calculate analytically this integral:
$$
{\rm J}\left(q\right)
=
\int_{0}^{\infty}{{\rm d}x \over x^{q}\left({\rm e}^{kx}-1\right)}
$$
where $-1\le q\le N$
with:
$N\in\mathbb{N}$ and ...

**1**

vote

**1**answer

161 views

### Lebesgue's integrability condition in several variables

The well known Lebesgue's condition of Riemann integrability says that a bounded function in one variable
$f\colon [a,b] \to \mathbb{R}$ is Riemann integrable if and only if it is continuous almost ...

**1**

vote

**0**answers

255 views

### exponential integral $e^{x/t^{3}}$ and floor function

Is it possible to give a closed form for the integral
$$ \int_{0}^{\infty} \frac{dt}{t^{3}}\rho (t)e^{-\frac{x}{n^{2}t^{2}}}$$
where $ \rho (t) = t- \left\lfloor t\right\rfloor $ is the fractional ...

**1**

vote

**0**answers

85 views

### Distribute Monte Carlo samples among dimensions

Simplified problem: Given a $d$-times nested convolution of an input function $g(x):\mathbb{R}\mapsto \mathbb{R}$ with the same band-limited smooth function $f(x):\mathbb{R}\mapsto \mathbb{R}$. I am ...

**1**

vote

**1**answer

166 views

### Lebesgue integrability and Kurzweil Henstock integrability

Why it's possible to integrate the function: $$f(x)=\frac{1}{x}\sin\left(\frac{1}{x^\alpha}\right)$$ using Kurzweil Henstok integral while it's not Lebesgue integrable because the singularity in ...

**1**

vote

**0**answers

262 views

### complex contour integral calculation after Möbius transformation

Good day to everyone.
In my scientific research I've got stuck with a contour integration problem.
I would like to evaluate the following integral:
$$I=\int_0^{\infty } \frac{e^{\frac{\alpha -\mathrm ...

**5**

votes

**2**answers

771 views

### Integral with Dirac delta (me or wolfram mathematica?) [closed]

I asked the question on math.stackexchange but didn't get an answer so I came here.
I tried to compute with Wolfram Mathematica the following integral
$$I=\int_0^\pi\int_{-\infty}^\infty x e^{-\mathrm ...

**6**

votes

**1**answer

230 views

### “Values” of divergent integrals

Are there existing theories of integration in which $I_0 = \int_0^{\infty} dx$ and $I_1 = \int_0^{\infty} x \ dx$ are well-defined infinite elements in a non-archimedean extension of the reals? I can ...

**0**

votes

**0**answers

181 views

### Convergence of the integral of step functions

This is a question about the proof of Lemma A in §16 of the book Functional Analysis by F. Riesz and B. Sz.-Nagy.
Lemma A: For every sequence of step functions $\{\varphi_n\}$ which decreases to ...

**2**

votes

**0**answers

238 views

### Definite integral probably equal to zeta with known (but unusable) closed form for the indefinite integral

Related to this and
this questions.
Basically got definite integral that experimentally equals
$\zeta(s)$ both numerically and symbolically.
Closed form for the indefinite integral is known, but I ...

**3**

votes

**0**answers

213 views

### When does this method for integrals of fractional/integer parts work?

In a question
Agno suggested an interesting way to compute $\{x\}$ and $\zeta(s)$.
Define
$$ F(x) = \{x\} = x - \lfloor x \rfloor = \frac{i \, \log\left(-e^{\left(-2 i \, \pi x\right)}\right)}{2 \, ...

**0**

votes

**2**answers

463 views

### Is there a probability density function satisfying the following conditions?

I find myself in need of the solution of this problem in finding a probability density function. I had asked this question in Math Stack Exchange but I did not get an answer so I am posting it here.
...

**1**

vote

**0**answers

142 views

### Equal maximum and minimum in a large-scale linear programming

For a linear optimization of an integral (with integral constraints), I perform a linear programming for the equivalent series. Maximum and minimum of the LP problem tend to be equal as I increase the ...

**2**

votes

**0**answers

182 views

### Expectation involving the ratio of normal pdf to normal cdf?

i need to calculate some expectations which involving the ratio of normal pdf to normal cdf.
Specifically, they are $E\{\phi(x)/\Phi(x)\}$ and $E\{x\phi(x)/\Phi(x)\}$ where $x\sim N(0,1)$.
Written ...

**4**

votes

**1**answer

233 views

### How to get an expression for this integral(Numerically/Analytically)

I have the following problem:
I need to evaluate the integral $$\int_{\cos(\alpha)}^{1} P_l(t)P_{l'}(t) dt $$ for $\alpha \in [0,\pi]$ and each combination of $l$ and $l'$, where $P_l$ is the l-th ...

**3**

votes

**1**answer

230 views

### Practical error-estimates for (adaptive) Newton-Cotes Quadrature

I am looking for practical error estimates for Newton-Cotes Quadrature rules.
Most books on numerical methods I have found mainly deal with theoretical error bounds/estimates for the respective ...

**1**

vote

**1**answer

101 views

### Integral estimate on a two dimensional Riemannian manifold

For my Master's thesis, I'd like to prove the following (but I'm not sure it's true):
On a two-dimensional Riemannian manifold (oriented and closed), for any smooth function $f$, it holds that
$$
...

**17**

votes

**1**answer

534 views

### An NP-hard $n$ fold integral

We are given rational numbers $[c_1, c_2, \ldots, c_n]$ and $v$ from the interval $[0,1]$.
Consider the $n$-fold integral
$$
J = \int_{\theta_1 \in I_1, \theta_2 \in I_2 \ldots, \theta_n \in I_n} ...

**4**

votes

**1**answer

429 views

### Numerical multivariate definite integration

I need to compute a set of multivariate definite integrals with infinite integration domain
$$\displaystyle \int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty} f(x_1,x_2, \ldots , x_n)\;\;dx_1 ...

**2**

votes

**0**answers

112 views

### Marginalizing multivariate normal over defined interval

Hello everyone,
I am trying to obtain an analytic expression for the following Gaussian integral
$$\frac{1}{\sqrt{(2 \pi)^n |\Sigma|}} \int \kern-0.2em \cdots \kern-0.2em \int d\mathbf{x}_{\sim i} ...

**2**

votes

**3**answers

215 views

### Speed of convergence for Weyl's Equidistribution theorem

If $f$ is a continuous periodic fonction on $[0,1]$ and $a\not\in\mathbb{Q}$, the Weyl's equidistribution theorem states that
$$\frac{1}{n}\sum_{k=0}^{n-1}f(ak)\rightarrow \int_0^1 f(x)dx.$$
Can we ...

**2**

votes

**0**answers

390 views

### Hubbard-Stratonovich Transformation

Hello,
The Hubbard-Stratonovich transformation
$\exp(x^2) = \frac{1}{\sqrt{4 \pi}} \int_{-\infty}^{+\infty} du \exp(-\frac{u^2}{4} - xu)$
allows one to wirte the exponential of a the square of a ...

**1**

vote

**1**answer

181 views

### Convergence of a sum to the integral

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a 1-periodic function. I am looking about the conditions on $(a,b)\in\mathbb{R}^2$ such that we have the property :
...

**0**

votes

**1**answer

207 views

### Interpreting numerical double integration as a matrix multiplication

I have a convex optimization problem of finding a function Q(x,y) as below:
Minimize $\int{k(x,y)Q(x,y)dxdy}$ subject to a list of constraints which are not relevant to the question, so I'm skipping ...

**0**

votes

**1**answer

531 views

### computing an integral involving standard normal pdf and cdf

recently, i need to compute this kind of integral:
$$ \int ^\infty _c \Phi(ax+b) \phi(x) dx$$
where a, b and c are all constants and $\Phi(x)$ denotes the CDF of standard normal distribution and ...

**11**

votes

**3**answers

858 views

### Applications of visual calculus

Mamikon's visual calculus (see Mamikon, Tom Apostol, Wikipedia) is a very beautiful and surprisingly efficient tool.
The basis is
Mamikon's theorem. The area of a tangent sweep is equal to the ...

**2**

votes

**2**answers

387 views

### How do these two Haar measures on SL(2,R) compare?

By using the Iwasawa decomposition, one obtains a (bi-invariant) Haar measure on $G:=\mathrm{SL}(2,\mathbb{R})$ which can be symbolically written as ...

**14**

votes

**2**answers

505 views

### Evaluation of an $n$-dimensional integral

I asked the same question on math.se but got no answer there. Since it pertains to my current research, I decided to ask here:
Let $n\in 2\mathbb{N}$ be an even number. I want to evaluate
$$I_n
:=
...

**4**

votes

**1**answer

412 views

### Identity involving Fresnel integrals

In the paper E. Mehlum, Appell and the apple (nonlinear splines in space), Technical
Report No. 1676 (1981), Central institute for industrial research, Oslo (reproduced in the book Mathematical ...

**2**

votes

**0**answers

205 views

### Coutour Integral of Gamma Functions

How do I solve the Integral
$$ \frac{1}{2\pi j} \oint
\frac{b^{ - s} \Gamma[2 + i - s] \Gamma[s] \Gamma[-1 - i + s]}{
(2 + i - s) \Gamma[3 + i - s]} \:\mathrm{d}s$$
This integral is an inverse ...

**3**

votes

**2**answers

527 views

### Stokes theorem for manifolds without orientation?

Hello!
In textbooks Stokes theorem is usually formulated for orientable manifolds (At least I couldn't find any version not using orientability). Is Stokes theorem: ...

**1**

vote

**2**answers

291 views

### Defining definite integral using indefinite integral.

Sometimes definite integral is defined using antiderivatives:
$$\int_{a}^b{f(t)dt}=F(b)-F(a)$$
where $F$ is any continuous function such that:
$$(\forall t\in[a,b]\setminus C)(F'(t) \space exists ...

**4**

votes

**1**answer

429 views

### Is there a closed form expression/series expansion for $\int_{\epsilon-i\infty}^{\epsilon+i\infty} e^{az+b^2z^2}\Gamma(z)\Gamma(1-z)dz$ ?

I've been trying to find a closed form expression/series expansion for the following integral without success:
$$F(a,b)=\int_{\epsilon-i\infty}^{\epsilon+i\infty} ...

**1**

vote

**1**answer

281 views

### Why is there a formula for symbolic differentiation (chain and product rules) but not for symbolic integration? [duplicate]

Possible Duplicate:
Why is differentiating mechanics and integration art?
There is a formula for the derivative of any product, composite or sum of functions, in terms of the derivatives of ...

**7**

votes

**3**answers

761 views

### Nontrivial trivial integrals

I posted this question to stackexchange and after 24 hours it's got five votes and no answers, so let's see if mathoverflow can say more than that.
Consider two propositions in geometry:
...

**0**

votes

**0**answers

595 views

### Real analytic functions

I am quite confused with some ideas regarding the Real analytic functions.
Just to introduce my questions:
A function $f$ is real analytic on an open set $D$ of the real line if for any $x_0\in D$ ...

**2**

votes

**2**answers

454 views

### Sum involving binomial coefficients

I have the following sum
$\sum_{j=1}^K {K \choose j} (-1)^{j+1}/j$. Now I can write this as the integral $\int_{-1}^0 \frac{(1+x)^K - 1}{x} dx$. However, I wonder whether there is a closed form ...

**1**

vote

**0**answers

170 views

### convergence of sets and limit of an integral

Let $X\subset\mathbb{R}$ and $Y\subset\mathbb{R}$ be compact sets.
Let $f:X\times Y\rightarrow\mathbb{R}$ be a $C^{1}$ function.
Let $s:Y\rightarrow X$ be a function (not necessarily continuous).
...

**0**

votes

**1**answer

276 views

### An infimum of integrals of a positive function.

Hi,
I have a question concerning integration theory I can't figure out, maybe someone can help:
Fix $\varepsilon>0$ and consider $\delta \colon [0,1] \to (0,\infty)$ measurable. Is it then true ...

**3**

votes

**1**answer

562 views

### definition of operator valued integral with spectral measure

I am trying to make sense of some operators that come up on Buchholz and Summers' work on warped convolutions (two works on arxiv: 2008 and 2011).
There, they work on a Hilbert space $H$ and on the ...

**0**

votes

**1**answer

500 views

### What are integration on fractal? [closed]

Who can explain the proof of the formula (2.12) given here: J. Phys. A: Math. Gen. 20 (1987) 3861-3875. Printed in the UK ...

**2**

votes

**1**answer

351 views

### Integral of Modified Bessel Function of the Second Type

Given the identity
$$ \int^\infty_0 K_v\left(\alpha\sqrt{x^2+z^2}\right) \frac{x^{2\mu+1}}{\left(\sqrt{x^2+z^2}\right)^v}\:\mathrm{d}x = \frac{2^\mu \Gamma(\mu+1)}{\alpha^{\mu+1}z^{v-\mu-1}} ...

**2**

votes

**1**answer

184 views

### hypervolume under the square of an n-simplex

I posted this question at math.stackexchange.com, reformulated and posted again both times without much luck. I also asked a math professor at my uni who suggested I post it here. Hopefully, it is ...

**1**

vote

**0**answers

88 views

### Integrating B-Spline composed with log

If $f$ is a real B-Spline and $a, b$ are real numbers, then is there a numerically stable way to evaluate the following expression?
$\int_a^b f (\log x) \mathrm{d}x$