The integration tag has no usage guidance.

**2**

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314 views

### How to perform this matrix integral?

Edit: some backgrouds added.
In quiver matrix model which is reviewed DV or CKR, the path integral reduce to the matrix integral
$$Z \sim \int \prod_{i=1}^r d\Phi_i \prod_{<a,b>} dQ_{ab} ...

**1**

vote

**1**answer

84 views

### Is sequential completeness of LCS strictly stronger that Riemann integrability of curves?

$\def\RiemInt{\,\text{-}\,\lower.5mm\hbox{$^{^{\rm Riem}}$}\kern-1mm\int}$
This is essentially a reformulation of this MSE question which has not received any answers for about three weeks. To ...

**1**

vote

**0**answers

125 views

### Are the two functions equal? If not, what relation are they?

We have
$$A=\frac{1}{2\pi}\int_{-\infty}^{\infty}\prod_{i=1}^{\infty}\frac{\sin(v2^{-k})}{v2^{-k}}e^{ivx}dx$$
$$B=\frac{1}{2}\sum_{k=1}^{n}\cos (\pi kx)\prod_{i=1}^{m}\frac{\sin(\pi k2^{-i})}{\pi ...

**3**

votes

**3**answers

324 views

### Positivity of the Coulomb energy in two dimensions

In dimensions $d\geq 3$ the Coulomb energy is always non-negative (since the Fourier transform of $\frac{1}{\|\cdot\|^{d-2}}$ is non-negative). What can one say about positivity properties of the ...

**21**

votes

**3**answers

1k 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
...

**2**

votes

**3**answers

257 views

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

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

**5**

votes

**1**answer

522 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

218 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

226 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

143 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

179 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

211 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

286 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

88 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

198 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

303 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

1k 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

238 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

236 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

253 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

227 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

493 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

169 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

284 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

243 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

331 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

102 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

580 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

576 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

171 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

238 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 ...

**3**

votes

**0**answers

767 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

189 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

275 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

734 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

1k 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

471 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

568 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

469 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

220 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

627 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

318 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

474 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

314 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

804 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:
...

**2**

votes

**2**answers

490 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

174 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

350 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

660 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

581 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 ...