The integration tag has no wiki summary.

**2**

votes

**1**answer

207 views

### Multiple ergodic averages with varying number of terms

Hi. I've been stuck on the following question for some time.
Consider a sequence of functions $\left( f_n \right)$ from an ergodic space $\left( \mathsf{X}, \mathsf{S}, \mu \right)$ to $\left[ 0,1 ...

**7**

votes

**1**answer

799 views

### Universally measurable sets and weak topology

After I posted this question, a couple of months ago, and got from MO-users several
good hints, I think i'm ready, after some study, to ask another related question (or rather, to focus on the main ...

**1**

vote

**3**answers

1k views

### How do you calculate the solid angle of a rectangular, axis aligned section of a surface defined by a two dimensional function?

I have $f(x,y) = \frac{1}{2} (1 - x^2 - y^2)$, which is a paraboloid centered around the origin (plot).
Now I want to calculate the solid angle (with the origin as the viewpoint) of the surface area ...

**1**

vote

**2**answers

674 views

### integration of a laplacian

Hi,
I solved for a Poisson equation with finite elements, using piecewise linear basis functions on 2d triangles.
Now, I want to evaluate the following expressions:
$$ \int_\Omega \Delta u ~dx$$
and
...

**1**

vote

**1**answer

239 views

### Integrate kˆ(n-1) / prod_{i=1…n} (kˆ(2)+x_iˆ{2}) dk between 0 and infinity, with x_i constants and n>=1? [closed]

[some formatting tweaked, and the question copied from the title to the main body, by YC]
Hi,
I've been struggling a lot to calculate this integral.
$$ \int_0^\infty \frac{k^{n-1}}{\prod_{i=1}^n ...

**1**

vote

**3**answers

473 views

### Undefined gamma function problem

Hello,
I'm trying to solve the following integral :
$\int_0^\infty \frac{1}{t^{d/2}}(e^{-\gamma t} - e^{-\delta t})dt$.
I know it equals
...

**2**

votes

**2**answers

714 views

### measurability of integrated functions

Hello everybody,
DISCLAIMER: I'm not a mathematician, but a computer scientist, so I hope the question is not trivial (or perhaps I hope so, in order to get a definitive answer). Anyway it's not a ...

**4**

votes

**1**answer

349 views

### Time-integral of a smooth, vector-valued function of a planar Brownian bridge

I'm looking for information on how to compute the distribution of the random vector
$$Z = \int_0^t f(B_s) ds$$
where $t>0$ is fixed, $B_s$ is a 2D Brownian bridge with $B_0 = 0$, $B_t=b \in ...

**0**

votes

**0**answers

472 views

### Integration of exponentials of quadratic forms (general Gaussian itegrals) over semi-finite domains

Hi
I'm interested in evaluating the integral of $e^{-s^T A s + b s}$ over a semi-infinite domain. $A$ is $\binom{N}{2}$x$\binom{N}{2}$ and has $N-1$ eigenvalues equal to $N$, and the rest 0. In ...

**4**

votes

**1**answer

389 views

### Is the 2d gauge integral equivalent to the Lebesgue integral for nonnegative functions?

Let $f$ be a function from $[0,1]\times [0,1]$ to $\mathbb{R}$.
Definition:
2dgauge$\displaystyle\int f \; = \; I$
$\Leftrightarrow$
For all neighborhoods $U$ of $I$, there exists a function ...

**2**

votes

**1**answer

449 views

### Integral and limit

During my research this integral has shown up
$ \frac{1}{2T} \int_{-T}^T \left( 1 - \frac{|\tau|}{T}\right)e^{-\alpha\tau^2}\cos(2\pi f_0 \tau) d\tau$
I tried to solved by taking the real part of a ...

**30**

votes

**3**answers

2k views

### Is there a systematic method for differentiating under the integral sign?

This MO question by Tim Gowers reminded me of a question I've wondered about for some time. In the delightful book Surely You're Joking, Mr. Feynman!, Feynman praises the technique of differentiating ...

**1**

vote

**2**answers

480 views

### limit of definite integral as $N \to \infty$

I'm interested in $\theta(N):=\int_0^1 (1-x)^{N-1} e^{xN} dx$. I'd like to show that $\theta(N)\sim c/\sqrt{N}$ as $N\to\infty$ and determine $c$. Any ideas?

**16**

votes

**3**answers

2k views

### Weak and Strong Integration of vector-valued functions

This is probably an elementary question, but outside my area of expertise, and I was unable to find any suitable reference:
Suppose $f:X\to E$ is a continuous function from a compact spaces (endowed ...

**16**

votes

**6**answers

2k views

### Why not evaluate integrals using ODE-solvers?

Hello!
I have a question about the relationship between numerical integration and the solution of ordinary differential equations (ODE). Suppose I want to evaluate the integral
$I(x) = \int_{0}^{x} ...

**5**

votes

**2**answers

1k views

### A tricky integral

Let $\alpha>0$ and $\beta\in\mathbb{R}$. I am looking for an explicit formula for the integral
$$\int_{-\infty}^{\infty} (1+x^2)^{-1/2}e^{-\alpha x^2}e^{-i \beta x}dx.$$
I tried several changes ...

**0**

votes

**1**answer

904 views

### Time scale calculus vs Lebesgue–Stieltjes calculus

About the same time, it seems, as I asked this question, a new post appeared on the wikipedia discussion page for Time scale calculus which suggests the Time scale derivative (aka Hilger derivative ...

**1**

vote

**3**answers

2k views

### “Riemann–Stieltjes derivative”?

Can you define a "derivative" operator such that its antiderivative F(x) of f(x) can be used in the sense of F(b)-F(a) to calculate the Riemann–Stieltjes integral of f(x)?
Perhaps it would be related ...

**0**

votes

**0**answers

479 views

### Integrating the product of two functions one of which has a positive non-integer power

I'm looking to integrate several functions having the form
$\int_0^T \frac{ sin(\omega \tau) }{\omega} \tau^{2H} d\tau$
where $2H \ge 0$ but may not be an integer. I'd like to know if the machinery ...

**0**

votes

**1**answer

321 views

### Linear Mapping and integration

I have been reading the paper - "Introduction to Quantum Fisher Information".
In section 1.2 the author talks about the linear map $\mathbb{J}_D$, which he defines as follows:
Let $D \in M_n$ be a ...

**1**

vote

**1**answer

1k views

### How to do integration using MCMC?

I want to evaluate $I = \int_V f(\vec{x}) d\vec{x}$. The classical Monte Carlo method is to sample uniformly from within the integration volume $V$, and then compute $I \approx V \frac{1}{N} ...

**1**

vote

**1**answer

211 views

### evaluating an integral related to the volume of Hessenberg orthogonal matrices

Consider the following integral,
$$
{1 \over 4\pi^{2}}\int_{0}^{2\pi}\int_{0}^{2\pi}
\sqrt{\, 9 -\sin^{2}\left(\theta_{1} \over 2\right)
\sin^{2}\left(\theta_{2} \over 2\right)\,}
\,{\rm ...

**12**

votes

**4**answers

2k views

### Integrals from a non-analytic point of view

I've mentioned before that I'm using this forum to expand my knowledge on things I know very little about. I've learnt integrals like everyone else: there is the Riemann integral, then the Lebesgue ...

**7**

votes

**3**answers

739 views

### Expectation of a simple function of multivariate gaussians iid rvs

I would like to compute analytically the following expected value:
$$ E\left( \frac{X_i^2}{\sum_j \lambda_j^2 X_j^2}\right) $$
where the $X_i \approx N(0,1)$ are iid.
It seems to be an elementary ...

**4**

votes

**1**answer

2k views

### Inverse of a function defined by an integral

Hi, I have a function defined by an integral as follows.
$$
z=f(w) = \int_0^w \frac{(\zeta-a_1)^{\alpha_1}(\zeta-a_2)^{\alpha_2}...}{(\zeta-b_1)^{\beta_1}(\zeta-b_2)^{\beta_2}...}\ d\zeta
$$
where $w$ ...

**0**

votes

**0**answers

395 views

### Integration involving the complete elliptic integral of the first kind K(k)?

Is there any reference showing how to do definite integrals involving the complete elliptic integral of the first kind K(k)?
Something like
$\int_0^1 K(k) dk $
$\int_0^1 k^nK(k) dk$
$\int_0^1 ...

**6**

votes

**2**answers

715 views

### Contour integration problem from probability

Can integrals of the form
$$
\int_{-\infty}^{\infty}{\exp\left(-\left[x - c\right]^{2}\right) \over 1 + x^{2}}\, {\rm d}x
$$
be computed in closed form using contour integration (or any other ...

**0**

votes

**1**answer

458 views

### Surface of the cut of an ellipsoid / Marginal density of a multivariate normal over an affine space

So I'm trying to get the marginal density of a multivariate normal over an affine space
if $A$ is a matrix in $\mathbb{R}^p \times \mathbb{R}^n$ for $p < n$ and $B \in \mathbb{R}^n$, $\Sigma$ is a ...

**19**

votes

**2**answers

1k views

### What are the obstructions for a Henstock-Kurzweil integral in more than one dimension?

I have recently come across the book The Kurzweil-Henstock Integral and its Differentials by Solomon Leader, in which, if I understand correctly, the HK integration process is modified in a way that ...

**23**

votes

**3**answers

3k views

### What is the standard notation for a multiplicative integral?

If $f: [a,b] \to V$ is a (nice) function taking values in a vector space, one can define the definite integral $\int_a^b f(t)\ dt \in V$ as the limit of Riemann sums $\sum_{i=1}^n f(t_i^*) dt_i$, or ...

**0**

votes

**1**answer

940 views

### Modified Dirichlet function Darboux integrable on [0,2]?

Hi,
Given this modified Dirichlet function: f(x) = 0 if x is in Q, else f(x) = x. I am wondering if this function is Darboux integrable on the interval [0, 2].
I managed to show that every lower ...

**9**

votes

**6**answers

3k views

### Numerical integration over 2D disk

I have a real-valued function $f$ on the unit disk $D$ that is fairly well behaved (real-analytic everywhere) and would like to find the integral $\int_D f(x,y)dxdy$ numerically. After much searching, ...

**5**

votes

**3**answers

836 views

### An Integral and derived double integral

Suppose that $f\left(x\right)\geq0$ is continuous on $\left[-\infty,\infty\right]$
and $\int_{-\infty}^{\infty}f\left(x\right)dx=1$. Is it true that
...

**0**

votes

**2**answers

2k views

### Area enclosed by x^4 + y^4 = 1 [closed]

Trying to solve for the area enclosed by $x^4+y^4=1$. A friend posed this question to me today, but I have no clue what to do to solve this. Keep in mind, we don't even know if there is a ...

**48**

votes

**8**answers

6k views

### Why is Lebesgue integration taught using positive and negative parts of functions?

Background: When I first took measure theory/integration, I was bothered by the idea that the integral of a real-valued function w.r.t. a measure was defined first for nonnegative functions and only ...

**19**

votes

**2**answers

1k views

### What theorem constructs an initial object for this category? (Formerly “Integrability by abstract nonsense”)

Tom Leinster has a note here about how you can realize L^1[0,1] as the initial object of a certain category. You should really read his note because it is only 2.5 pages and is much more charming than ...

**11**

votes

**2**answers

848 views

### Is there a notion of integration over the algebraic numbers?

For reasons which are hard to articulate (due to they not being very clear in my mind), but having to do with the eprint From Matrix Models and quantum fields to Hurwitz space and the absolute Galois ...

**3**

votes

**1**answer

1k views

### Approximating a multiple sum with an integral

Hi,
I want to approximate a multiple sum of the form
$$\sum_{x_1+x_2+\cdots+x_m \leq n}e^{g(x_1,x_2,\ldots,x_m)},$$
where each $x_i$ is an integer between $0$ and $n$,
by an integral
...

**3**

votes

**1**answer

2k views

### Fourier transforms and the Kurzweil-Henstock integral

Is it possible to define Fourier transforms on locally compact commutative groups using the Kurzweil-Henstock integral instead of the Lebesgue integral?

**1**

vote

**2**answers

562 views

### Defined Almost Everywhere

How can one prove that the convolution of f \in L^1 and g \in L^p is defined almost everywhere? Here f and g are measurable functions in R^n.
In general what techniques are there for showing some ...

**4**

votes

**1**answer

745 views

### Kullback-Leibler divergence of scaled non-central Student's T distribution

What is the Kullback-Leibler divergence of two Student's T distributions that have been shifted and scaled? That is, $\textrm{D}_{\textrm{KL}}(k_aA + t_a; k_bB + t_b)$ where $A$ and $B$ are Student's ...

**10**

votes

**4**answers

6k views

### Visualization of Riemann–Stieltjes Integrals

The Riemann–Stieltjes integral $\int_a^b f(x)\,dg(x)$ is a generalization of the Riemann integral. It is e.g. heavily used as a starting point for stochastic integration. The approximating ...

**8**

votes

**4**answers

2k views

### An integral that somehow equals pi^2/6 and involves dilogarithms?

I am attempting to show that
$$ \sum_{k \ge 1}^\infty {k^2 x^k \over (1+x^k)^2} \sim (1-x)^{-3} {\pi^2 \over 6} $$
as $x$ approaches 1 from below. The sum can be approximated by the integral
$$ ...