# 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, ...

981
questions

**6**

votes

**2**answers

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### 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

1k 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 ...

**3**

votes

**3**answers

3k 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

495 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

727 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 ...

**2**

votes

**1**answer

4k 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} \sum_{i=1}...

**1**

vote

**1**answer

234 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 d}...

**22**

votes

**4**answers

4k 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

830 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

5k 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 $w$ ...

**1**

vote

**2**answers

1k 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 \...

**7**

votes

**2**answers

873 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

528 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 ...

**28**

votes

**3**answers

3k 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 ...

**34**

votes

**3**answers

4k 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 ...

**13**

votes

**5**answers

9k views

### The Fundamental Theorem of Calculus in Lebesgue Theory

I am interested to what extent the famous identity
$$
\int_a^b f'(x) \ dx=f(b)-f(a)
$$
is true for a function $f:[a,b]\to \mathbb C$ continuous on $[a,b]$ and differentiable on $(a,b)$. One famous ...

**0**

votes

**2**answers

2k views

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

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

**14**

votes

**7**answers

5k 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

894 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
$\int_{-\infty}^{\infty}\left|x\right|f\left(x\...

**4**

votes

**1**answer

873 views

### Intersection volume of two tori (Quantitating the orientational freedom of catenane rings?)

At the risk of posting too low-level a question...
Please consider two tori, with tube radii $r_1$ and $r_2$, and center-of-the-hole to center-of-the-tube radii $c_1$ and $c_2$. I'd like to find an ...

**1**

vote

**2**answers

4k 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 ...

**75**

votes

**8**answers

12k 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 ...

**30**

votes

**2**answers

2k 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

1k 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

2k 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
$$\int_{x_1+x_2+\...

**24**

votes

**4**answers

12k views

### About the Riemann integrability of composite functions

When I was teaching calculus recently, a freshman asked me the conditions of the Riemann integrability of composite functions.
For the composite function $f \circ g$, He presented three cases:
1) ...

**5**

votes

**1**answer

2k views

### Fourier transforms via Kurzweil-Henstock integral on locally compact commutative groups

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

**-1**

votes

**2**answers

1k views

### Defined Almost Everywhere [closed]

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 ...

**7**

votes

**1**answer

2k 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 ...

**17**

votes

**5**answers

13k 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 Riemann–...

**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
$$ \...