The integration tag has no wiki summary.

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### Finding Kuramoto Model coupling strength with limits?

The following is an equation describing the coupled phases of N oscillators according to the Kuramoto model:
$$
1 = K \int_{-\pi/2}^{\pi/2}\cos^{2}\left(\theta\right)\,
{\rm ...

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**1**answer

188 views

### Practical way to check for geometric convergence

Target distribution is multimodal, 24 dimensions, continuous state space. For MCMC integration (MH sampler) I use a manually tuned proposal distribution.
When I measure the convergence rate ...

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384 views

### Reference for integral of functions taking values in a topological vector space.

(Note that I am interested in the Gelfand-Pettis integral specifically, as opposed to, for example, the Bochner integral.) I have tried Googling things like "integral topological vector space", ...

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4k views

### Proof without words for surface area of a sphere [closed]

I love the book Proofs Without Words by Roger B. Nelsen. One of the proofs I liked the most was this: Area under one arch of a cycloid is 3 times the area of the wheel that traces it. You break the ...

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**1**answer

271 views

### A particular kind of Cauchy Principal Value integral

I am sorry to bother the community with such a narrow question, it may perhaps be a little specific. As I study Random Matrix Theory, I often have to solve integrals of the form
$$\mathcal{P} ...

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**1**answer

342 views

### Modern version of an inequality of R. M. Gabriel for contour integrals

I am currently reading the 1998 article Dynamics of the Binary Euclidean Algorithm:
Functional Analysis and Operators by Brigitte Vallée, which cites a 1928 article by R. M. Gabriel for the following ...

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**1**answer

100 views

### Dertivative of a Special Function with respect to Order

The marcum Q-function is defined by
$$ Q_m(a,b) = \int^\infty_b x \left(\frac{x}{a}\right)^{m-1} \exp\left(-\frac{x^2+a^2}{2}\right)
I_m\left(a x\right)
\:\mathrm{d} x,$$
where $m\in\mathbb{N}$ , ...

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452 views

### integrate of functions involving floor

Is there any exact formula or at least exact inequalities for the following intehral
$$
\int_{2}^{x}{{\rm d}t \over \left\lfloor\vphantom{\large h}%
\log(x)/\log(t)\right\rfloor
\log\left(t\right)}
...

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**1**answer

374 views

### A multiple definite integral.

I'm unable to find an easy way to compute the following multiple definite integral.
Apologies if it is trivial.
Let $C$ be a $N \times 1$ real vector.
Let $M$ be a $N \times n$ real matrix.
Let ...

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2k views

### About the definition of Borel and Radon measures

I am trying to understand the notion of Radon measure, but I am a little bit lost with the different conventions used in the litterature.
More precisely, I have a doubt about the very definition of ...

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451 views

### Certain compact subset of $L_1$

Let $(\Omega,\Sigma, \mu)$ be a probability measure and $X$ a Banach space. I am interested in subsets $F\subseteq L_\infty (\mu,X)$ that satisfy these two compactness conditions:
$F$ is a ...

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3k views

### Is $x \, \tan(x)$ integrable in elementary functions?

I'm teaching Calculus and my students asked me to calculate the integral of $x \, \tan(x)$.
I spent quite a lot of effort to do this, but I'm now even not sure if the integral could be presented in ...

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504 views

### Gauss Legendre Method for Implicit Integration

Methods that are usually adopted for time integration in transport phenomena problems are either:
Euler (explicit, first-order accurate)
$\frac{dY}{dt}=f(t,Y)$
$Y^{n+1}=Y^n+\Delta t f(t,Y^n)$
...

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219 views

### Integrating a product

By trying to find a marginal distribution I came accross integration of the product series. For the sake of generality, lets assume the integral is of following form:
$$\int \prod_{k=1}^{n}\left ( ...

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952 views

### Borel sets preserved under open maps?

Given open map f: $R^n$ to $R^n$ such that each open set $U\in R^n$, $f(U)$ is also open. Are Borel sets in $R^n$ preserved under f?
Motivation: Pre-image of Borel sets under continuous map is a ...

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342 views

### Integration in several variables and elementary applications

This fall I'm teaching the "second half" of the standard entry-level undergraduate multivariable calculus course: the focus is on double and triple integrals, path integrals, Green's theorem, Stokes' ...

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462 views

### Defining the integral of a function using the product measure

Imagine that we're trying to define the expression
$$\int_U f(x)dx$$
in a rigorous way.
Assume that $f:X \rightarrow \mathbb{R}^{\geq 0}$ where $(X,\mu)$ is a measure space, and suppose that $U$ is a ...

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2k views

### Counterexamples to differentiation under integral sign?

I'm exploring differentiation under the integral sign (I want to be much faster and more assured in doing this common task). So one thing I'm interested in is good counterexamples, where both ...

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### Asymptotic behaviour/upper bound for $\int_0^{\infty} \exp(-c x^a+K x^b)dx$ for $a>b>0$ as $K\rightarrow \infty$?

What is the asymptotic behaviour or an upper bound for $\int_0^{\infty} \exp(-c x^a+K x^b) \, dx$, for $a>b>0,$ as $K\rightarrow \infty$?
Or any good reference for tools to tackle this question?
...

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### Boundedness of Riemann-like sums on unbounded interval

Hi
I am trying to find suitable conditions (integrability, growth...) on a function $f:\mathbb{R}\to \mathbb{R}$ such that:
\begin{equation}
\sum_{k\in\mathbb{Z}}f(kh)h= \mathcal{O}(1),\qquad h\to ...

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493 views

### Rain droplets falling on a table

Suppose you have a circular table of radius $R$. This table has been left outside, and it begins to rain at a constant rate of one droplet per second. The drops, which can be considered points as they ...

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788 views

### Integration on high dimensional sphere

Hi, I need to integrate a function on an n-dimensional sphere surface. One way is to use the triangle function like: http://en.wikipedia.org/wiki/N-sphere#Spherical_volume_element, however, it is too ...

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### convergence of multiple integral

I am searching for some theorems and books about convergence of multiple integrals of the form:
$$
\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\;f(x,y)\;\mathrm{d}x\,\mathrm{d}y.
$$
In particular, ...

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6k views

### Integration of the product of pdf & cdf of normal distribution [closed]

Denote the pdf of normal distribution as $\phi(x)$ and cdf as $\Phi(x)$. Does anyone know how to calculate $\int \phi(x) \Phi(\frac{x -b}{a}) dx$? Notice that when $a = 1$ and $b = 0$ the answer is ...

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101 views

### Variation of a Function

Let $g$ be a function of finite $q$ Variation and $f$ be a function of finite $p$ Variation, and $\frac{1}{q}+\frac{1}{p}>1$.
What can be said about the variation of $H$ with ...

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355 views

### Fourier coefficients of a rational function

Any ideas how to compute or to approximate integral
$$\int_{0}^{1}\frac{(x+a)^{2q}(x+b)^{2q}}{(x-1)^{4q}+x^{4q}}\exp({-2\pi i x y})dx$$
where $q \in \mathbb{N}$ and $a,b =-2,-1,0,1$, $y \in (0,1)$

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197 views

### Evaluating the integral $\int_{a}^{+\infty} \frac{\exp(-bx)}{x+c} Ei(x) dx$

I'm trying to evaluate or simplify this integral:
$$I_{a,b,c} = \int_{a}^{+\infty} \frac{\exp(-bx)}{x+c} Ei(x) dx $$
with $a,b,c \in \mathbb{R}_+^*$.
and $ Ei(x) =\int_{-\infty}^{x} ...

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378 views

### Question about Riemann integral and total variation [closed]

Let $g$ be Riemann integrable on $[a,b]$, $f(x)=\int_a^xg(t)dt$ for $x∈[a,b]$.
How to show that the total variation of $f$ is equal to $∫_a^b|g(x)|dx$?

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301 views

### Is there a good comparative study of the Banach integral?

The Banach integral is elegant in its definition, and I am intrigued as to why it is so rarely seen. Is it in practice difficult to calculate from the definition? And are there any other problems with ...

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434 views

### One-sided Cauchy principal value

What is the notion of a principal value of an integral when the singularity appears at one endpoint? Namely,
$ PV \int_a^b f(t) dt = ? $,
where the integral is convergent in the upper limit, but ...

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689 views

### Action Integral

In the theory of action-angle variables, you wind up having to solve integrals with a characteristic square-root behavior near the endpoints to express the action in terms of the orbital quantities. ...

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### Is $C_c(\mathbb{R}^2,\mathbb{R}^2)$ dense in the irrotational square integrable functions?

Let $L_D(\mathbb{R}^n)^n$ be the set of square integrable functions which are the weak derivative of a locally square integrable function. That is
$$L_D(\mathbb{R}^n)^n=\{Du\colon u\in ...

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746 views

### Integration in the surreal numbers

In the appendix to ONAG (2nd edition), Conway points that the definition of integration (using Riemann sums as left and right options) gives the "wrong" answer : $\int_0^\omega \exp(t)\thinspace ...

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297 views

### Questions on calculating volume using n-1 forms

Is there an n-1 form on $R^n$ which calculates the volume of n-manifolds? Similarly, is there such a 1 form on $S^2$, and $RP^2$? I thought this has something to do with the orientation, is that ...

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### Semi implicit DAE integration using an implicit Runge Kutta scheme

I'm looking for some references regarding integration of DAEs in the form
$M(t) \frac{dy}{dt} + G(y(t),t) + f(t) = 0, \quad y \in R^n, M(t) \in R^{nxn}$
using a high order implicit Runge Kutta ...

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403 views

### Lebesgue integral with respect to vector measures?

Good evening,
I'm reading some papers of Jim Agler and Nicholas Young, in which they prove a formula of integral representation with respect to a vector measure, but the integration is in the sense ...

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798 views

### What is the Dunford Integral and why is it useful?

Wikipedia http://en.wikipedia.org/wiki/Pettis_integral defines the Pettis Integral for Banach space valued functions wrt to some measure space by duality.
It calls the Pettis & Bochner integral ...

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237 views

### derivative of a special function in integral form

What is the derivative of $Q_m\left(\frac{\alpha}{x^a},\frac{\beta}{x^b}\right)$ with respect to $x$, i.e,
$$\frac{\partial}{\partial x}Q_m\left(\frac{\alpha}{x^a},\frac{\beta}{x^b}\right),
\quad
...

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### On explicit eigenfunctions

Given an algebraic surface $S$ defined by an algebraic equation such
as $x^{4}+2y^{4}+3z^{4}=1$, how would one find the third smallest
eigenvalue $\mu_{3}$ for the differential equation $\Delta ...

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### Why do I need densities in order to integrate on a non-orientable manifold?

Integration on an orientable differentiable n-manifold is defined using a partition of unity and a global nowhere vanishing n-form called volume form. If the manifold is not orientable, no such form ...

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### Naive definition of surface area doesn't work?

A first stab at a definition of surface area might go like this:
Let S be a surface. Select finitely many points from S and make a bunch of triangles having these points as vertexes. Add up the ...

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194 views

### Does integral of A^T(t)*A(t) converge if det(A(t)) does not converge to 0?

Suppose you have an nxn parametric square matrix A(t). I am wondering if I can prove this:
$(lim\_{t\rightarrow\infty}det(A(t)) \ne 0) \Rightarrow (\int^{\infty}A^T(t)A(t)dt$ Does not Converge $)$

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506 views

### Contour Integral with Gamma functions and 2F1

Given the following contour integral
$$\frac{1}{2\pi j}\int^{c+j\infty}_{c-j\infty} \frac{\Gamma(-1+a+s)\Gamma(b+s)}{\Gamma(3+a-s)}\cos(-1+a+s)\,
{}_2F_1\Big(-1-a+s,-1+a+s;\frac{1}{2};z\Big) y^s\: ...

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### Is the pairing between contours and functions perfect (modulo the kernel given by Stokes' theorem)?

Let $s: \mathbb C^n \to \mathbb C$ be a homogeneous degree-$d$ polynomial which is nonsingular (in the sense that the hypersurface it defines in $\mathbb{CP}^{n-1}$ is smooth; equivalently the ...

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574 views

### On two-dimensional Gaussian integrals

Fix $\epsilon, 0\leq \epsilon\leq 1/2.$ Let $Z_1,Z_2$ be zero mean, unit variance Gaussian random variables which are jointly Gaussian with $\mathbb{E}Z_1Z_2=-(1-2\epsilon)\leq 0.$
Then,
...

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204 views

### Integrating gamma products and quotients over a vertical line

The Wolfram functions collection contains a small number of integrals of products and quotients of terms $\Gamma(a_i\pm t)$ over a vertical line, all of which can be evaluated in terms of only gamma ...

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341 views

### Products of trigonometric functions with increasing frequencies

I am looking at weighted $L_2$ norms of a class of Littlewood polynomials, related to Walsh and Rademacher functions which made me look for pseudo-closed forms or computationally efficient expressions ...

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### How to do integrals involving two Bessel functions and another function?

I often encounter the integrals in the following form:
$\int_0^\infty{\rm Bessel}(ax)\cdot{\rm Bessel}(bx)\cdot f(cx)dx$,
where Bessel can be $J$, $N$, $H^{(1)}$, $H^{(2)}$, $I$, or $K$; and $f(x)$ ...

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292 views

### Residue Cancellation

I am trying to understand how to apply the residue theorem to solve
$\frac{1}{2\pi j}\int^{\gamma+j\infty}_{\gamma-j\infty}\Gamma(n-s)\Gamma(s)\Gamma(1-s) {}_1F_1(s;b;c) ...

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591 views

### Integral with confluent hypergeometric function

Can we get a closed form for the following contour integral?. Let us assume that n is a non-negative integer,
$\frac{1}{2\pi ...