# Tagged Questions

The integration tag has no wiki summary.

**0**

votes

**1**answer

231 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
...

**1**

vote

**0**answers

152 views

### 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 ...

**11**

votes

**5**answers

2k views

### 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 ...

**1**

vote

**2**answers

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 $)$

**5**

votes

**1**answer

477 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\: ...

**10**

votes

**1**answer

598 views

### 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 ...

**2**

votes

**1**answer

499 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,
...

**1**

vote

**0**answers

186 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 ...

**3**

votes

**1**answer

325 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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votes

**2**answers

1k views

### 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 $f(x)$ ...

**1**

vote

**0**answers

284 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) ...

**4**

votes

**1**answer

536 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 ...

**1**

vote

**0**answers

429 views

### Integral with modified bessel function and exponentials

I am trying to solve the following equations
$\int^\infty_0 \exp\left(-\frac{\alpha}{x+1}\right)\exp(-c x) x^{\frac{n-1}{2}} I_{n-1}\left(\sqrt{\beta x}\right)\ \mathrm{d}x $
and
$\int^\infty_0 ...

**3**

votes

**0**answers

144 views

### An isoperimetric inequality for “order” polytopes

I am looking for an isoperimetric inequality for order-like polytopes.
An order polytope $K\in \mathbb{R}^n$ is defined by a set of linear inequaities:
$$ \forall i \; 0\leq x_i \leq 1 $$
and
$ ...

**8**

votes

**1**answer

1k views

### About the surface area vs. volume of polytopes

Given a convex body $K\in\mathbb{R}^n$, represented by a set of linear inequalities (intersection of halfspaces), I am interested in understanding how much of its volume can be close to its perimeter ...

**8**

votes

**2**answers

590 views

### An identity involving an infinite integral with a sinh in the denominator

I recently encountered the rather appealing looking integral, which appears in the theory of random matrices :
$$\int_{-\infty}^{\infty}\prod_{j=1}^{p-1}(j^{2}+z^{2})\frac{zdz}{\mathrm{sinh}(2\pi z)} ...

**2**

votes

**1**answer

343 views

### A Bessel integral

Today I came across the integral
$\int_a^\infty e^{-bx} I_n(x) dx$
where $I_n$ is the modified Bessel function of the first kind. There is a solution for $a=0$, provided in Gradshteyn and Ryzhik, ...

**2**

votes

**2**answers

894 views

### Definite Integral ∫_{0}^{∞} dx exp(−x^2−a exp(b x^2))

I've been trying without success to do $$\int_0^\infty dx\; \exp(-x^2) \exp(-a\exp(bx^2)).$$
It's not in my integral tables. Wolfram online integrator won't do it. It doesn't seem to be amenable to a ...

**1**

vote

**1**answer

167 views

### Integrating with sub-level sets

This is a simple question, and I'm sure it was a homework assignment at some point (assuming it's true) but it's one that I'm puzzled over. Suppose I have a compact domain $D \subset \mathbb{R}^n$ ...

**1**

vote

**0**answers

251 views

### Gauge integral of the derivative of a function except on a set of measure 0.

For the entire question, the interval I am integrating over is $[0,1]$.
Background: In order to exhibit an isometry from $L^2[0,1]$ into $l^2$, I need to either assume absolute continuity over some ...

**0**

votes

**1**answer

121 views

### Average values of <OR> integrating over nonincreasing simplex

Hi,
I am trying to compute a distribution by integrating over all non-increasing categorical distributions of a given size $n$.
For instance, for $n=2$, each categorical distribution must follow ...

**1**

vote

**2**answers

446 views

### High dimensional beta integral (a typo in Stein's book “singular integrals”)

Hello,
When I read Stein's book of Singular Integrals, at p. 118, there is an obvious mistake:
$$
\int_{R^n} |x-y|^{-n+\alpha} ...

**-1**

votes

**1**answer

589 views

### does equi-integrability implies uniform convergence?

A collection $\{f_n\}$ of real valued functions is said to be HK-equi-integrable on $I=[a,b]$, if there exists a gauge $\delta$ on $I$ such that for every $\epsilon>0$, there exists a $\delta$-fine ...

**5**

votes

**3**answers

729 views

### Can distribution theory be developed Riemann-free?

I imagine most people who frequent MO have been indoctrinated into the point of view that the Riemann integral can be safely discarded once one has taken the time to develop the Lebesgue integral. ...

**1**

vote

**3**answers

311 views

### do numerical integration with fixed abscissas

Can I do an integral, possibly using gaussian quadrature, when the abscissas are fixed (for reasons that I don't want to get into right now), i.e. is it possible to calculate the weights for fixed ...

**0**

votes

**0**answers

218 views

### More multinomial type integrals over the hypercube

The question is related to my previous question about integrating the multinomial over the hypercube and the motivation for this question is the same, but the integral is a bit different. Here it is,
...

**3**

votes

**2**answers

484 views

### Is there a corresponding Hahn decomposition theorem for the real-valued Radon measures?

Hello,
As we know that a signed measure $\mu$ on $R$ can be decomposed to the positive part $\mu_+$ and negative one $\mu_-$ by the Hahn decomposition theorem.
My question is whether each ...

**3**

votes

**2**answers

1k views

### Best Numerical Method for Evaluating a Hilbert transform

I have to evaluate a Hilbert transform for some $\mathcal{L}^p(\mathbb{R},\mathbb{C})$-function ($1\leq p<\infty$). I know there are a number of algorithms out there to do it, but I don't have a ...

**0**

votes

**1**answer

408 views

### ($n$-dimensional) Inverse Fourier transform of $\frac{1}{\| \mathbf{\omega} \|^{2\alpha}}$

Note: I first posted question on math.stackexchange and I got one reply, which was a bit helpful (I'm still trying to understand it fully), but did not explore the two solution cases that I mentioned. ...

**5**

votes

**3**answers

949 views

### Integration by parts for a general negative-definite self-adjoint operator.

I suspect I am asking a very stupid question.
Suppose you have self-adjoint negative-definite operator $L$ densely defined on a space $L^2(\pi)$, with $Lf = \nabla \cdot ( A(x)\nabla f)$, for some ...

**-1**

votes

**1**answer

356 views

### linear versus non-linear integral equations

I'm having trouble solving an integral equation. It appears to me to be a homogenous fredholm equation of the second kind. However, I'm being told that this can't be a fredholm equation, because it ...

**7**

votes

**3**answers

1k views

### Rationale for Hadamard's finite part of a divergent integral

(Note: I asked this question a few days ago on math.stackexchange but didn't get any responses. I've therefore decided to post it here instead.)
I have a problem justifying throwing away the ...

**2**

votes

**2**answers

743 views

### The easiest symbolic integration method to try implementing.

Hello! I wonder how hard is it to implement more or less general symbolic integration algorithm (number of lines in a certain language)? Maybe someone here did this or knows some good blog posts ...

**1**

vote

**1**answer

308 views

### Can you dispense with the use of the existence of Positive integrable function in Henstock theory especially in Fubini theorem.

In Henstock-Kurzweil integral in the proof of Fubini theorem you need a strictly positive integrable function for rectangles of infinite volume. How to deal with such situation in general setting . ...

**0**

votes

**0**answers

358 views

### Is an integrable map from a measure space to a Banach space always measurable?

Is every integrable mapping defined in a general measure space to a Banach space measurable?
The answer is yes if it is function (real valued). The answer is yes if it is a mapping into a ...

**82**

votes

**5**answers

6k views

### Source and context of $\frac{22}{7} - \pi = \int_0^1 (x-x^2)^4 dx/(1+x^2)$?

Possibly the most striking proof of Archimedes's inequality $\pi < 22/7$ is an integral formula for the difference:
$$
\frac{22}{7} - \pi = \int_0^1 (x-x^2)^4 \frac{dx}{1+x^2},
$$
where the ...

**4**

votes

**4**answers

2k views

### Generalized Gauss-Green theorem

I am looking for a generalized version of the Gauss-Green theorem also known as the divergence theorem:
http://en.wikipedia.org/wiki/Divergence_theorem
A quick search on MathSciNet suggests that ...

**0**

votes

**0**answers

212 views

### integrating a character of a non-archimedean local field

By way of motivation, this computation comes from a proof in Bump's book Automorphic Forms and Representations where he shows that the Weil index of the reduced norm of a four-dimensional central ...

**75**

votes

**17**answers

16k views

### Why is differentiating mechanics and integration art?

It is often said that "Differentiation is mechanics, integration is art." We have more or less simple rules in one direction but not in the other (e.g. product rule/simple <-> integration by ...

**14**

votes

**1**answer

965 views

### A mass spring model for hair simulation

A strand of hair is represented by a set of particles connected by springs.
The velocity for a particular particle is calculated implicitly using the following formula:
...

**4**

votes

**1**answer

337 views

### Integrating the multinomial over a hypercube

I have come across an integral of the form
$$\int_{b}^{a}\cdots\int_{b}^{a} \left( \sum_{i=1}^{n}x_i\right)^mdx_1d x_2\dots dx_n.$$
I have a solution that makes use of the partition function, but I ...

**2**

votes

**1**answer

208 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

854 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

2k 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

750 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

248 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

502 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

733 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

354 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

498 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 ...