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8
votes
2answers
552 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
1answer
321 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
2answers
866 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
1answer
149 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
0answers
232 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
1answer
115 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
2answers
418 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
1answer
563 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
3answers
698 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
3answers
284 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
0answers
209 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, ...
2
votes
2answers
416 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
2answers
872 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
1answer
399 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
3answers
906 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
1answer
351 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
3answers
912 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 the throwing away the ...
2
votes
2answers
664 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
1answer
305 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
0answers
335 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 ...
75
votes
5answers
5k 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
4answers
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
0answers
180 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 ...
64
votes
16answers
14k 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
1answer
919 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
1answer
314 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
1answer
206 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
1answer
758 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
3answers
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
2answers
627 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
1answer
238 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
3answers
461 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
2answers
701 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
1answer
342 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
0answers
460 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
1answer
384 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
1answer
445 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
3answers
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
2answers
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
3answers
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
6answers
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
2answers
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
1answer
891 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
2answers
1k 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
0answers
475 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
1answer
306 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
1answer
983 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
1answer
209 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
4answers
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 ...
4
votes
1answer
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 $w$ ...