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1
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0answers
185 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
1answer
319 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 ...
5
votes
2answers
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
0answers
283 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
1answer
527 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
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0answers
415 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
0answers
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
1answer
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
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2answers
586 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
339 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
890 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
166 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
250 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
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
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2answers
442 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
588 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
726 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
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3answers
309 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
215 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
476 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
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
1answer
407 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
946 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
355 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
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
2answers
734 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
307 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
356 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 ...
80
votes
5answers
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
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
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0answers
210 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 ...
74
votes
17answers
15k 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
956 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
336 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
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1answer
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
1answer
831 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
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
2answers
733 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
242 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
497 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
722 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
351 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
484 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
392 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
457 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
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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
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2answers
486 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
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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 ...
17
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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 ...