## New answers tagged integration

1

Okay, so I think I may have found the answer myself. So, really, the absolute value symbol is a trick. You can get rid of it by pulling out an $i$, and then you have
$$\mathcal{I}:=\frac{1}{2\pi i}\int_a^b dx \frac{\log(x e^{-t_1})}{z-x}\frac{\sqrt{(z-a)(z-b)}}{\sqrt{(x-a)(x-b)}}$$
What you have to do is take a dumbbell contour around taking a clockwise &...

4

You can rewrite the integral as
$$
\int_0^{2\pi} \left(\sum_{j=0}^{n-1}\cos\Big(\sin t+\tfrac tn+2\pi \tfrac jn\Big)\right)\,dt.
$$
But $\sum_{j=0}^{n-1}\cos\big(a+2\pi \tfrac jn\big)=0$ for all $a$.
In particular, the equality holds if $\sin t$ is replaced by any $2\pi$-periodic function.

10

Diestel and Uhl commented on this in Measure theory and its applications:

1

You just need to use a higher precision, using exact numbers when possible, as shown in the Mathematica work below (with $k:=\kappa$ and $t:=\theta$). (However, this question is indeed better suited for Mathematica SE.)

3

It looks like you care only about the order of magnitude (i.e., an answer up to a constant factor), in which case it is fairly easy.
First, ignore all coefficients. Setting them to $1$ just changes the answer at most constant number of times. Now, suppose we have the denominator of the form $\sum_{(\alpha,\beta)} x^\alpha r^\beta$ where $\alpha$ is a multi-...

2

The accuracy issue with the evaluation of the hypergeometric function can be avoided for integer $\kappa$, since then the full expression reduces to an error function (see my answer to your previous question).
I tried this out for $\kappa=5$. Then
$$\mathbb{E}\left[ a \mathcal{Q} \left( \sqrt{b } \gamma \right) \right]=\frac{a}{48 b^4 {\theta}^8} \left[\sqrt{...

4

Let's consider the second integral, which can be written in the following form:
$$
I(p, q, i, j, k, l; a, b, c, d)
:=
\int_0^\infty
dt\,
\exp(-p t^2)
t^q
j_i(a t) j_j(b t) j_k(c t) j_l(d t)
$$
where in your case, $i = j = k = 1$, $l = 0$, and $q = -1$.
These kinds of integrals (as well their generalization to a product of arbitrarily many spherical Bessel ...

1

The best evaluation, a single sum, that I can derive is
$$ \int_0^{\pi/2} \big( P_n^m(\cos{\theta}) \big)^2\ d\theta = \frac{\pi}{2} \frac{(2m)!}{m!^4} \Big( \frac{(n+m)!}{(n-m)!} \Big)^2
{}_4F_3
\bigl( \begin{smallmatrix} m+1/2, & m+1/2, & m-n, &m+n+1 \\ m+1 & m+1 & 2m+1 \end{smallmatrix} | 1\bigr) $$
I derived it from the answer ...

3

Thanks to the comment by Johannes, the solution can indeed be obtained by using the following identities:
\begin{equation}
P_\ell(z)
=
\frac{1}{2^\ell}
\sum\limits_{k=0}^{\left\lfloor \frac{\ell}{2}\right\rfloor}
(-1)^k
\begin{pmatrix} \ell \\ k \end{pmatrix}
\begin{pmatrix} 2\ell - 2k \\ \ell \end{pmatrix}
z^{\ell - 2k}
\tag{3}
\label{3}
\end{equation}
and:
...

3

To evaluate
\begin{align}
c_{n,m}&=\int_0^{1/2}\frac{x(x-1/2)}{\sin^2(2\pi x)}\, \sin(2\pi(2m+1)nx)\,dx\\
&=-\frac{1}{8}\int_0^{1}\frac{x(1-x)}{\sin^2(\pi x)}\, \sin(\pi(2m+1)nx)\,dx
\end{align}
we first notice by changing $x\to 1-x$, that the integral vanishes if $n$ is even. In the following $c_{2n+1,m}$ is calculated.
We apply twice a result ...

2

Besides some condition ensuring that the map is well defined a simple and natural condition for the continuity is a Lipschitz condition with respect to the second variable, i.e., $$|f(x,y)-f(x,z)|\le c|y-z|$$ for some constant $c$ independent of $x,y,z$. This is a standard assumption for (a version of) the Picard-Lindelöf theorem for ODEs.
I repeated my ...

1

$\newcommand\Ga\Gamma$
Without loss of generality $a=1$. Let then $Q:=\mathcal Q$, $k:=\kappa>0$, and $t:=\theta\sqrt b>0$, so that $\sqrt b\,\gamma$ has the gamma distribution with parameters $k,t$. Let also $c:=\Ga(k)t^k$. Then, letting $f$ denote the standard normal pdf, we have
$$c\,EaQ(\sqrt b\,\gamma)=\int_0^\infty dy\,y^{k-1} e^{-y/t}Q(y)
=\...

1

Mathematica gives an answer in terms of a hypergeometric function:
$$\frac{a 2^{-\frac{\kappa}{2}-\frac{3}{2}} b^{-\frac{\kappa}{2}} {\theta}^{-k} \, _2F_2\left(\frac{\kappa}{2}+\frac{1}{2},\frac{\kappa}{2};\frac{1}{2},\frac{\kappa}{2}+1;\frac{1}{2 b {\theta}^2}\right)}{\sqrt{\pi } \Gamma \left(\frac{\kappa}{2}+1\right)}-\frac{a 2^{-\frac{\kappa}{2}-2} \...

0

An solution for this integral can be found at Mathmatica.SE, which is reproduced next.
After applying the change of variable technique with $x=2^r-1$ we get
$$f=\frac{e^{-\frac{\sqrt{r}}{b}} r^{\frac{d}{2}-1} \log _2(r+1)}{2 \left(b^d \Gamma (d)\right)} $$
$$\text{Integrate}[f,\{r,0,\infty \},\text{Assumptions}\to d\in \mathbb{R}\land b\in \mathbb{R}\land d&...

6

As you've pointed out, the only parameter that matters here is the angle $\theta$ between $x$ and $y$. To see how, consider instead the Gaussian integral:
$$
I(x,y)=\frac{1}{(2\pi)^{(d+1)/2}}\int_{u\in\mathbb{R}^{d+1}}\max(0,x^Tu)\cdot\max(0,y^Tu)\exp(-\frac{1}{2}u\cdot u)du
$$
The integral you are interested in is obtained by changing to $(d+1)$ dimensional ...

2

I'll suggest here another possible characterisation, expanding on a suggestion of the OP in one of the comments. Again, this is an assertion that certain known properties of line integration characterise it uniquely; this doesn't provide a "new" construction of line integration. Unlike my previous answer, here all the action takes place on the one ...

10

I don't know if it's exactly what you're looking for, but line integration is the unique way to assign a real number $I(\omega,c)\in\mathbb{R}$ to every pair of a smooth $1$-form $\omega$ on a smooth manifold $M$ with boundary and smooth path $c\colon[0,1]\to M$ such that:
(adjunction) if $f\colon M\to N$ is a smooth map of smooth manifolds with boundary, $\...

4

A nice version of the manifold version of Fubini's theorem is in Differentialgeometrie und Fasserbündel by Rolf Sulanke and Peter Wintgen:
Let $\phi \in C^1(M,N)$, where $M,N$ are smooth manifolds of dimensions $m,n$, respectively, with $m \ge n$. Let $\omega \in \Omega^{m-n}(M)$ and $\eta \in \Omega^n(N)$, and let $f : M \to {\bf R}$ be measurable (meaning, ...

1

I think this is
$$
-\int_{0+\epsilon}^{1-\epsilon}
{\frac {2\;{\rm W} \left(-\frac12\,\ln \left( 1-{x
}^{2} \right) \right)}{\ln \left( 1-{x}^{2} \right) }}\,{\rm d}x
$$
not elementary.

0

Hint: you may try $n=2$ and follow in general this :\begin{align} \int_0^\infty \left(\frac{\log(x)}{e^x}\right)^n\,dx&=\int_0^\infty e^{-nx}\log^n(x)\,dx\\\\ &=\frac1n\sum_{k=0}^n\binom{n}{k}\log^{n-k}(n)\int_0^\infty e^{-x}\log^n(x)\,dx\\\\ &=\frac1n\sum_{k=0}^n\binom{n}{k}\log^{n-k}(n) \left.\left(\frac{d^n \Gamma(x+1)}{dx^n}\right)\right|_{x=...

16

Let
$$f(a):=\int_0^\infty x^{a-1}e^{-2x}\,dx.$$
Then
$$f''(1)=\int_0^\infty \ln^2x\,e^{-2x}\,dx,$$
which is the integral in question.
On the other hand, $f(a)=2^{-a}\,\Gamma(a)$, and hence
the integral in question is
$$f''(1)=\frac{\ln^2 2}2 - \Gamma'(1)\ln2
+ \Gamma''(1)/2
=\frac{\pi^2}{12}+ \frac{(\gamma +\ln2)^2}2=1.6293\dots,$$
where $\gamma=0.57721\dots$...

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