10

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


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, $\...


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 ...


4

A nice version of the manifold version of Fubini's theorem is in Differentialgeometrie und Fasserb√ľndel 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, ...


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.


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 ...


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-...


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

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{...


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 ...


2

Call $I$ the integral, then $$I=\sum_{k} \int_{k\pi}^{(k+1)\pi} xe^{-x^6 \sin^2x}dx=\sum_{k} \int_{0}^{\pi} (z+k\pi)e^{-(z+k\pi)^6 \sin^2z}dz=:\sum_k I_k.$$ Then $$ I_k \le (k+1)\pi \int_0^\pi e^{-k^6 \pi^6 \sin^2 z}dz=2(k+1)\pi\int_0^{\pi/2} e^{-k^6 \pi^6 \sin^2 z}dz \le 2(k+1)\pi\int_0^{\pi/2} e^{- 4 k^6 \pi^4 z^2}dz, $$ since $\sin z \ge (2/\pi)z$ in $...


2

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 &...


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 ...


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.)


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} \...


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 ...


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