4
votes
How to find the coefficient of $x^k$ in the expression $\prod_{p=2}^n (1+xp) $
From representation
$$\prod_{p=2}^n (1+xp) = (-x)^{n+1} (-1/x)_{n+1} (1+x)^{-1}=
\sum_{i\geq 0} s_1(n+1,i) (-x)^{n+1-i}\cdot \sum_{j\geq0} (-x)^j,
$$
it follows that the coefficient of $x^{k-1}$ in ...
4
votes
Conjectured closed form of $\int\limits_0^1 \frac{\ln y \operatorname{Li}_2 (-y)}{1-y^2} \, dy$
Use Package MultipleZetaValues (Version 1.2.0) for Mathematica, developed by Kam Cheong Au,
You can get it and install using this command
...
4
votes
Accepted
Prove the limit of the integral
$\newcommand{\ga}{\gamma}$Let
\begin{equation*}
L(s):=\int_0^{\pi/2} \frac{\sin^2(sx)}{\sin^2x}\,f(x)\,dx- \frac\pi2f(0)s
-\frac{f'(0)}2\ln s,
\end{equation*}
\begin{equation*}
R:=\int_{0}...
3
votes
Densities, pseudoforms, absolute differential forms and measures, differential forms, etc
I am not the greatest expert on the details of this stuff, but since nobody else tried so far, let me have an attempt:
Prelude: Measures
Since you mention measures, I start with that, though this is ...
3
votes
Accepted
Calculating an integral involving Haar measure on orthogonal projections
The average diverges. To check this, try $n=2$, $m=1$; the matrix $U={u_1\choose u_2}$ has probability density
$$P(u_1,u_2)=\frac{2}{\pi}\delta(1-u_1^2-u_2^2).$$
Then, taking $v={1\choose 0}$,
$$\int \...
1
vote
Exponential trigonometric integral
Here are my comments as an answer: Based on my short study as well as also on the comment by @TheSimpliFire, I am quite sure that the best one can get is the integral representation below, or a ...
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