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 ...
Max Alekseyev's user avatar
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 ...
Jorge Zuniga's user avatar
  • 2,210
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}...
Iosif Pinelis's user avatar
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 ...
mlk's user avatar
  • 1,974
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 \...
Carlo Beenakker's user avatar
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 ...
Fred Hucht's user avatar
  • 2,705

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