da luca.barletta » 24/10/2006, 12:46
1) Consider the "triangular wave" function:
$f(x)=|x|={(x,,0<=x<=pi),(-x,,-pi<=x<0):}$
and such as $f(x)=f(x+2kpi), kinZZ$.
Since f(x) is even, we can compute its cosine transformation:
$a_0=2/piint_0^pi f(x)dx = pi$
$a_k = 2/piint_0^pi f(x)cos(kx)dx = {(0,,k=2n , ninNN),(-4/(k^2pi),,k=2n+1 , ninNN):}$
so we have:
$pi/2-4/pisum_(n=0)^(+infty) cos((2n+1)x)/(2n+1)^2$
Note that this result is consistent thank to the Dirichlet's th.
In particular, being $f(x)=x, x in [0,pi]$, we can write:
$x = pi/2-4/pisum_(n=0)^(+infty) cos((2n+1)x)/(2n+1)^2 0<=x<=pi$
and from the position $x=0$ we get
$0=pi/2-4/pisum_(n=0)^(+infty) 1/(2n+1)^2$
that is
$sum_(n=0)^(+infty) 1/(2n+1)^2 = pi^2/8$
2) Of course the following holds:
$int_{0}^{1}(lnx)/(x^2-1)dx=pi^2/8$
I'm just finding an elegant way to put down all the formulas...
Excuse for my english. I've taken some lessons from Biscardi.
Ultima modifica di
luca.barletta il 24/10/2006, 12:57, modificato 1 volta in totale.
Frivolous Theorem of Arithmetic:
Almost all natural numbers are very, very, very large.