User:GFauxPas/Sandbox

Welcome to my sandbox, you are free to play here as long as you don't track sand onto the main wiki. --GFauxPas 09:28, 7 November 2011 (CST)

User:GFauxPas/Sandbox/Zeta2/lnxln1-x/existence

User:GFauxPas/Sandbox/Zeta2/lnxln1-x/integrand

User:GFauxPas/Sandbox/Zeta2/lnxln1-x/evaluation

User:GFauxPas/Sandbox/Zeta2/FourierSeries/

User:GFauxPas/Sandbox/Zeta2/Informal Proof

$\mathcal L \left\{{}\right\}$

DiffEQ Ongoing Project
Objective: To analyze different characterizations of $\zeta(2)$.

Laplace Transform
Let $f\left({t}\right)$ be a real function, where $t \ge 0$.

The Laplace transform of $f$ is defined as:


 * $\displaystyle \mathcal L \left\{{f\left({t}\right)}\right\} = F\left({t}\right) = \int_0^{\to +\infty}e^{-st}f\left({t}\right) \, \mathrm dt$

wherever this improper integral converges.

Theorem
$\displaystyle \mathcal L \left\{{e^{at}}\right\} = \frac 1 {s-a}$

where $a \in \C$ is constant, and $\operatorname{Re}\left({s}\right) > \operatorname{Re}\left({a}\right)$.

Proof
Because $\operatorname{Re}\left({s}\right) > \operatorname{Re}\left({a}\right)$:

Theorem

 * $\displaystyle \mathcal L \left\{{\sin at}\right\} = \frac a {s^2 + a^2}$

Proof
Also:

So: