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 $\zeta(2)$.

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

The real 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.

Here $F$ is a function of the real variable $s$ and $e^t$ is the real exponential function.

"Real" sounds like real-valued here but it isn't, I need a better name for "on reals"

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

Proof
is $\displaystyle \int \operatorname{Im}\,\left({f\left({s}\right)}\right)\, \mathrm ds = \operatorname{Im}\,\left({\int f\left({s}\right)\,\mathrm ds}\right)$ a theorem?

I need to be careful with domains here and whether or not I need to invoke the complex Laplace transform...

--GFauxPas (talk) 12:06, 5 May 2014 (UTC)