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

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


 * $\mathcal L \left\{{e^{at}f\left({t}\right)}\right\} = F\left({s-a}\right)$

Theorem
Let $\mathcal Lf\left({x}\right) = F\left({x}\right)$ be the Laplace Transform of $f$.

Let $e^t$ be the exponential.

Let $a \in \R$ be constant.

Then:


 * $\displaystyle \mathcal L \left\{{e^{at} f\left({t}\right)}\right\} = F\left({s-a}\right)$

everywhere that $\mathcal Lf$ exists, for $\operatorname{Re}\left({s}\right) > a$