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

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

Basis for the Induction
By hypothesis $f$ is of exponential order $a$.

Thus Laplace Transform of Derivative can be invoked:


 * $\mathcal L \left\{{f'\left({t}\right)}\right\} = s \mathcal L \left\{{f\left({t}\right)}\right\} - f\left({0}\right)$ is proved in Laplace Transform of Derivative.

This is the basis for the induction.

Induction Hypothesis
Fix $n \in \N$ with $n \ge 1$.

Assume:


 * $\displaystyle \mathcal L \left\{{f^{\left({n}\right)} \left({t}\right)}\right\} = s^n \mathcal L \left\{{f\left({t}\right)}\right\} - \sum_{j \mathop = 1}^n s^{j-1} f^{\left({n-j}\right)}\left({0}\right)$

This is our induction hypothesis.

To change: make the induction hypothesis $n-1$.

Induction Step
To change: make the induction step $n$.

By hypothesis $f$ is of exponential order $a$. This is our induction step: