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

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

Theorem
Let $\mathcal O_1$ be an odd real function defined on some symmetric set $S$.

Let $\mathcal O_2$ be an odd real function defined on some symmetric set $S'$.

Let $\mathcal O_1\mathcal O_2$ be their pointwise product, defined on the intersection of the domains of $\mathcal O_1$ and $O_2$.

Then $\mathcal O_1\mathcal O_2$ is even.

That is:


 * $\forall x \in S \cap S': \left({\mathcal O_1\mathcal O_2}\right)\left({-x}\right) = \left({\mathcal O_1\mathcal O_2}\right)\left({x}\right)$.

Proof
The result follows from the definition of an even function.