# Product of Even and Odd Functions

## Theorem

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

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

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

Then $\mathcal O\mathcal E$ is odd.

That is:

$\forall x \in S \cap S': \left({\mathcal O\mathcal E}\right)\left({-x}\right) = - \left({\mathcal O\mathcal E}\right)\left({x}\right)$.

## Proof

 $\displaystyle \left({\mathcal O\mathcal E}\right)\left({-x}\right)$ $=$ $\displaystyle \mathcal O\left({-x}\right)\mathcal E\left({-x}\right)$ Definition of Pointwise Multiplication of Real-Valued Functions $\displaystyle$ $=$ $\displaystyle -\mathcal O\left({x}\right)\mathcal E\left({x}\right)$ as $\mathcal O$ is odd and $\mathcal E$ is even $\displaystyle$ $=$ $\displaystyle -\left({\mathcal O\mathcal E}\right)\left({x}\right)$

The result follows from the definition of an odd function.

$\blacksquare$