Product of Even and Odd Functions

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

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

Let $\OO \EE$ be their pointwise product, defined on the intersection of the domains of $\OO$ and $\EE$.

Then $\OO \EE$ is odd.

That is:


 * $\forall x \in S \cap S': \map {\paren {\OO \EE}} {-x} = - \map {\paren {\OO \EE} } x$.

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

Also see

 * Product of Odd Functions
 * Product of Even Functions