Product of Even and Odd Functions

Theorem
Let $\mathcal O$ be an odd real function defined on some symmetric real interval $\mathbb I$.

Let $\mathcal E$ be an even real function defined on some symmetric real interval $\mathbb I'$.

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:


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

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

Also see

 * Product of Odd Functions
 * Product of Even Functions