Product of Even and Odd Functions

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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.

$\blacksquare$


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