Product of Even and Odd Functions

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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

\(\ds \map {\paren {\OO \EE} } {-x}\) \(=\) \(\ds \map \OO {-x} \map \EE {-x}\) Definition of Pointwise Multiplication of Real-Valued Functions
\(\ds \) \(=\) \(\ds -\map \OO x \map \EE x\) as $\OO$ is odd and $\EE$ is even
\(\ds \) \(=\) \(\ds -\map {\paren {\OO \EE} } x\)

The result follows from the definition of an odd function.

$\blacksquare$


Also see