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.
$\blacksquare$