ThmDex – An index of mathematical definitions, results, and conjectures.

Result R4287 on D3997: Even euclidean real function

Finite product of even complex functions is even

Formulation 0

Let $f_1, \dots, f_N : \mathbb{R}^K \to \mathbb{C}$ each be an D3997: Even euclidean real function such that

(i)	$\prod_{n = 1}^N f_n$ is the D4652: Pointwise function product of $f_1, \dots, f_N$

Then $\prod_{n = 1}^N f_n$ is an D3997: Even euclidean real function.

Proofs

Proof 0

Let $f_1, \dots, f_N : \mathbb{R}^K \to \mathbb{C}$ each be an D3997: Even euclidean real function such that

(i)	$\prod_{n = 1}^N f_n$ is the D4652: Pointwise function product of $f_1, \dots, f_N$

If $x \in \mathbb{R}^K$, then \begin{equation} \left( \prod_{n = 1}^N f_n \right)(-x) = \prod_{n = 1}^N f_n(-x) = \prod_{n = 1}^N f_n(x) = \left( \prod_{n = 1}^N f_n \right)(x) \end{equation} $\square$