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

Result R2963 on D4693: Function odd part

Odd function equals its odd part

Formulation 0

Let $f : \mathbb{R}^n \to \mathbb{R}^m$ be an D3998: Odd euclidean real function.
Let $f_{\mathsf{odd}} : \mathbb{R}^n \to \mathbb{R}^m$ be the D4693: Function odd part of $f$.

Then \begin{equation} f = f_{\mathsf{odd}} \end{equation}

Proofs

Proof 0

Let $f : \mathbb{R}^n \to \mathbb{R}^m$ be an D3998: Odd euclidean real function.
Let $f_{\mathsf{odd}} : \mathbb{R}^n \to \mathbb{R}^m$ be the D4693: Function odd part of $f$.

If $x \in \mathbb{R}^n$, then \begin{equation} \begin{split} f_{\mathsf{odd}}(x) & = \frac{f(x) - f(-x)}{2} \\ & = \frac{- f(- x) - f(-x)}{2} \\ & = - \frac{f(- x) + f(-x)}{2} \\ & = - f(-x) \\ & = f(x) \end{split} \end{equation} $\square$