even odd

Evenness

A function f is called "even" if it holds that:

The integral over the range (-a,a) of an even function f is:

$∫\limits_{-a}^{a} f[x]⋅dx=2⋅∫\limits_{0}^{a} f[x]⋅dx$

A function f is called "odd" if it holds that:

The integral over the range (-a,a) of an odd function f is 0:

$∫\limits_{-a}^{a} f[x]⋅dx=0$

For a function f to be both even and odd, it has to hold that:

Which means:

The constant function returning zero is the only function that is both even and odd.

It is possible for a function to be neither even nor odd, but:

Any function f can be split into one even part and one odd part:

The first outer term is even, the second outer term is odd.

Author: Danny (remove the ".nospam" to send)

Last modification on: Sat, 04 May 2024 .