Even and odd functions

Even and odd functions

A function ${\displaystyle f(t)}$ is said to be even if and only if ${\displaystyle f(-t)=f(t)}$ .

A function ${\displaystyle g(t)}$ is said to be odd if and only if ${\displaystyle g(-t)=-g(t)}$.

The product of even functions is even, the product of odd functions is even, and the product of an even function and an odd function is odd. The sum of even functions is another even function and the sum of odd functions is an odd function. Functions that are neither even nor odd can be represented as the sum of an even and an odd function.

The integral of an even function ${\displaystyle f(t)}$ on symmetric limits has the property

${\displaystyle \int _{-T}^{T}f(t)\;dt=2\int _{0}^{T}f(t)\;dt}$

and is, in general nonzero, whereas the integral of an odd function ${\displaystyle g(t)}$ on symmetric limits vanishes

${\displaystyle \int _{-T}^{T}f(t)\;dt=0.}$