Contents
Periodic and Non-Periodic functions
Definition
A function is defined as periodic if it can be moved along the x axis to a place where it exactly matches its original form. In mathematical terms, x(t) is periodic if and only if:
$ \,\! x(t+T)=x(t) $
Examples of periodic and non-periodic functions
Periodic examples:Basically any trigonometric function
$ \,\!cos(t)=cos(t+2\pi) $
$ \,\!sin(t)=sin(t+4\pi) $
This example can be shown to be periodic by drawing a graph, or simply computing values
$ \,\!cos(\pi+2\pi)=cos(\pi)=-1 $
also, any square, triangle, or sawtooth waves are periodic
Non-Periodic examples
any algebraic function:
$ \,\!f(t)=2x+5 $
$ f(t)=\frac{2x^3+5}{4^x-x} $
$ \,\!f(t)=log(x)+e^{x+2} $
any power, exponential or logarithmic function, without a periodic portion, are non-periodic as well.
Periodicity Test for Exponentials
If $ \,\!\frac{\omega_0}{2\pi} $ is rational, then function is periodic
so $ \,\!e^{2} $ is not periodic because $ \,\!\frac{2}{2\pi}=\frac{1}{\pi} $ is not rational
