Periodic Functions

# Periodic Functions

We will now define a very special type of function known as a periodic function.

 Definition: A function $f$ is said to be $p$-Periodic, ($p > 0$) if $f(x + p) = f(x)$ for all $x \in D(f)$.

The most familiar periodic functions are the trigonometric functions. Namely, $\sin x$, $\cos x$, and $\tan x$ are all $2 \pi$-periodic since $\sin (x + 2\pi) = \sin x$, $\cos (x + 2\pi) = \cos x$, and $\tan (x + 2\pi) = \tan x$ for all $x$ in their respective domains. Graphically, this means that shifting the graphs of any of these functions horizontally by any integer multiple of $2\pi$ yields the same graph. The graphs of $\sin x$ (red), $\cos x$ (yellow), and $\tan x$ (blue) are given as a reference below: Another example of a periodic function is the ceiling function denoted $\left \lceil x \right \rceil$ whose output is the smallest integer greater than or equal to $x$. For example, $\left \lceil \pi \right \rceil = 4$. Similarly, the floor function denoted $\left \lfloor x \right \rfloor$ outputs the largest integer less than or equal to $x$. For example, $\left \lceil \pi \right \rceil = 3$. These functions are both $1$-periodic. The graphs of the ceiling function (orange) and the floor function (green) are given as a reference below: From the five examples given above, we see that a periodic function need not be bounded (e.g. $\tan x$) and need not be continuous (e.g. $\left \lceil x \right \rceil$ and $\left \lfloor x \right \rfloor$)

For a sixth example of a periodic function, let $C \in \mathbb{R}$. Then $f(x) = C$ is a periodic function with period $p$ for all $p > 0$. Of course constant functions being periodic is not that interesting of a concept!