4.3: Periodic Functions

We first need to define a periodic function. A function is called periodic with period $p$ if $f(x+p)=f(x)$, for all $x$, even if $f$ is not defined everywhere. A simple example is the function $f(x)=\sin(bx)$ which is periodic with period $(2π)∕b$. Of course it is also periodic with periodic $(4π)∕b$. In general a function with period $p$ is periodic with period 2p3p…. This can easily be seen using the definition of periodicity, which subtracts p from the argument

$$f(x+3p) = f(x+2p) = f(x+p) = f(x). \nonumber$$

The smallest positive value of p for which f is periodic is called the (primitive) period of f.

Exercise $\PageIndex{1}$

What is the primitive period of $\sin(4x)$?

Answer

$\frac{π}{2}$.