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.

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