In physics and engineering we see periodic phenomena such as vibrations, the motion of the tide, planetary, and alternating current (AC).

We say a function $f$ is periodic with period $T$ if $x+T$ lies in the domain of $f$ whenever $x$ lies in the domain of $f$ and if for every $x$ in the domain of $f$

\[

f(x+T)=f(x)

\]

\[

f(x+T)=f(x)

\]

- The condition “$x+T$ lies in the domain of $f$ whenever $x$ lies in the domain of $f$” does not say that $f$ needs to be defined for all $x$. For example,
- The period of a periodic function is not unqiue. In fact, if $f$ is periodic with $T$, it is also periodic with periods $2T$, $3T,\dots$ , or $-T$ and $-2T,\dots$ because

\[

f(x+2T)=f((x+T)+T)=f(x+T)=f(x)

\] and

\[

f(x-T)=f(x-T+T)=f(x),

\] and so on. In general:

If $f$ is periodic with period $T$ then

\[

f(x+nT)=f(x)

\] for every integer $n$.

\[

f(x+nT)=f(x)

\] for every integer $n$.

The smallest **positive** period of a periodic function (if exists) is called the (**fundamental**)** period **of the function.

- A constant function $f(x)\equiv3$ is periodic, and every number $T$ is its period

\[

f(x+T)=f(x)=3.

\] Therefore, there is no smallest period, and hence $f$ does not have a fundamental period.