Define: Period & Fundamental Period

Question

How period of a periodic function is different from its fundamental period?

Distinction & similarity between period & fundamental period.

Authenticated definitions of period & fundamental period of a function.

I know period of $sin(x) = 2n\pi$, $n$ belongs to $N$ and fundamental period of $sin(x) = 2\pi$.

But I don't find a good reference for them. Please look @ fundamental period.

Thnx

Answer 1

If, $\exists T \ne 0$ such that $\forall x, f(x + T) = f(x)$, then $f(x)$ is a periodic, and $T$ is a period.

If there exists a smallest positive number $T$ such that $\forall x, f(x + T) = f(x)$, then $T$ is the fundamental period.

Answer 2

A function $f: \mathbb{R} \to \mathbb{R}$ is periodic if there exists a $T \neq 0$ for which $f(x+T) = f(x)$ for all $x\in \mathbb{R}$. Such a $T$ is called a period. If there is a minimum period, $T_0$, then this is called the fundamental period. (Here we mean minimum in absolute value, $|T_0| < |T|$ for all periods $T$ of $f$.)

Note that any number of the form $nT_0$ for $n\in \mathbb{Z}$ is a period for $f$, and that the fundamental period must always divide any other period. That is if $T$ is a period for $f$ then $T/T_0 \in \mathbb{Z}$.

If this were not the case then $T/T_0 = N + r$ for some $N \in \mathbb{Z}$ and $0 < r< 1$, and $$f(x)=f(x+T) = f(x + (T_0 N + T_0 r)) = f(x + T_0 r)$$ thus $T_0 r$ is a period that is smaller than $T_0$ in absolute value which is a contradiction.

Thus if $T$ is a period for $f$ and $T_0$ a fundamental period for $f$, then we have $T=nT_0$ for some $n\in\mathbb{Z}$. This is why $T_0$ is called the fundamental period.

I would add that there are periodic functions without a fundamental period. For example, a constant function is periodic with no fundamental period. Nor does the Dirichlet indicator function $$D(x) = \left\{ \begin{array}{cc} 1& x \in \mathbb{Q}\\ 0 & x \not \in \mathbb{Q}\end{array}\right.$$ have a fundamental period. I believe a sufficient condition for a periodic function to have a fundamental period is that the function be non constant and continuous for at least one point.

Answer 3

There is a U.S. presidential election every 20 years. For example, there were elections in 1880, 1900, 1920, 1940, 1960, 1980, and 2000. So the U.S. elections have a period of 20 years; they recur every 20 years.

But also, U.S. elections recur every four years (for example, 1980, 1984, 1988, 1992, 1996, and 2000) and their 20-year recurrence is a simple result of their recurrence every four years; if something happens every four years, like the elections, then it must also happen every twenty years, like the elections.

The elections have a period of twenty years, because they recur every twenty years. But they have a fundamental period of four years, because they recur every four years, and not any smaller amount.