General Periodicity Property

Theorem

Let $f: X \to X$ be a periodic function, where $X$ is either the set of real numbers $\R$ or the set of complex numbers $\C$.

Let $L$ be a periodic element of $f$.

Then:

$\forall n \in \Z: \forall x \in X: \map f x = \map f {x + n L}$

That is, after every distance $L$, the function $f$ repeats itself.

Corollary

Let $f: \R \to \R$ be a real function.

Then $L$ is a periodic element of $f$ if and only if:

$\forall x \in \R: \map f {x \bmod L} = \map f x$

where $x \bmod L$ is the modulo operation.

Proof

Let $X = \mathbb C$.

There are two cases to consider: either $n$ is not negative, or it is negative.

Since the Natural Numbers are Non-Negative Integers, the case where $n \ge 0$ will be proved using induction.

Case 1

Basis for the Induction

The case for which $n = 0$ is trivial, because:

$x + 0 \cdot L = x$

Induction Hypothesis

For some $n \in \Z_{\ge 0}$, suppose that:

$\map f x = \map f {x + n L}$

Induction Step

For the induction step, let $n \to n + 1$.

Then:

 $\displaystyle \map f {x + \paren {n + 1} L}$ $=$ $\displaystyle \map f {x + \paren {L + n L} }$ $\displaystyle$ $=$ $\displaystyle \map f {\paren {x + L} + n L}$ Complex Addition is Associative $\displaystyle$ $=$ $\displaystyle \map f {x + L}$ Induction Hypothesis $\displaystyle$ $=$ $\displaystyle \map f x$ Basis for the Induction

Case 2

If $n < 0$, then:

 $\displaystyle \map f {x + n L}$ $=$ $\displaystyle \map f {\paren {x + n L} - n L}$ Negative of Negative Number is Positive and Case 1 $\displaystyle$ $=$ $\displaystyle \map f {x + \paren {n L - n L} }$ Complex Addition is Associative $\displaystyle$ $=$ $\displaystyle \map f x$

Combining the results above, it is seen that for all $n \in \Z$:

$\map f x = \map f {x + n L}$

The proof for when $X = \R$ is nearly identical to the above proof.

$\blacksquare$