General Antiperiodicity Property

Theorem

Let $f: X \to X$ be an antiperiodic function, where $X$ is either $\R$ or $\C$.

Let $L$ be an antiperiodic element of $f$.

Let $n \in \Z$ be an integer.

If $n$ is even, then $n L$ is a periodic element of $f$.

If $n$ is odd, then $n L$ is an antiperiodic element of $f$.

Proof

Suppose that $X = \C$.

Case 1

If $n$ is even, then:

\(\ds \map f {x + n L}\)

\(=\)

\(\ds \map f {x + \paren {2 k} L}\)

for some $k \in \Z$

\(\ds \)

\(=\)

\(\ds \map f {x + \paren {k 2} L}\)

Complex Multiplication is Commutative

\(\ds \)

\(=\)

\(\ds \map f {x + k \paren {2 L} }\)

Complex Multiplication is Associative

\(\ds \)

\(=\)

\(\ds \map f x\)

by Double of Antiperiodic Element is Periodic and the General Periodicity Property

$\Box$

Case 2

If $n$ is odd, then:

\(\ds \map f {x + n L}\)

\(=\)

\(\ds \map f {x + \paren {2 k + 1} L}\)

for some $k \in \Z$

\(\ds \)

\(=\)

\(\ds \map f {x + \paren {\paren {2 k} L + L} }\)

Complex Multiplication Distributes over Addition

\(\ds \)

\(=\)

\(\ds \map f {\paren {x + \paren {2 k} L} + L}\)

Complex Addition is Associative

\(\ds \)

\(=\)

\(\ds -\map f {x + \paren {2 k} L}\)

\(\ds \)

\(=\)

\(\ds -\map f x\)

by Case 1

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

$\blacksquare$