General Antiperiodicity Property

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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 $f$.


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


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.