Multiplication is Superfunction

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Theorem

The function $f: \C \to \C$, defined as:

$\map f z = z \times c$

is a superfunction for any complex number $c$.


Proof

Define $h: \C \to \C$ by $\map h z = z + c$.

Then:

\(\ds \map h {\map f z}\) \(=\) \(\ds \map h {z \times c}\)
\(\ds \) \(=\) \(\ds z \times c + c\)
\(\ds \) \(=\) \(\ds \paren {z + 1} \times c\)
\(\ds \) \(=\) \(\ds \map f {z + 1}\)

Thus $\map f z = z \times c$ is a superfunction and $\map h z = z + c$ is the corresponding transfer function.

$\blacksquare$