Exponential Function is Superfunction

Theorem

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

$\map f z = c^z$

is a superfunction for any complex number $c$.

Proof

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

Then:

\(\ds \map h {\map f z}\)

\(=\)

\(\ds \map h {c^z}\)

\(\ds \)

\(=\)

\(\ds c^z \times c\)

\(\ds \)

\(=\)

\(\ds c^{z + 1}\)

\(\ds \)

\(=\)

\(\ds \map f {z + 1}\)

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

$\blacksquare$