Complex Power by Complex Exponential is Analytic

Lemma
Let $\psi, \eta \in \C$ be constant.

Let $\map f z = z^\psi \, \map \exp {-\eta z}$, where:


 * $z^\psi$ denotes $z$ to the power of $\psi$, defined on its principal branch


 * $\map \exp {-\eta z}$ denotes the complex exponential function.

Then $f$ is analytic on any simply connected domain that does not contain the origin nor any points on the negative real axis.

Proof
Let $z$ be written in exponential form:


 * $z = r \map \exp {i \theta}$

where:
 * $r > 0$
 * $\theta \in \hointl {-\pi} \pi$

Let $\psi = a + i b, \eta = c + i d$.

By the definition of $f$:

Define:

Then:

Define:

Then:

We check the Polar Form of Cauchy-Riemann Equations.

As a preliminary:

Then: