Complex Power by Complex Exponential is Analytic

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

Let $f \left({z}\right) = z^\psi e^{-\eta z}$, where:


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


 * $e^{-\eta z}$ is the complex exponential.

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 polar form:


 * $z = r e^{i \theta}$

where:
 * $r > 0$, and:


 * $\theta \in \left({-\pi \,.\,.\, \pi}\right)$

Let $\psi = a + ib, \eta = c + id$.

By the definition of $f$:

Define:

Then:

Define:

Then:

Then:

We check the polar form of the Cauchy-Riemann Equations.

As a preliminary:

Then: