Complex Power by Complex Exponential is Analytic

Lemma
Let $q \in \C$ be constant.

Let $f(z) = z^q e^{-z}$, where:


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


 * $e^{-z}$ 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 $q = a + bi$.

Let $z$ be written in polar form:


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

where:
 * $r > 0$, and:


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


 * $\theta \in \left({0\,.\,.\,2\pi}\right)$.

By the definition of $f$,

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

Therefore: