Definition:Order of Pole/Simple Pole

Theorem
Let $f: \C \to \C$ be a complex function.

Let $x \in U \subset \C$ be such that $f$ is analytic in $U \setminus \left\{{x}\right\}$, with a pole at $x$.

Let the order of the pole at $x$ be $1$.

Then $x$ is a simple pole.