Definition:Isolated Singularity/Pole
Definition
Let $U$ be an open subset of a Riemann surface.
Let $z_0 \in U$.
Let $f: U \setminus \set {z_0} \to \C$ be a holomorphic function.
Definition 1
Let $z_0$ be an isolated singularity of $f$.
Then $z_0$ is a pole if and only if:
- $\ds \lim_{z \mathop \to z_0} \cmod {\map f z} \to \infty$
Definition 2
Let $z_0$ be an isolated singularity of $f$.
Then $z_0$ is a pole if and only if $f$ can be written in the form:
- $\map f z = \dfrac {\map \phi z} {\paren {z - z_0}^k}$
where:
- $\phi$ is analytic at $z_0$
- $\map \phi {z_0} \ne 0$
- $k \in \Z$ such that $k \ge 1$.
Order of Pole
Let $z_0 \in U \subset \C$ be such that $f$ is holomorphic in $U \setminus \set {z_0}$, with a pole at $z_0$.
By Existence of Laurent Series there is a series:
- $\ds \map f z = \sum_{n \mathop \ge n_0}^\infty a_j \paren {z - z_0}^n$
The order of the pole at $z_0$ is defined to be $\size {n_0} > 0$.
Simple Pole
Let $z_0 \in U \subset \C$ be such that $f$ is holomorphic in $U \setminus \set {z_0}$, with a pole at $z_0$.
Let the order of the pole at $z_0$ be $1$.
Then $z_0$ is a simple pole.
Examples
Example: $\frac 1 {\paren {z - 3}^2 \paren {z + 1} }$
Let $f$ be the complex function:
- $\forall z \in \C \setminus \set {-1, 3}: \map f z = \dfrac 1 {\paren {z - 3}^2 \paren {z + 1} }$
Then $f$ has:
- a pole of order $2$ at $z = 3$
- a simple pole at $z = -1$.
Example: $\frac {z^2 + 1} z$
Let $f$ be the complex function:
- $\forall z \in \C \setminus \set 0: \map f z = \dfrac {z^2 + 1} z$
Then $f$ has:
- a simple pole at $z = 0$.
Example: $\frac {\sin z} {\paren {z - \pi} \paren {z - 2}^4}$
Let $f$ be the complex function:
- $\forall z \in \C \setminus \set 2: \map f z = \dfrac {\sin z} {\paren {z - \pi} \paren {z - 2}^4}$
Then $f$ has a pole of order $4$ at $z = 2$
Note that at $z = \pi$ there is no pole as the numerator is $0$ at that point.
Also see
- Results about poles in the context of Complex Analysis can be found here.