Pole (Complex Analysis)/Examples

Examples of Poles in the context of Complex Analysis

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.