Laurent Expansion of Isolated Essential Singularity

Theorem

Let $f$ be a complex function with an essential singularity at $z_0 \in \C$.

Let $z_0$ also be an isolated singularity.

Then there exists a Laurent expansion for $f$ with a principal part with infinitely many terms.

Proof

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Examples

Example: $\map \exp {\frac 1 z}$

Let $f$ be the complex function defined as:

$\forall z \in \C \setminus \set 0: \map f z = \map \exp {\dfrac 1 z}$

Then $f$ has an essential singularity at $z = 0$.

The Laurent expansion of $f$ is given by:

\(\ds \map f z\)

\(=\)

\(\ds \sum_{k \mathop \ge 0} \dfrac 1 {k! \, z^k}\)

\(\ds \)

\(=\)

\(\ds 1 + \dfrac 1 z + \dfrac 1 {2! \, z^2} + \cdots\)

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): singular point (singularity): 1.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): singular point (singularity): 1.