Laurent Expansion of Isolated Essential Singularity/Examples/Exponential of Reciprocal

Examples of Use of Laurent Expansion of Isolated Essential Singularity

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.