Laurent Expansion of Isolated Essential Singularity/Examples
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Examples of Use of Laurent Expansion of Isolated Essential Singularity
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:
\(\ds \map f z\) | \(=\) | \(\ds \sum_{k \mathop \ge 0} \dfrac 1 {k! \, z^k}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1 + \dfrac 1 z + \dfrac 1 {2! \, z^2} + \cdots\) |