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

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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:

\(\ds \map f z\) \(=\) \(\ds \sum_{k \mathop \ge 0} \dfrac 1 {k! \, z^k}\)
\(\ds \) \(=\) \(\ds 1 + \dfrac 1 z + \dfrac 1 {2! \, z^2} + \cdots\)


Proof

Let $U \subseteq \C$ be an open set.

Let $f : U \to \C$ be a complex function.


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So $f$ has a point at $z=0 $, for which $f$ is not analytic.

We have that $f$ has a singular point at $z = 0$.

$f$ cannot be extended to a holomorphic function $f: U \to \C$

So $f$ has a singular point at $z = 0$ which is not a removable singularity.

We have that $f$ has a singular point at $z = 0$ which is not a pole.

Therefore $f$ has a singular point which is neither a removable singularity nor a pole at $z = 0$

Hence the result by definition of removable singularity.

Sources

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