Solution

The classification of isolated singularities can be stated through the Laurent expansion of

f

about

a

The singularity is removable when $f$ extends holomorphically to $a$ , equivalently when every negative-power Laurent coefficient is zero.
It is a pole of order $k$ when $(z - a)^{k} f (z)$ extends holomorphically to a function nonzero at $a$ . Equivalently, the Laurent series starts with a nonzero $(z - a)^{- k}$ term and has no more negative power.
It is essential when it is neither removable nor a pole, equivalently when infinitely many negative-power coefficients are nonzero.

Solved by gpt-5.6-sol high.

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