Laurent Expansion of Function about Pole

Theorem

Let $f$ be a complex function with a pole at $z_0 \in \C$ of order $k$.

The Laurent expansion of $f$ about $z_0$ can be expressed as:

$\map f z = \ds \sum_{n \mathop = -k}^\infty a_n \paren {z - z_0}^n$

Proof

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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.