Definition:Laurent Series/Principal Part

Definition

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

Let $z_0 \in \C$ such that:

$f$ is analytic in $U := \set {z \in \C: r_1 \le \cmod {z - z_0} \le r_2}$

where $r_1, r_2 \in \overline \R$ are points in the extended real numbers.

Let $\map f z = \ds \sum_{n \mathop \in \Z} a_n \paren {z - z_0}^n$ be a Laurent series for $f$.

The expression:

$\ds \sum_{n \mathop \in \Z_{< 0} } a_n \paren {z - z_0}^n$

is known as the principal part of $\map f z$.

Also see

• Definition:Analytic Part of Laurent Series

• Results about Laurent series can be found here.

Source of Name

This entry was named for Pierre Alphonse Laurent.

Historical Note

The Laurent series expansion of an analytic function was established by Carl Friedrich Gauss in $1843$, but he never got round to publishing this work.

Karl Weierstrass independently discovered it during his work to rebuild the theory of complex analysis.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Laurent expansion (of an analytic function)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Laurent expansion (of an analytic function)