Definition:Laurent Series

Not to be confused with Definition:Formal Laurent Series.

Definition

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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.

A Laurent series is a summation:

$\forall z \in \C: r_1 < \cmod {z - z_0} < r_2: \map f z = \ds \sum_{n \mathop \in \Z} a_n \paren {z - z_0}^n$

where:

$a_n = \dfrac 1 {2 \pi i} \ds \int_C \map f z \paren {z - z_0}^{n + 1} \rd z$

$C$ is a circle with center $z_0$ and radius $r$ for $r_1 < r < r_2$

$\ds \int_C \map f z \paren {z - z_0}^{n + 1} \rd z$ is the contour integral over $C$.

such that the summation converges to $f$ in $U$.

Principal Part

The expression:

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

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

Analytic Part

The expression:

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

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

Also known as

A Laurent series is also commonly known as a Laurent expansion.

Also see

• Definition:Complex Singular Point

• 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)