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Laurent Series

Last Updated : 14 Nov, 2025

The Laurent series is an expansion of a complex function that includes both positive and negative powers of (z βˆ’ z0). It generalizes the Taylor series, allowing representation of functions with singularities. The series is valid in an annular region around a point z0 and consists of two parts:

  • Regular part: terms with non-negative powers (like the Taylor series).
  • Principal part: terms with negative powers, representing the function’s behavior near singularities.

Formula for Laurent Series

Formally, for a function f(z) defined on an annulus A = {z ∈ C : r < |z - z0| < R}, the Laurent series expansion of f(z) about a point z0​ is given by:

Where,

  • an are the coefficients determined by:
  • C is a closed contour around z0 within the region of analyticity.

The series can be split into two parts:

  • Principal Part: The terms with negative powers of
  • Regular Part: The terms with non-negative powers of

Convergence of Laurent Series

Convergence of the Laurent series occurs on an annulus; defined as {z: r1 < | z – z0 | < r2}.

For a Laurent series to converge, the positive and negative degree terms of the power series must converge. This convergence is uniform on compact sets within the annulus. As a result, the series defines a holomorphic function on this region.

Radius of Convergence

Radius of convergence of a power series is the distance within which the series converges to a finite value.

The convergence of a Laurent series depends on the distance from the point z0 and can be divided into three regions:

  • Interior of the Inner Radius (r1): The series converges for |z - z0| < r1 only if an = 0 for all n < 0 , reducing it to a Taylor series.
  • Annulus ( r1 < |z - z0| < r2 ): The series converges in this annular region. This is the most general case for Laurent series, where both positive and negative powers of (z - z0) are present.
  • Exterior of the Outer Radius ( r2 ): The series converges for |z - z_0| > r_2 if a_n = 0 for all n \geq 0 , turning it into a series in negative powers of (z - z_0).

Convergence Criteria

Some of the common criteria for convergence of series are:

Cauchy-Hadamard Theorem: For a series , define:

The series converges in the annulus r1 < |z - z0| < r2 .

Absolute Convergence: If the Laurent series converges at some point z1 , then it converges absolutely at every point z such that |z - z0| = |z1 - z0|.

Uniform Convergence: The series converges uniformly on compact subsets within the annulus r1 < |z - z0| < r2.

Laurent Series vs Taylor Series

Some of the key differences between laurent and taylor series are listed in the following table:

Laurent SeriesTaylor Series
Represents a function as a series with both positive and negative powers of (z - z0)Represents a function as a series with only non-negative powers of (z - z0).
f(z)=βˆ‘βˆžβ€‹n=βˆ’βˆžan​(zβˆ’z0​)nf(z)=βˆ‘βˆžβ€‹n =0an​(z βˆ’z0​)n
Annulus( r1 < |z - z0| < r2 )z - z0

Contains both a principal part (negative powers) and an analytic part (non-negative powers).

Contains only the analytic part (non-negative powers).

Can handle isolated singularities within the annulusCannot handle singularities; requires function to be analytic

where Ξ³ is a contour around z0.

where fn(z0) is the nth derivative at z0.

Always exists for functions with isolated singularities, providing a unique representation.Exists only for analytic functions within the radius of convergence.

Applications of Laurent Series

The applications of the Laurent Series are as follows:

  • Complex analysis: Helps in studying the behavior of complex functions near singularities.
  • Residue calculus: Used for evaluating complex integrals via the residue theorem.
  • Engineering: Applied in signal processing and control theory for stability analysis.
  • Physics: Utilized in quantum mechanics and electrodynamics for potential expansions.
  • Mathematical modeling: Assists in solving differential equations and in the analysis of stability and bifurcation.

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Solved Questions of Laurent Series

Example 1: Find the Laurent series for f(z) = z+1/z around z0 = 0.

Solution:

The given function can be expressed as:

Hence, the Laurent series is:

This series is valid in the region 0<∣z∣<∞.

Example 2: Find the Laurent series for f(z) = z/z2 + 1 around z0 = i.

Solution:

Using partial fractions, the function can be decomposed as:

The term 1/z + i is analytic at z=i and can be expanded using a geometric series:

Therefore, the Laurent series is:

The region of convergence is 0 < ∣zβˆ’i∣ < 2.

Example 3: Find the Laurent series for f(z) = z+1/z around 𝑧0 = v0 and determine the region of convergence.

Solution:

The given function is:

Here, the function is already in the form of a Laurent series. We can write:

The term 1 represents the analytic part with non-negative powers of z.

The term 1/𝑧 represents the principal part with negative powers of z.

Region of Convergence:

The series is valid for 0 < ∣z∣ < ∞.

Example 4: Find the Laurent series for f(z) = z/ z2 + 1 around z0 = i. Identify the region where your answer is valid and the singular part.

Solution:

First, use partial fractions to decompose f(z):

Expanding 1/z+i around z0=i:

The Laurent series is:

Singular Part:

The term 1/z βˆ’ i represents the principal part.

Region of Convergence:

The series is valid for 0<∣zβˆ’i∣<2.

Practice Questions on Laurent Series

Question 1: Determine the Laurent series for f(z)= 1/z(zβˆ’1) around z0=0.

Question 2: Compute the Laurent series for f(z)= 1/z2 + 4 around z0=2i.

Question 3: Find the Laurent series for f(z)= e2/z3 around z0 =0.

Question 4: Find the Laurent series of about z = 0 for ∣z∣ < 2and ∣z∣ > 2 .

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