Compute Laurent series expansions with step-by-step solutions. Analyze complex functions, singularities, residues, and annulus regions using professional complex analysis tools. Complete residue calculation and singularity analysis—100% free.
General Laurent series Complete coefficient calculation
∫
Residue Focus
Residue calculation Contour integration
⚡
Singularity Analysis
Pole identification Essential singularity
⭕
Annulus Region
Region of convergence Visual analysis
Number of Terms
10
Precision ℹ️
Decimal precision for coefficient calculation. Higher precision gives more accurate results but takes longer to compute.
6 decimals
Display Format
Complex form
Laurent Series Results
Laurent Series Expansion:
f(z) = ∑n=-∞∞ aₙ(z - z₀)ⁿ
Term n
Coefficient aₙ
Term Value
Contribution
Complex Plane with Singularities
Annulus Region
z₀
● Inner radius: r = 0
● Outer radius: R = 1
Region: r < |z - z₀| < R
Coefficient Magnitudes
Residue Analysis
Step-by-Step Solution
1
Original Function
f(z) = 1/(z - 1)
This is a complex rational function with a simple pole at z = 1. The function is analytic everywhere except at this singularity.
2
Identify Singularities
Singularities: z = 1
The denominator z - 1 = 0 gives z = 1 as the only singularity. Since the numerator is constant and nonzero, this is a simple pole (order 1).
3
Choose Annulus Region
0 < |z| < 1
For expansion about z₀ = 0, the nearest singularity is at |z| = 1. Thus, the Laurent series converges in the annulus 0 < |z| < 1 (punctured disk).
4
Compute Laurent Series
f(z) = -∑n=0∞ zⁿ = -(1 + z + z² + z³ + ...)
Using geometric series: 1/(1 - z) = ∑ zⁿ for |z| < 1. Since we have 1/(z-1) = -1/(1-z), we get f(z) = -∑ zⁿ. This is actually a Taylor series (no negative powers) because we're expanding in a region that doesn't contain the pole.
5
Residue Calculation
Res(f, 1) = 1
For a simple pole at z = a, the residue is lim_{z→a} (z-a)f(z). Here, lim_{z→1} (z-1)×1/(z-1) = 1. The residue is the coefficient a_{-1} in the Laurent series about z = 1.
Understanding Laurent Series
The Laurent series is a fundamental concept in complex analysis that generalizes the Taylor series to functions with singularities. It represents a complex function as a power series that includes both positive and negative powers, allowing analysis of functions near their singular points.
🔍 What Makes Laurent Series Special?
Unlike Taylor series which only converge in disks and require analytic functions, Laurent series:
Converge in annular regions (rings) centered at expansion points
Include negative powers to handle singularities
Classify singularities based on coefficient patterns
Enable residue calculation for contour integration
Handle functions that aren't analytic everywhere
📊 Laurent Series vs Taylor Series
Feature
Taylor Series
Laurent Series
Powers
Only non-negative: (z-z₀)ⁿ, n ≥ 0
All integers: (z-z₀)ⁿ, n ∈ ℤ
Convergence Region
Disk centered at z₀
Annulus (ring) centered at z₀
Function Requirement
Analytic at z₀
May have isolated singularity at z₀
Applications
Local approximation of analytic functions
Analysis near singularities, residue calculation
🎯 Types of Isolated Singularities
Pole (Order m)
Finite number of negative power terms. Highest negative power is -m. Residue is coefficient of (z-z₀)⁻¹.
Essential Singularity
Infinite number of negative power terms. Function behaves wildly near singularity (Picard's Theorem).
Removable Singularity
No negative power terms (all aₙ = 0 for n < 0). Function can be "redefined" to be analytic.
🔬 The Residue Theorem
The residue theorem is one of the most powerful tools in complex analysis:
∮C f(z) dz = 2πi × ∑ Res(f, zₖ)
The integral of f(z) around a closed contour C equals 2πi times the sum of residues at singularities inside C.
📈 Real-World Applications
Electromagnetic Theory: Solving Laplace's equation in annular regions
Quantum Mechanics: Partial wave analysis in scattering theory
Fluid Dynamics: Potential flow around obstacles
Signal Processing: Z-transform analysis (discrete-time Fourier transform)
Control Theory: Stability analysis of linear systems
Number Theory: Analytic continuation of zeta functions
💡 Pro Tip: Choosing the Right Annulus
The Laurent series expansion depends critically on the annulus region. For a function with singularities at distances r₁, r₂, r₃,... from z₀ (sorted), valid annuli are:
1. 0 < |z-z₀| < r₁ (if z₀ is a singularity)
2. r₁ < |z-z₀| < r₂
3. r₂ < |z-z₀| < r₃
... and so on.
Different annuli give different Laurent expansions for the same function!
Frequently Asked Questions
What's the difference between Laurent series and Taylor series?
+
Taylor series contain only non-negative powers (z-z₀)ⁿ and converge in disks centered at z₀. They require the function to be analytic at z₀. Laurent series contain both positive and negative powers and converge in annular regions (rings). They can represent functions with isolated singularities at z₀. Every Taylor series is a Laurent series with all negative coefficients equal to zero.
How do I find the residue from a Laurent series?
+
The residue of a function at an isolated singularity z₀ is simply the coefficient a₋₁ of the (z-z₀)⁻¹ term in its Laurent series expansion about z₀. For simple poles, residue = lim_{z→z₀} (z-z₀)f(z). For poles of order m > 1, use the formula: Res(f,z₀) = 1/(m-1)! lim_{z→z₀} d^{m-1}/dz^{m-1} [(z-z₀)ᵐf(z)].
Can a function have multiple different Laurent series?
+
Yes! A function can have different Laurent series expansions about the same point z₀, depending on the annulus chosen. For example, 1/(z(z-1)) has three different Laurent series about z₀=0: one for 0<|z|<1, one for 1<|z|<∞, and one for |z|>1. Each expansion is valid in its specific annulus and has different coefficients.
What is an essential singularity?
+
An essential singularity is an isolated singularity that is neither removable nor a pole. In the Laurent series, it has infinitely many negative power terms. Near an essential singularity, the function exhibits wild behavior: according to Picard's Great Theorem, in every neighborhood of an essential singularity, f(z) takes on every complex value (with at most one exception) infinitely often. Example: e^{1/z} has an essential singularity at z=0.
How does the Residue Theorem help evaluate real integrals?
+
The Residue Theorem transforms difficult real integrals into easier residue calculations. Common applications include: 1) Trigonometric integrals ∫₀²π R(cosθ, sinθ)dθ via z=e^{iθ}, 2) Improper integrals ∫_{-∞}^{∞} f(x)dx by closing contour in upper/lower half-plane, 3) Fourier integrals ∫_{-∞}^{∞} f(x)e^{iax}dx using Jordan's Lemma. The key is to identify appropriate contours and calculate residues at enclosed poles.
What's the practical importance of Laurent series in engineering?
+
Laurent series are crucial in: 1) Control Systems - Analyzing stability via poles in the complex plane, 2) Signal Processing - Z-transforms for discrete-time systems, 3) Electromagnetics - Solving boundary value problems in annular regions, 4) Fluid Mechanics - Potential flow with sources/sinks, 5) Quantum Scattering - Partial wave expansions. They provide the mathematical foundation for understanding system behavior near singular points.
Related Calculators
Explore these other powerful mathematical tools on DerivativeCalculus.com: