Singularities and Residues
Poles, removable and essential singularities.
Key idea
Poles, removable and essential singularities.
In depth
An isolated singularity is removable (limit exists), a pole (blows up like 1/(z − a)ⁿ) or essential (wild behaviour). The residue is the coefficient a₋₁ of the Laurent series, measuring the integral around the singularity.
Worked example
f(z) = 1/(z − 2) has a simple pole at 2 with residue 1. sin z/z has a removable singularity at 0.
Real-world uses
- Calculating the effects of poles in systems and the flow around sharp edges
Common mistake
Mixing up the order of a pole with its residue.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Singularities and Residues?
Poles, removable and essential singularities. It is a topic at the expert level in Analysis & Topology.
Where is Singularities and Residues used in real life?
Real-world examples: Calculating the effects of poles in systems and the flow around sharp edges.
What should I learn before Singularities and Residues?
Make sure you are comfortable with Laurent Series, the previous topic in Analysis & Topology.
What is a common mistake with Singularities and Residues?
Mixing up the order of a pole with its residue.
How can I practise Singularities and Residues?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Singularities and Residues?
Move on to Residue Theorem next.
Related expert topics in Analysis & Topology
- Axioms of the Real Numbers – ℝ is a complete ordered field.
- Completeness and Supremum – Every nonempty bounded-above set has a least upper bound.
- Archimedean Property – For any real x there is an integer n > x.
- Density of Rationals – Between any two reals there is a rational.
- Sequences and Convergence (ε–N) – aₙ → L if for every ε>0 eventually |aₙ − L| < ε.
- Limit Theorems for Sequences – Algebra of limits and squeeze theorem.