Residue Theorem
Integral equals 2πi times the sum of residues.
Key idea
Integral equals 2πi times the sum of residues.
In depth
∮_C f dz = 2πi Σ Res(f, aₖ) for the poles inside C. It reduces integrals to finding residues, which are often easy: for a simple pole, Res = lim (z − a)f(z).
Worked example
∮_{|z|=2} 1/(z² + 1) dz: poles ±i with residues 1/(2i) and −1/(2i), sum 0, integral 0.
Formula
Real-world uses
- Evaluating integrals in physics, signal processing and inverse Laplace transform problems
Common mistake
Including poles outside the contour.
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 Residue Theorem?
Integral equals 2πi times the sum of residues. It is a topic at the expert level in Analysis & Topology.
What is the formula for Residue Theorem?
The key formula is ∮ f dz = 2πi Σ Res.
Where is Residue Theorem used in real life?
Real-world examples: Evaluating integrals in physics, signal processing and inverse Laplace transform problems.
What should I learn before Residue Theorem?
Make sure you are comfortable with Singularities and Residues, the previous topic in Analysis & Topology.
What is a common mistake with Residue Theorem?
Including poles outside the contour.
How can I practise Residue Theorem?
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 Residue Theorem?
Move on to Real Integrals by Residues 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.