Cauchy’s Integral Theorem
Integral around a closed loop of a holomorphic function is 0.
Key idea
Integral around a closed loop of a holomorphic function is 0.
In depth
If f is analytic on and inside a simple closed contour, then ∮f dz = 0. It follows from Green’s theorem and the Cauchy–Riemann equations. Integrals between two points are path-independent in simply connected domains.
Worked example
∮ z² dz = 0 for any closed loop.
Formula
Real-world uses
- Explaining why closed-loop integrals vanish
- Used in electromagnetic theory
Common mistake
Applying it when f has a singularity inside 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 Cauchy’s Integral Theorem?
Integral around a closed loop of a holomorphic function is 0. It is a topic at the expert level in Analysis & Topology.
What is the formula for Cauchy’s Integral Theorem?
The key formula is ∮ f dz = 0.
Where is Cauchy’s Integral Theorem used in real life?
Real-world examples: Explaining why closed-loop integrals vanish; Used in electromagnetic theory.
What should I learn before Cauchy’s Integral Theorem?
Make sure you are comfortable with Contour Integrals, the previous topic in Analysis & Topology.
What is a common mistake with Cauchy’s Integral Theorem?
Applying it when f has a singularity inside the contour.
How can I practise Cauchy’s Integral 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 Cauchy’s Integral Theorem?
Move on to Cauchy’s Integral Formula 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.