Cauchy’s Integral Formula

Key idea

Values inside a loop are determined by boundary values.

In depth

For f analytic inside C, f(a) = (1/2πi)∮ f(z)/(z − a) dz. Values inside are determined by values on the boundary, and derivatives follow: f⁽ⁿ⁾(a) = n!/(2πi) ∮ f/(z − a)ⁿ⁺¹ dz.

Worked example

∮_{|z|=2} e^z/(z − 1) dz = 2πi · e.

Formula

Real-world uses

• Reconstructing a function from its boundary values in imaging and fluid flow

Common mistake

Using it when the point a is outside the contour (the integral is 0).

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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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.