Contour Integrals

Key idea

Integrals of complex functions along paths.

In depth

∫_C f(z) dz is computed by parametrising the path z(t) and integrating f(z(t))z′(t) dt. The value generally depends on the path, unless f is analytic in a region containing the loop.

Worked example

∮ dz/z around the unit circle = ∫₀²π (1/e^(it))·ie^(it) dt = 2πi.

Formula

Real-world uses

• Evaluating difficult real integrals in physics and working out the flow around an aircraft wing

Common mistake

Forgetting the z′(t) factor.

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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