Complex An.·Course
Complex Analysis
Complex analysis: complex numbers, analytic functions, Cauchy integrals, Laurent series, and residues
Part of the Mathematics track — finish it for a verifiable diploma →
§ 01 — Orientation
New here?
What you'll be able to do, who this is for, how long it takes, and where to begin.
By the end, you will be able to
- 01Work with analytic functions.
- 02Use Cauchy's integral theorems.
- 03Expand functions in series and classify singularities.
- 04Compute real integrals with residues.
- 05See why complex analysis is so powerful.
Who this is for
How long it takes
- Quick orientationSkim the opening module and the cheatsheet to get the shape of it.~2 h
- Full read-throughRead every article once, in order.~2 h
- Mastery pathRead, take the quizzes, and space out your reviews.~2 weeks
- APA
Stoa. (2026). Complex Analysis [Online course]. Stoa. https://stoa.school/course/complex-analysis
- MLA
Stoa. “Complex Analysis.” Stoa, 2026, https://stoa.school/course/complex-analysis.
- Chicago
Stoa. “Complex Analysis.” Stoa. Accessed September 1, 2026. https://stoa.school/course/complex-analysis.
§ 02 — Curriculum
5 modules.
Each module is a small unit. Most read in sequence — but a determined reader can begin anywhere.
- M IHolomorphic FunctionsCauchy–Riemann conditions and elementary functions of a complex variable3 articles
18 minBegin → - M IIIntegration in the Complex PlaneCauchy integral, Cauchy’s formula, and Morera’s theorem3 articles
18 minBegin → - M IIILaurent Series and SingularitiesLaurent series expansion, isolated singularities, and their classification3 articles
18 minBegin → - M IVSpecial Methods and FunctionsIntegrals with logarithms, series summation, and special functions3 articles
18 minBegin → - M VEntire Functions and the Laplace TransformWeierstrass theorem, Mittag-Leffler theorem, and applications to ODEs3 articles
18 minBegin →
§ 03 — Learning outcomes
4 outcomes.
Study analyticity, the Cauchy–Riemann equations, and conformal mappings
Apply Cauchy’s theorem and the Cauchy integral formula
Expand functions into Taylor and Laurent series and classify singular points
Evaluate integrals using the residue method and apply it to physical problems
§ 04 — Practices