Complex analysis
Graduate · Math
Syllabus focus
Topics typically covered
Click a topic for the full text and related unit practice.
Graduate Complex Analysis — scope drawn from open graduate complex analysis texts and typical US graduate complex analysis syllabi.
Holomorphic and meromorphic functions
- Review of analytic functions — Review of analytic functions
- Open mapping theorems — Open mapping theorems
- Singularities and residues — Singularities and residues
- The argument principle — The argument principle
- Normal families and Montel's theorem — Normal families and Montel's theorem
- Riemann mapping theorem (proof outline) — Riemann mapping theorem (proof outline)
- Elliptic functions (introduction) — Elliptic functions (introduction)
Several complex variables (introduction)
- Holomorphic functions of several variables — Holomorphic functions of several variables
- Hartogs phenomenon — Hartogs phenomenon
- Domains of holomorphy (overview) — Domains of holomorphy (overview)
- Plurisubharmonic functions (introduction) — Plurisubharmonic functions (introduction)
- Cauchy–Riemann equations in higher dimensions — Cauchy–Riemann equations in higher dimensions
- Stein manifolds (preview) — Stein manifolds (preview)
Riemann surfaces and advanced topics
- Riemann surfaces: definitions and examples — Riemann surfaces: definitions and examples
- Uniformization theorem (statement) — Uniformization theorem (statement)
- Divisors and line bundles — Divisors and line bundles
- Sheaves (introduction) — Sheaves (introduction)
- Hodge theory on Riemann surfaces (optional) — Hodge theory on Riemann surfaces (optional)
- Nevanlinna theory (optional capstone) — Nevanlinna theory (optional capstone)
Study units
Each unit includes a study guide, worksheets, review, practice test, and answer key. One unit is free; subscribe for the full class.
- Holomorphic and meromorphic functions
Review of analytic functions
Coming soon - Several complex variables (introduction)
Holomorphic functions of several variables
Coming soon - Riemann surfaces and advanced topics
Riemann surfaces: definitions and examples
Coming soon
Notes
Graduate Complex Analysis — scope drawn from open graduate complex analysis texts and typical US graduate complex analysis syllabi. Topic outline: `content/topics/graduate/complex_analysis.json`.