Theoretical / proof-based
Complex analysis · Graduate · Math
Learning objectives from the Complex analysis syllabus, grouped by unit. Click an objective for study materials.
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)
Learning objectives
Click an objective for study materials.
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)
Definitions and structure
- 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
Proofs and reasoning
- Riemann mapping theorem (proof outline) — Riemann mapping theorem (proof outline)
- Elliptic functions (introduction) — Elliptic functions (introduction)
- Holomorphic functions of several variables — Holomorphic functions of several variables
- Hartogs phenomenon — Hartogs phenomenon
- Domains of holomorphy (overview) — Domains of holomorphy (overview)
Abstraction and generalization
- 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: definitions and examples — Riemann surfaces: definitions and examples
- Uniformization theorem (statement) — Uniformization theorem (statement)
Modeling and computation
- Apply review of analytic functions in engineering contexts — Apply review of analytic functions in engineering contexts
- Apply open mapping theorems in engineering contexts — Apply open mapping theorems in engineering contexts
- Apply singularities and residues in engineering contexts — Apply singularities and residues in engineering contexts
- Apply the argument principle in engineering contexts — Apply the argument principle in engineering contexts
- Apply normal families and montel's theorem in engineering contexts — Apply normal families and montel's theorem in engineering contexts
Data and technology
- Use software to explore complex analysis problems numerically — Use software to explore complex analysis problems numerically
- Interpret computational results against analytic predictions — Interpret computational results against analytic predictions
- Build spreadsheets or scripts for routine calculations — Build spreadsheets or scripts for routine calculations
- Visualize functions, fields, or datasets tied to course topics — Visualize functions, fields, or datasets tied to course topics
- Connect course methods to lab, industry, or research workflows — Connect course methods to lab, industry, or research workflows
Problem-solving practice
- Riemann mapping theorem (proof outline) — Riemann mapping theorem (proof outline)
- Elliptic functions (introduction) — Elliptic functions (introduction)
- Holomorphic functions of several variables — Holomorphic functions of several variables
- Hartogs phenomenon — Hartogs phenomenon
- Domains of holomorphy (overview) — Domains of holomorphy (overview)
Multi-Unit Problems
Course-level sets that combine skills across study units (coming soon).
Browse Multi-Unit ProblemsWhat each unit includes
Open a unit below for full materials. Typical resources:
- Study guide
- Exam Strategy
- Common Mistakes
- Worksheets
- Word problems
- Mixed Practice
- Multi-Unit Problems
- Review
- Practice test
- Answer key
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
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