Complex analysis
Undergraduate · Math
Syllabus focus
Topics typically covered
Click a topic for the full text and related unit practice.
Undergraduate Complex Analysis — scope drawn from open complex analysis notes (e.g. Beck et al. A First Course in Complex Analysis) and typical US undergraduate syllabi.
Complex numbers and analytic functions
- Complex numbers, polar form, and Euler's formula — Complex numbers, polar form, and Euler's formula
- Limits, continuity, and differentiability in the complex sense — Limits, continuity, and differentiability in the complex sense
- Cauchy–Riemann equations and harmonic functions — Cauchy–Riemann equations and harmonic functions
- Elementary analytic functions: exp, log, powers, and trig functions — Elementary analytic functions: exp, log, powers, and trig functions
- Branch cuts and multivalued functions — Branch cuts and multivalued functions
Integration and series
- Contour integrals and parametrization — Contour integrals and parametrization
- Cauchy's integral theorem and integral formula — Cauchy's integral theorem and integral formula
- Liouville's theorem and the maximum modulus principle — Liouville's theorem and the maximum modulus principle
- Taylor and Laurent series — Taylor and Laurent series; classification of singularities
- Residue theorem and evaluation of real integrals — Residue theorem and evaluation of real integrals
Conformal mapping and advanced topics
- Conformal maps and the Riemann mapping theorem (statement) — Conformal maps and the Riemann mapping theorem (statement)
- Möbius transformations and the extended complex plane — Möbius transformations and the extended complex plane
- Argument principle and Rouché's theorem (introduction) — Argument principle and Rouché's theorem (introduction)
- Schwarz lemma and normal families (optional capstone) — Schwarz lemma and normal families (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.
- Complex numbers and analytic functions
Complex numbers, polar form, and Euler's formula
Coming soon - Integration and series
Contour integrals and parametrization
Coming soon - Conformal mapping and advanced topics
Conformal maps and the Riemann mapping theorem (statement)
Coming soon
Notes
Undergraduate Complex Analysis — scope drawn from open complex analysis notes (e.g. Beck et al. A First Course in Complex Analysis) and typical US undergraduate syllabi. Topic outline: `content/topics/undergraduate/complex_analysis.json`.