Doctorate Complex Analysis Syllabus

Complex Numbers

  • The field
  • Geometry, linear fractional (Möbius) transformations, Riemann sphere
  • Complex functions, analytic functions, Cauchy-Riemann equations
  • Power series, exponential and trigonometric functions
  • Conformality

Cauchy Theory

  • Line integrals
  • Cauchy's theorem and formula, residues, singularities, calculation of integrals, maximum modulus principle
  • Taylor and Laurent series
  • Liouville's theorem, fundamental theorem of algebra, open mapping theorem, Rouche's formula
  • Schwarz's Lemma, Jensen's formula, Weierstrass' theorem

Representation Theorems

  • Partial fractions
  • Infinite products, entire functions, Hadamard's theorem
  • Theorems of Mittag-Leffler, Weierstrass & Runge


  • Harmonic functions, reflection principle
  • Poisson integral
  • Dirichlet problem

Special Functions

  • Gamma function, Riemann function


  • Normal families, Riemann mapping theorem
  • Analytic continuation, monodromy theorem
  • Picard's theorem


Conway: Functions of One Complex Variable
Ahlfors: Complex Analysis
Rudin: Real and Complex Analysis
Hille: Analytic Function Theory
Heins: Complex Function Theory
Churchill: Complex Variables and Applications
Veech: A Second Course in Complex Variables