# 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

#### Harmonicity

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

#### Special Functions

• Gamma function, Riemann function

#### Miscellaneous

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

#### References

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