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