Complex Variables (one-term) Fall 2019

Links & Materials
Lecture Notes
  • Lecture 1 (Field of complex numbers. Riemann sphere)
  • Lecture 2 (Differentiable maps. Holomorphic functions. Cauchy-Riemann equations)
  • Lecture 3 (Polynomial and rational functions. Power series. Radius of convergence)
  • Lecture 4 (Differentiation of power series. Complex logarithm. Integration along curves)
  • Lecture 5 (Goursat's theorem. Cauchy's theorem. Existence of a primitive in open disk)
  • Lecture 6 (Winding numbers. Cauchy's integral formula)
  • Lecture 7 (Cauchy's integral formula for higher derivatives. Morera's theorem. Removable singularities)
  • Lecture 8 (Taylor's series. Taylor's formula with a remainder)
  • Lecture 9 (Poles. Essential singularities. Casorati-Weierstrass theorem)
  • Lecture 10 (Argument principle. Open mapping theorem. Maximum principle)
  • Lecture 11 (Simple connectivity. General form of Cauchy's theorem)
  • Lecture 12 (Schwarz lemma. Ascutomorphism groups of basic regions)
  • Lecture 13 (Residue calculus)
  • Lecture 14 (Laurent series. Harmonic functions)
  • Lecture 15 (Harmonic functions. Poisson formula. Schwarz theorem)
  • Lecture 16 (Mittag-Lefller theorem)
  • Lecture 17 (Infinite products)
  • Lecture 18 (Gamma function)
  • Lecture 19 (Zeta function)
  • Lecture 20 (Riemann mapping theorem I)
  • Lecture 21 (Riemann mapping theorem II)
  • Lecture 22 (Analytic continuation. Monodromy theorem)
  • Lecture 23 (Applications of confromal mappings)
  • Lecture 24 (Elliptic functions)
Homework Assignments