#### 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

- Homework 1 (due Tuesday, September 17)
- Homework 2 (due Tuesday, September 24)
- Homework 3 (due Tuesday, October 1)
- Homework 4 (due Tuesday, October 8)
- Homework 5 (due Thursday, October 17)
- Homework 6 (due Tuesday, October 27)
- Homework 7 (due Tuesday, November 5)
- Homework 8 (due Tuesday, November 12)
- Homework 9 (due Tuesday, November 19)
- Homework 10 (due Tuesday, November 26)
- Homework 11 (due Tuesday, December 3)