Class Log
Numbers refer to sections of the textbook "Complex analysis" by E. Stein and R. Shakarchi.
Monday 4/30/12. Review problems
Friday 4/27/12. Example applications of general form of Cauchy's theorem.
Wednesday 4/25/12. Characterizations of simple connectedness.
Monday 4/23/12. Proof of general form of Cauchy's theorem.
Friday 4/20/12. The winding number. General form of Cauchy's theorem and Residue theorem.
Wednesday 4/18/12. General solution of Dirichlet problem for the disc (Poisson's formula). More examples of conformal mappings.
Tuesday 4/17/12. The Dirichlet problem for harmonic functions. Basic examples and solution by conformal mapping.
Monday 4/16/12. No class (Patriots' day).
Friday 4/13/12. Harmonic functions: The real and imaginary parts of a holomorphic function are harmonic; A harmonic function on a simply connected domain is the real part of a holomorphic function; Mean value property; Maximum principle for harmonic functions; A harmonic function on a bounded domain is uniquely determined by its boundary values.
Wednesday 4/11/12. Conformal equivalence of the upper half plane and rectangle revisited: description in terms of elliptic functions (cf. Chapter 8, Section 4.5).
Monday 4/9/12. Description of the field of meromorphic functions on a complex torus of dimension 1 in terms of the Weierstrass P-function (Chapter 9, Theorem 1.8). The Schwarz reflection principle (Chapter 2, Section 5.4).
Friday 4/6/12. The Weierstrass P-function (Chapter 9, Section 1.2).
Wednesday 4/4/12. The sum of the residues of a meromorphic 1-form on a compact Riemann surface equals zero. For a meromorphic function on a compact Riemann surface the number of zeroes equals the number of poles (counted with multiplicities).
Monday 4/2/12. Chapter 9, Section 1: Elliptic functions. Complex torus of dimension 1. The notion of a Riemann surface.
Friday 3/30/12. Chapter 8, Section 4: Conformal mappings onto polygons.
Wednesday 3/28/12. Chapter 8, Section 3.3: Proof of Riemann mapping theorem.
Monday 3/26/12. Chapter 8, Section 3.2: Montel's theorem.
Friday 3/16/12. Chapter 2, Section 5.2: A uniform limit of holomorphic functions is holomorphic.
Wednesday 3/14/12. Basic properties of fractional linear transformations: 3-transitivity, preserve cross ratio, preserve the set of circles and lines. Circles and lines in the extended complex plane correspond to circles on the Riemann sphere.
Monday 3/12/12. Review for midterm exam on Wednesday 3/14/12.
Friday 3/9/12. Automorphisms of the Riemann sphere.
Wednesday 3/7/12. Chapter 8, Section 2.2: Automorphisms of the upper half plane.
Monday 3/5/12. Chapter 8, Section 2.1: The Schwarz lemma and automorphisms of the disc.
Friday 3/2/12. Chapter 8, Section 1: Conformal mappings. Examples.
Wednesday 2/29/12. Chapter 3, Section 5 (continued): Cauchy's theorem in simply connected domains, and Section 6: The complex logarithm.
Monday 2/27/12. Chapter 3, Section 5: Homotopic paths, simple connectedness.
Friday 2/24/12. Chapter 3, Section 4: The argument principle, Rouche's theorem, Open mapping theorem, and Maximum principle.
Wednesday 2/22/12. Chapter 3, Section 3 (continued): Behaviour of functions at infinity, the meromorphic functions on the Riemann sphere are the rational functions.
Monday 2/20/12. No class (Presidents' day).
Friday 2/17/12. Chapter 3, Section 3 (continued) : Laurent series. Meromorphic functions. The Riemann sphere.
Wednesday 2/15/12. Chapter 3, Section 3 (continued): Removable singularities, poles, and essential singularities.
Monday 2/13/12. Chapter 3, Section 3: Singularities and meromorphic functions.
Friday 2/10/12. Chapter 3, Section 2.1: Computing real integrals using the residue formula.
Wednesday 2/8/12. Chapter 3, Section 2: The residue formula.
Monday 2/6/12. Chapter 2, Section 4 (continued): Analytic continuation. Chapter 3, Section 1: Zeroes and poles.
Friday 2/3/12. Chapter 2, Section 4 (continued): Holomorphic functions have power series expansions, Liouville's theorem, Fundamental theorem of algebra.
Wednesday 2/1/12. Chapter 2, Section 4: Cauchy's integral formulas.
Monday 1/30/12. Chapter 2, Sections 1 and 2: Goursat's theorem (Cauchy's theorem for triangles), Cauchy's theorem in the disc.
Friday 1/27/12. Chapter 1, Section 3: Integration along curves, statement of Cauchy's theorem and sketch proof using Green's theorem.
Wednesday 1/25/12. Chapter 1, Section 2.3: Power series, radius of convergence.
Monday 1/23/12. Chapter 1, Sections 1.1-2.2: Complex numbers and Holomorphic functions.