Class Log
References are to the textbook "Complex analysis" by E. Stein and R. Shakarchi.
Monday 4/30/18. Harmonic functions (continued): More examples of solutions to the Dirichlet problem. General solution for the disc. Solution for domain with piecewise smooth boundary by conformal mapping.
Friday 4/27/18. Harmonic functions (continued): Mean value property, Maximum principle, Uniqueness of solutions to the Dirichlet problem for bounded domains. Examples.
Wednesday 4/25/18. More applications of Cauchy's theorem and the residue theorem. Harmonic functions: the real and imaginary parts of a holomorphic function are harmonic functions. A harmonic function on a simply connected domain is the real part of a holomorphic function.
Monday 4/23/18. Characterization of simply connected domains (continued). Sample applications of the general form of Cauchy's theorem and the residue theorem.
Friday 4/20/18. Proof of general form of Cauchy's theorem. Characterizations of simply connected open subsets of the plane. Reference: Ahlfors, Complex Analysis, Section 4.2; Stein and Shakarchi, Appendix B.
Wednesday 4/18/18. Winding number. Statement of general form of Cauchy's theorem and Residue theorem. Reference: Lang, Complex Analysis, Chapter IV.
Tuesday 4/17/18. Chapter 9, Section 1.2. Description of elliptic functions in terms of Weierstrass P-function. Embedding of complex torus in complex projective plane as cubic complex curve via P and P'.
Friday 4/13/18. Chapter 9, Section 1.2. The Weierstrass P-function.
Wednesday 4/11/18. Chapter 9, Sections 1 and 1.1. Elliptic functions.
Monday 4/9/18. Chapter 8, Section 3.3, Proof of Riemann mapping theorem.
Friday 4/6/18. Chapter 8, Section 3.2, Montel's theorem (continued).
Wednesday 4/4/18. Chapter 8, Section 3.2. Montel's theorem.
Monday 4/2/18. Chapter 8, Section 2. Automorphisms of the upper half plane. Chapter 2, Section 5.2. If a sequence of holomorphic functions converges uniformly on compact sets to a function f, then f is holomorphic (and the derivatives of the sequence of functions converge uniformly on compact sets to the derivative of f).
Friday 3/30/18. Chapter 8, Section 1. A holomorphic bijection has nowhere zero derivative and so holomorphic inverse. Chapter 8, Section 2. Schwarz Lemma. Automorphisms of the unit disc.
Wednesday 3/28/18. The cross ratio. Mobius transformations preserve the cross ratio. The cross ratio is real iff the 4 points lie on a circle or line. Mobius transformations map circles or lines to circles or lines.
Monday 3/26/18. Chapter 8, Section 1. Further examples of holomorphic bijections. Mobius transformations. 3-transitivity.
Friday 3/23/18. Chapter 8, Section 1. Holomorphic bijection from the upper half plane to the unit disc. Interpretation in terms of stereographic projection. More examples of holomorphic bijections.
Wednesday 3/21/18. Chapter 3, Section 5. Existence of primitives for holomorphic functions on simply connected domains. Chapter 3, Section 6. Example: the complex logarithm. Chapter 8, introduction. Statement of the Riemann mapping theorem.
Monday 3/19/18. Chapter 3, Section 5. Homotopy of curves (continued). Simply connected domains.
Friday 3/9/18. More discussion of maximum principle. Chapter 3, Section 5. Homotopy of curves.
Wednesday 3/7/18. Chapter 3, Section 4. Rouché's theorem. Open mapping theorem. Maximum principle.
Monday 3/5/18. Residue theorem applied in chart at infinity. Chapter 3, Section 4. Argument principle.
Friday 3/2/18. Chapter 3, Section 3. f is meromorphic on the Riemann sphere iff f is a rational function. The automorphisms of the Riemann sphere are the Mobius transformations.
Wednesday 2/28/18. Chapter 3, Section 3. Meromorphic functions are holomorphic maps to the Riemann sphere. Singularities at infinity. Examples.
Monday 2/26/18. Chapter 3, Section 3. The Riemann sphere / extended complex plane. Riemann surfaces. Complex tori. (A nice introductory reference for Riemann surfaces is R. Miranda, Algebraic curves and Riemann surfaces.)
Friday 2/23/18. Chapter 3, Section 3. Essential singularities. Casorati-Weierstrass theorem. Laurent expansions (cf. p. 109, Problem 3).
Wednesday 2/21/18. Chapter 3, Section 3. Riemann's removable singularity theorem.
Monday 2/19/18. Presidents' day.
Friday 2/16/18. Chapter 3, Section 2. Evaluation of real integrals via residues (continued).
Wednesday 2/14/18. Chapter 3, Section 2. Computation of residues. Evaluation of real integrals via residues.
Monday 2/12/18. Chapter 3, Sections 1 and 2. Zeroes and poles of holomorphic functions. Residues. Cauchy's residue theorem.
Friday 2/9/18. Chapter 2, Section 4. Liouville's theorem. Fundamental theorem of algebra. Zeroes of holomorphic functions are isolated.
Wednesday 2/7/18. Snow day.
Monday 2/5/18. Chapter 2, Section 4. Cauchy's integral formulas. Cauchy inequalities. Existence of power series expansions for holomorphic functions.
Friday 2/2/18. Chapter 2, Sections 1 and 2. Goursat's theorem (Cauchy's theorem for a triangle). Existence of local primitives (=> Cauchy's theorem for the disc).
Wednesday 1/31/18. Chapter 1 Section 3; Chapter 2, Section 1. Fundamental theorem of calculus. Sketch proof of Cauchy's theorem (via Green's / Stokes' theorem).
Monday 1/29/18. Chapter 1, Section 3. Integration along curves.
Friday 1/26/18. Chapter 1, Section 2.3. Power series, radius of convergence. Complex exponential and trigonometric functions.
Wednesday 1/24/18. Chapter 1, Section 2.2. Cauchy--Riemann Equations. Conformal property.
Monday 1/22/18. Chapter 1, Sections 1.1-2.2: Complex numbers and Holomorphic functions.