| Week | Topics | |
|---|---|---|
| 1 | Jan 20 Jan 22 |
Intro; Complex Numbers §1.1 Complex differentiability and Cauchy-Riemann equations §1.2 |
| 2 | Jan 27 Jan 29 |
Conformality; Power series §1.2 Complex line integrals §1.3 |
| 3 | Feb 3 Feb 5 |
Cauchy's theorem via Green's Theorem; Definite integrals §2.3; Goursat's Proof §2.1 Cauchy integral formula and formula for derivatives §2.4 |
| 4 | Feb 10 Feb 12 |
Removing f' continuous assumption; Cauchy inequalities, Liouville's Thm, Fund Thm of Algebra, Analyticity, Isolation of zeros §2.4 Identity principle, Mean Value Property, Maximum Modulus principle, Harmonic functions, Dirichlet problem. |
| 5 | (Feb break) Feb 19 |
Morera's Theorem §2.5.1; Uniform convergence on compact sets §2.5.2; Runge approximation §2.5.5
|
| 6 | Feb 24 Feb 26 |
Homotopy, simply connected, anti-derivatives §3.5; Complex logarithm §3.6 Mercator projection and log; Zeros and poles §3.1; Riemann removable singularity theorem |
| 7 | Mar 3 Mar 5 PRELIM |
Residues §3.1; Residue formula and definite integrals §3.2
|
| 8 | Mar 10 Mar 12 |
Laurent series (see Problem 3.3); Types of singularities §3.3 Argument principle, Rouche's Theorem, Open mapping theorem §3.4 |
| 9 | Mar 17 Mar 19 |
Inverse function theorem, Local Mapping theorem, biholomorphisms Mobius transformations, Automorphisms of C and Riemann sphere |
| 10 | Mar 24 Mar 26 |
Mobius triple transitivity, Disc and upper half plane are biholomorphic, Blaschke maps Schwarz lemma, automorphisms of disc and upper half plane §8.2 |
| Spring Break | Mar 30 - Apr 3 |
|
| 11 | Apr 7 Apr 9 |
Riemann mapping theorem, Equicontinuity and Arzela-Ascoli §8.3 Injectivity preserved under limits, Finish proof of RMT §8.3 |
| 12 | Apr 14 Apr 16 |
Schwarz-Christoffel maps: rectangles §8.4.1 General Schwarz-Christoffel maps §8.4.2; Extension to boundary §8.4.3 |
| 13 | Apr 21 Apr 23 |
Schwarz reflection principle §2.5.4; Schwarz-Christoffel maps are biholomorphisms §8.4.4 Schwarz-Christoffel map to rectangle and reflection §8.4.5; Ellipse arc length and elliptic integrals; Riemann surfaces |
| 14 | Apr 28 Apr 30 |
Peirce quincuncial projection as a doubly periodic meromorphic function; Riemann surface associated to plane algebraic curve; Lattices and definition of Weierstrass ℘ function §9.1 Double periodicity of ℘, Order of elliptic function, zeros of ℘', Differential equation for ℘ §9.1 |
| 15 | May 5 | [THIS MATERIAL NOT ON EXAM] Compactification of cubic curve in projective space; (℘, ℘') gives biholomorphism from torus to cubic; Addition law on cubic; Sum formula for ℘, and translation to ℘^{-1}; Elliptic integral identities -- return to reals
|
| Date TBA | FINAL |