Lecture |
Date |
Topics |

1 | 1/13/97 | algebra and analytic geometry of complex numbers, the Riemann sphere |

2 | 1/15/97 | power series and their convergence |

3 | 1/17/97 | differentiability and analyticity, differentiability of power series |

4 | 1/20/97 | elementary functions in the complex domain, mapping properties |

5 | 1/22/97 | conformality |

6 | 1/24/97 | Cauchy-Riemann equations, partial z and partial z bar, harmonic functions, existence of harmonic conjugates |

7 | 1/27/97 | Mobius transformations |

8 | 1/29/97 | Path integrals, Cauchy's formula and theorem for a disc, analytic implies power series at each point, |

9 | 1/31/97 | Cauchy's estimate, Maximum Modulus Principle, Liouville's Theorem, Fundamental Theorem of Algebra, isolation and finite multiplicity of roots |

10 | 2/3/97 | homotopy of paths and Cauchy's theorem, motivation for winding numbers |

11 | 2/5/97 | winding numbers and Cauchy's integral formula, zero counting, Open Mapping Theorem |

12 | 2/7/97 | Morera's Theorem, theorem on existence of primitives, Goursat's Theorem, isolated singularities |

13 | 2/10/97 | poles, essential singularities, Casorati-Weierstrass Theorem, Laurent expansion |

14 | 2/12/97 | The Residue Theorem, residue integrals |

15 | 2/14/97 | Argument Principle, Rouche's Theorem, Weierstrass Theorem |

16 | 2/17/97 | Hurwitz's Theorem and Montel's Theorem |

17 | 2/19/97 | Schwarz's Lemma and applications |

18 | 2/21/97 | The Riemann Mapping Theorem |

19 | 2/24/97 | Complements on conformal mapping, mean value property and maximum priniciple harmonic functions on a disk |

20 | 2/26/97 | digression: statement of the little and great Picard theorems and integrability of functions with isolated singularities, Poisson kernel, Dirichlet problem on a disk |

21 | 2/28/97 | Harnack's inequality and principle, subharmonic and superharmonic functions |

22 | 3/3/97 | Perron's approach to the solution of the Dirichlet problem |

23 | 3/5/97 | Schwarz reflection principle, some loose ends, midsemester evaluation |

24 | 3/7/97 | Midterm Exam |

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