Update (4/27/2012): By this point in the course, we are not really following the syllabus below very closely. The notes listed do not have that much resemblance to the corresponding descriptions anymore, so all the notes are listed at the bottom of this page with more descriptive naems.

The following tables lists the proposed schedule for the class. All chapter sections
refer to *Complex Analysis* by Stein and Shakarchi.

Week | Date | Chapter | Topic | Notes |
---|---|---|---|---|

1 | 3/26 | 1.1 | Introduction | Class 1 |

3/28 | 1.1 | Arithmetic with complex numbers | Class 2 | |

3/30 | 1.1 | Topology and limits of complex numbers. | Class 3 | |

2 | 4/2 | Real analysis/calculus review: differentiability classes of functions, derivatives as linear transformations | Classes 4, 5, 6, 7 | |

4/4 | 1.2 | Examples of complex-valued functions | ||

4/6 | 1.2 | The derivative of a complex function, the Cauchy-Riemann equations | ||

3 | 4/9 | 1.2 | The Cauchy-Riemann equations and related ideas | |

4/11 | 1.3 | Power series | Class 8, 9 | |

4/13 | 1.3 | Contour integration | Class 10, 11 | |

4 | 4/16 | 2.1 | Goursat's Theorem | Class 12, 13, 14 |

4/18 | 2.2 | Cauchy's Theorem | Class 15 | |

4/20 | 2.4 | The Cauchy Integral Formula | ||

5 | 4/23 | 2.3, 2.4 | Consequences of the Cauchy Integral Formula | Class 16, 17, 18, 19 |

4/25 | 2.4, 2.5.1 | More consequences of the Cauchy Integral Formula | ||

4/27 | 2.5.1, 2.5.2, 2.5.4 | Even more consequences of the Cauchy Integral Formula (Morera's Theorem, the Schwarz Reflection Principle) | Class 20, 21, 22, 23 | |

6 | 4/30 | 3.1 | Zeros and singularities | Class 24, 25 |

5/2 | 3.2 | The Residue Formula | ||

5/4 | 3.2 | Applications of the Residue Formula | ||

7 | 5/7 | 3.2, 3.3 | More applications of the Residue Formula | |

5/9 | 3.3 | Types of singularities: removable, poles, essential | ||

5/11 | 3.3 | More on singularities, the extended complex plane | ||

8 | 5/14 | 3.4 | The argument principle, Rouche's Theorem | |

5/16 | 3.4 | Applications of the argument principle: the open mapping theorem, maximum modulus principle, and a quantitative fundamental theorem of algebra | ||

5/18 | 3.5 | Topology: simply connected domains and homotopy | ||

9 | 5/21 | 3.5 | Holomorphic functions in simply connected domains, the complex logarithm | |

5/23 | 5.1 | Jensen's formula, density of zeros of entire functions | ||

5/25 | 5.2 | Functions of finite order | ||

10 | 5/28 | 5.3, 5.4 | Infinite products | |

5/30 | 5.4, 5.5 | The Hadamard Factorization Theorem |

- Introduction
- Arithmetic of complex numbers
- Topology and limits in \( \mathbb C \)
- Holomorphic functions and the Cauchy-Riemann equations
- Power series
- Contour integration
- Goursat's Theorem, a preliminary version of Cauchy's Theorem
- Uniform convergence of functions
- The Cauchy Integral Formula and some of its consequences
- More consequences of the Cauchy Integral Formula (Morera's Theorem, Mean Value properties, Maximum-Modulus principle, Schwarz reflection principle)
- Zeros and singularities of holomorphic functions.
- The residue formula and applications, homotopy of curves