Math 503: Complex Analysis, in the Fall, 1997, semester | |||||
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Preface | Background | Syllabus | Homework | Links to 503 technical pages | The text |

Students who have already had a graduate course in complex
analysis should * not * enroll in this course. The course is not
aimed at them and would be mostly a waste of their time. Their
presence might distort the dynamics of the classroom for its intended
audience.

Students should use the text vigorously. I believe it can be read and that the author has worked out a large number of examples in reasonable detail. Students should look at the textbook's problems for every topic covered in class and should know how to do most of them.

The midterm lasted two hours, and was given on Friday, October 24. There were problems given to prepare for the midterm. The exam itself and some answers are available.

There were also preparation problems for the final exam, and here also the exam itself (given for two hours on Wednesday, December 17) and some answers are available.

I originally wanted to require each student to "write up" at least
one problem during the semester using TeX. I've been persuaded *
not* to require this, but I'm willing to advise and encourage
students' efforts towards this goal.

Every student should make at least one oral presentation (perhaps a solution of a homework problem) in class during the semester. Again, having this as a requirement may be unrealistic, but I want to encourage both oral and written exposition of mathematics.

The assignments are textbook problems unless otherwise indicated.
(For example, "FL" means the
**F**irst **L**ecture notes.)

Date assigned |
Date due |
Problems |
---|---|---|

9/3/97 | 9/8/97 | FL p.1:ii; FL p.3: ii & iii |

9/10/97 | 9/15/97 9/17/97 |
1:4.28, 4.37, 4.38, 4.61, 5.23, 5.27 |

9/17/97 | 9/24/97 | 3:6.9, 6.12, 6.22, 6.27 |

9/24/97 | 10/1/97 | 3:6.48, 6.53; 4:4.17, 4.18 |

Topic | Reading | Date | |
---|---|---|---|

1 | The first lecture | Notes | 9/3/97 9/8/97 |

2 | The algebra of complex numbers and some elementary functions | I | 9/10/97 |

3 | A bit about topology, especially continuity | II | 9/15/97 |

4 | Complex differentiability; the Cauchy-Riemann equations | III:1-2 & 5 | 9/15/97 |

5 | Partial derivs & differentiability | | 9/17/97 |

6 | Harmonic conjugates; meeting logarithm | III:3,4 | 9/22/97 |

7 | Paths and integrals along paths | IV:1-2 | 9/24/97 |

8 | Primitives and integrals; Green's Theorem implies ... | IV:2 | 9/29/97 |

9 | Technicalities & the Goursat proof | V:1 | 10/1/97 |

10 | A Cauchy Theorem | V:1 | 10/6/97 |

11 | Winding number and another Cauchy Theorem | V:2 | 10/8/97 |

12 | Derivatives and consequences: Morera, Schwarz reflection, Cauchy estimates, Liouville, Maximum Modulus, Schwarz Lemma, etc. | V:3 | 10/13/97 |

13 | 10/15/97 | ||

14 | More consequences: Schwarz, Max Mod, unif conv, etc. |
10/20/97 | |

15 | Branches of logarithm & arctan; their Riemann surfaces. Automorphisms of the unit disk | V:4 | 10/22/97 |

16 | Homotopy, contactibility, simple connectivity: a homotopy Cauchy theorem | V:6-7 | 10/27/97 |

17 | The hard part of a homology Cauchy Theorem | V:5 | 10/29/97 |

18 | Discussion of real analyticity; beginnings of complex power series | VI:1 | 11/3/97 |

18 | Weierstrass M-test; u.c.c. convergence | VII:1-3 | 11/5/97 |

19 | Power series and Taylor series | VII:3 | |

20 | Laurent series | VII:3 | |

21 | Behavior near a point; discrete level sets; analytic continuation. | VIII:1 | |

22 | Isolated singularities | VIII:2 | |

23 | The Residue Theorem; some computations | VIII:3 | |

24 | Consequences: Rouche's Theorem, the argument principal, the open mapping theorem | VIII:3 | |

25 | Conformal mapping; automorphism groups of some domains | IX:1 | |

26 | The extended plane; Moebius mappings | VIII:4 & IX:2 | |

27 | More conformal maps | IX:2 | |

28 | Normal families | VII:4 | |

29 | The Riemann Mapping Theorem | IX:3 | |

30 | D-bar & analytic alternatives to homo{logy/topy}; Harmonic functions | ?? | |