David Dumas

University of Illinois at Chicago

Spring 2016

The inversion of the UIC circle mark in its boundary. (Animated.) |

Main Text |
Complex Analysis by Lars Ahlfors, 3ed. Hardcover: McGraw-Hill, 1979. ISBN-13: 978-0070006577 Paperback: McGraw-Hill, 1980. ISBN-13: 978-0070850088 |
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Supplementary Texts |
Gamelin Bak and Newman |

Lectures | MWF 10:00am in 308 Stevenson Hall |

david@dumas.io | |

Office hours |
Monday 2-3 and Wednesday 11-12 (including the final exam week) |

Office | 503 SEO |

CRN | 19436 |

- Course Syllabus
- Challenge problems - Last updated Thursday, April 14
- Practice Midterm
- Midterm and solutions
- Practice final exam and solutions
- Final exam and solutions

Midterm Exam | Mon, Feb 29 | In class |
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Final Exam | Fri, May 6 | 10:30am-12:30pm in 308 Stevenson Hall |

There will be a total of **eight** problems on the exam, and you will be asked to complete any **five** of them.

Homework | 40% |
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Midterm Exam | 20% |

Final Exam | 40% |

When computing your homework grade, the lowest weekly homework score will be dropped, and the remaining scores will be averaged with your challenge problem scores. (That is, *each challenge problem is worth as much as a weekly homework assignment!*)

**Suggested exercises — Elliptic Functions**

Sec 7.3.2(p274): 1

Sec 7.3.3(p276): 1,2,3,4**Homework 14 due Monday, Apr 25 — Conformal mapping again**

Sec 6.2.2(p238): 3,5,6**Homework 13 due Monday, Apr 18 — Compactness**

Sec 5.2.5(p206): 3

Sec 5.5.5(p227): 1,2,3

and one additional problem.**Homework 12 due Monday, Apr 11 — Products**

Sec 5.2.2(p193): 1,2,3

Sec 5.2.3(p197): 1,3

Sec 5.2.4(p200): 1,3**Homework 11 due Monday, Apr 4 — Series**

Sec 5.1.1(p179): 5

Sec 5.1.2(p184): 1,3

Sec 5.1.3(p186): 2,3,4

Sec 5.2.1(p190): 3,5**Homework 10 due Monday, Mar 28 — Harmonic functions**

Sec 4.6.2(p166): 1,2

Sec 4.6.4(p171): 1,2,3,4**Homework 9 due Monday, Mar 14 — Residues**

Sec 4.5.2(p154): 1,2,3

Sec 4.5.3(p161): 1,3a,3e,3g,5**Homework 8 due Monday, Mar 7 — Homological Cauchy Theorem**(PDF)

(This homework consists of Sec 4.4.7(p149) #2,4,5 and two additional problems.)**Homework 7 due Monday, Feb 29 — Geometric properties of holomorphic functions**

Sec 4.3.3(p133): 1,3,4

Sec 4.3.4(p136): 1,2,5,6**Homework 6 due Monday, Feb 22 — Cauchy's theorem and applications**

Sec 4.2.2(p120): 1,2

Sec 4.2.3(p123): 1,4,5

Sec 4.3.2(p129): 2,4,5**Homework 5 due Monday, Feb 15 — Conformal mapping, contour integrals**

Sec 3.4.2(p96): 2,3,7

Sec 4.1.3(p108): 2,3,5,6

**Homework 4 due Monday, Feb 8 — Linear fractional transformations**

Sec 3.3.1(p78): 2,4

Sec 3.3.2(p80): 1,2,4

Sec 3.3.3(p83): 5,6

Sec 3.3.5(p88): 3

**Homework 3 due Monday, Feb 1 — Elementary functions and branches**

Sec 2.3.2(p44): 1,4

Sec 2.3.4(p47): 5,6,7,8

Sec 3.2.2(p72): 1

**Homework 2 due Monday January 25 — Power series**

Sec 2.1.4(p32): 4,5

Sec 2.2.3(p37): 2,4,6

Sec 2.2.4(p41): 3,7**Homework 1 due Wednesday January 20 — The basics**

Sec 1.1.5(p11): 1

Sec 1.2.1(p15): 2,4

Sec 1.2.3(p17): 5

Sec 1.2.4(p20): 3,5

Sec 2.1.2(p28): 2,3

Unless otherwise noted, all problems refer to the main textbook, Complex Analysis, 3^{rd} Edition, by Lars Ahlfors. Problems are listed using notation like

Sec 1.2.4(p20): 3, 5which means

In Chapter 1, section 2.4, the list of exercises begins on page 20.

Complete problems 3 and 5 on this list.

Please make sure to write clearly and that the assignment number and your name appear at the top of the first page. Staple your homework if it spans several sheets of paper. Typeset solutions are welcome, but not required.

After that, we will for the most part proceed linearly through the textbook at a steady rate. The sections under discussion will be announced in each lecture, and it is up to you to keep up with the corresponding reading.

Any sections to be skipped or material to be covered out of order will be announced in advance. The following announcements of this type have been made:

**No lectures about section 3.1.**

Read as necessary, depending on your knowledge of topology.**Skip section 3.4.3 (Riemann surfaces).****Skip sections 5.3, 5.4, 6.3, 6.4, and 6.5.**

That is, after discussing the gamma function we will move on to normal families (5.5), the Riemann mapping theorem (6.1), Schwarz-Christoffel mappings (6.2), and then elliptic functions (chapter 7)

We covered a proof of the integral formula for the gamma function which is different from the one in Ahlfors. The suggested reading for the material for this is:

- Whittaker and Watson,
*A Course of Modern Analysis*, 4ed. Cambridge University Press, 1927.

Section 12.2 (pp241-243).

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