Math 535: Complex Analysis

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
Supplementary Texts	Gamelin Bak and Newman
Lectures	MWF 10:00am in 308 Stevenson Hall
Email	david@dumas.io
Office hours	Monday 2-3 and Wednesday 11-12 (including the final exam week)
Office	503 SEO
CRN	19436

Documents

• Course Syllabus
• Challenge problems - Last updated
• Practice Midterm
• Midterm and solutions
• Practice final exam and solutions
• Final exam and solutions

Important Dates

There will be one midterm exam and a cumulative final exam.

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

About the final exam

The final exam will take place on Friday 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.

Grading

Your course grade will be determined on the following basis:

Homework	40%
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!)

Weekly homework

• 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

Homework assignments should be submitted to the mailbox of the grader, John Kopper, by 4:00pm on the due date. The mailroom is located on the 3^rd floor of SEO. Late homework is not accepted.

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, 5

which 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.

Reading

By the end of the first week, read chapter 1 and the first two sections from chapter 2.

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).

Other Resources

• Learning TeX/LaTeX? There are countless online tutorials, references, and resources, but these simple templates might help you get started:
  • LaTeX homework template

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