Math 535: Complex Analysis
University of Illinois at Chicago
Complex Analysis by Lars Ahlfors, 3ed.
Hardcover: McGraw-Hill, 1979. ISBN-13: 978-0070006577
Paperback: McGraw-Hill, 1980. ISBN-13: 978-0070850088
Bak and Newman
MWF 10:00am in 308 Stevenson Hall
Monday 2-3 and Wednesday 11-12
(including the final exam week)
There will be one midterm exam and a cumulative final exam.
||Mon, Feb 29
||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.
Your course grade will be determined on the following basis:
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
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 3rd
floor of SEO. Late homework is not accepted.
Unless otherwise noted, all problems refer to the main textbook, Complex Analysis, 3rd Edition, by Lars Ahlfors. Problems are listed using notation like
Sec 1.2.4(p20): 3, 5
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.
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).
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