MATH 411 - Honors Complex Variables (Fall 2016)

Time and Location

Time: MWF 3:00 - 3:50 pm
Room: CAB 377


Instructor

Dr. Arno Berger (CAB 683, berger@ualberta.ca)


Office hours

W 4:15 - 6:00 pm, R 3:00 - 6:00 pm or by appointment.


General information

Please see this PDF document for all relevant details concerning MATH 411. (An abbreviated version of this document will be distributed in class.)


Course notes

Be prepared to take careful notes in class, as no set textbook will be used.

The course will loosely follow the fine notes by Drs. Runde and Bowman (PDF, 4.7MB) which you are very welcome to use. Be aware, however, that notation and terminology may differ from those used in class.


Material covered in class (Course Diary)

I plan to keep an up-to-date list of the topics, examples etc. covered in class.

Lecture # Date Material covered / special events Remarks/ additional material
    WELCOME TO
MATH 411 !!

 
1 Fri
2
Sep
The complex numbers: Algebraic foundations.
Mon
5
Sep
No class - Labour Day.  
2 Wed
7
Sep
Geometric properties.  
3 Fri
9
Sep
Sequences and series of complex numbers.  
4 Mon
12
Sep
A brief review of topological concepts.
The extended complex plane.
 
5 Wed
14
Sep
Limits and continuity.  
6 Fri
16
Sep
Linear functions. Complex differentiability. Handout - Lemma II.6
7 Mon
19
Sep
The Cauchy-Riemann equations.  
8 Wed
21
Sep
Power series: Examples.  
9 Fri
23
Sep
More on power series. Path integrals.  
10 Mon
26
Sep
The Cauchy Integral Theorem.  
11 Wed
28
Sep
Goursat's Lemma. More general paths. Handout - arbitrary paths
12 Fri
30
Sep
The Cauchy Integral formula. Thursday office hours extended to 3:00 - 6:00 pm.
13 Mon
3
Oct
Sequences of analytic functions. Weierstrass' Theorem.  
14 Wed
5
Oct
Morera's Theorem.  
15 Fri
7
Oct
Liouville's Theorem. Fundamental Theorem of Algebra. Identity Theorem.  
Mon
10 Oct
No class. Happy Thanksgiving!
16 Wed
12
Oct
The Open Mapping Theorem and Maximum Modulus Principle.  
17 Fri
14
Oct
Zeros. Isolated singularities.  
18 Mon
17
Oct
The Casorati-Weierstrass Theorem.  
19 Wed
19
Oct
Laurent series.  
Fri
21
Oct
MIDTERM TEST 1. Good luck!!
20 Mon
24
Oct
More on Laurent series.  
21 Wed
26
Oct
Examples.  
22 Fri
28
Oct
The Index (or Winding Number).  
23 Mon
31
Oct
The General Cauchy Integral Theorem.  
24 Wed
2
Nov
Simply connected regions. Homotopy.  
25 Fri
4
Nov
Residues.  
      Enjoy Reading Week.
26 Mon
14
Nov
The Residue Theorem.  
27 Wed
16
Nov
Applications of the Residue Theorem. Handout - Sec. IV.2.
28 Fri
18
Nov
More applications of the Residue Theorem.  
29 Mon
21
Nov
A few theoretical consequences of the Residue Theorem.  
30 Wed
23
Nov
Conformal mappings.  
Fri
25
Nov
MIDTERM TEST 2. Good luck!!
31 Mon
28
Nov
Groups of conformal automorphisms. Examples.  
32 Wed
30
Nov
Principles of (pre-)compactness.  
33 Fri
2
Dec
The Arzelà-Ascoli Theorem. Montel's Theorem. Handout - Arzelà-Ascoli Theorem.
34 Mon
5
Dec
The Riemann Mapping Theorem.  
35 Wed
7
Dec
Proof of the Riemann Mapping Theorem.
Concluding remarks.
 
Good bye and good luck !!
Tue
20
Dec
Final exam !

Please see box on the left for details.
Good luck !!
       

Homework

Fortnightly homework assignments will be posted here. Unless stated otherwise, the deadline for homework submission is 4:00 pm on Friday. Please submit your solutions into the designated MATH 411 assignment box on the third floor of CAB.

Three words about cheating:

    Don't Do It !!

Midterm test

Two midterm tests will be held on Friday October 21 and November 25, 2016, respectively, in class.

Some details about the first midterm:

  • Duration: 50 minutes.
  • Material covered: up to, and including Proposition III.38.
  • NO textbooks, notes, calculators, formula sheets etc.!
  • NO cell-phones, i-pods, or other electronics!
  • Please bring a valid ID with you.
  • Good luck!

To help you prepare for the midterm test, here is a practice version. The real test will be similar.

And now for the real thing ...

The format of the second midterm will be identical with that of the first. It will cover the material up to, and including Section IV.2 (Applications of the Residue Theorem). To help you prepare for the second midterm test, here again is a practice version. As always, the real test will be very similar.

And here is what really happened ...

Final exam

The final exam will be held on Tuesday, December 20, 2016 at 2:00 pm, in CAB 269.

Some details about the final:

  • Duration: 2 hours.
  • Material covered: Basically everything, but with more emphasis on Chapters IV and V.
  • NO textbooks, notes, calculators, formula sheets etc.!
  • NO cell-phones, i-pods, or other electronics!
  • Please bring a valid ID with you.
  • Good luck!

To help you prepare for the final, here is a practice version. The real exam will be very similar.

Other material

Dr. Runde's MATH 217/317 notes (PDF, 2.1MB) are a great resource for reviewing basic facts from calculus.

General information about the department's honors program is available here.