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MATH 411 - Honors Complex Variables (Fall 2015)

Time and Location

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


Instructor

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


Office hours

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

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

Homework

Weekly homework assignments will be posted here. Unless stated otherwise, the deadline for homework submission is 3:00 pm on Thursday. 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

The first midterm test will be held on Wednesday, October 14, 2015, in class.

Some details about the midterm:

  • Duration: 50 minutes.
  • Material covered: up to, and including Theorem III.26.
  • 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 very similar.

And now for the real thing ...

The format of the second midterm will be identical with the first. It will cover the material up to, and including Theorem IV.7 (Residue Theorem). To help you prepare for the second midterm test, here is again 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 Friday, December 11, 2015 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.