Math 300: Introduction to Complex Variables

1. Location and Time

2. Instructor Information

3. Textbook

We will cover sections from Chapters 1–6. See the outline below.

4. Outline and Timetable

4.1. Part 1: Complex numbers and analytic functions (11 hours)

1.1   The algebra of complex numbers
1.2   Point representation of complex numbers
1.3   Vectors and polar forms
1.4   The complex exponential
1.5   Powers and roots
1.6   Planar sets
1.7   The Riemann sphere
2.1   Functions of a complex variable
2.2   Limits and continuity
2.3   Analyticity
2.4   The Cauchy-Riemann equations
2.5   Harmonic functions

4.2. Part 2: Elementary functions and complex integration (13 hours)

3.1   Polynomials and rational functions
3.2   Exponential, trigonometric and hyperbolic functions
3.3   The logarithm
3.5   Complex powers and inverse trigonometric functions
4.1   Contours
4.2   Contour integrals
4.3   Independence of path
4.4   Cauchy's integral theorem
4.5   Cauchy's integral formula
4.6   Bounds for analytic functions

4.3. Part 3: Series expansions and residue theory (11 hours)

5.1   Sequences and series
5.2   Taylor series
5.3   Power series
5.4   Convergence
5.5   Laurent series
5.6   Zeros and singularities
5.7   The point at infinity
6.1   The residue theorem
6.2   Trigonometric integrals
6.3   Improper integrals
6.7   Argument principle
7.3   Moebius transformations
7.4   Moebius transformations, ctd.

5. Homework, Tests and Grades:

There will be weekly homework assignments, usually due on Mondays. Late homework will not be accepted. A selection of problems will be graded. I will drop the lowest homework score.

Graded homework will be available for pickup at the Math Learning Centre

There will be two midterm exams, on Friday October 6, and Friday, November 10. There are no make-up midterms. If you miss a midterm for a valid medical reason, the weighting for the final will be adjusted. Other than this, no re-negotiating of the weights of the different components of the overall grade will be considered.

There will be a final exam during the December exam period.

The following applies to all exams in Math 300: No calculators, notes, books, electronic devices or aids of any kind.

Your grade will be computed as follows:

Final Exam: 50%
Midterm 1: 20%
Midterm 2: 20%
Homework (lowest score dropped): 10%

6. Assignments and Notes:

Check back here for homework assignments and solutions, notes and links as the term progresses.

Date Reading Topics Problems
Wed Sept 6 1.1 Introduction, arithmetic operations 1.1: 6(a), 8, 10, 20(a,b,c) 30 due: Mon Sept 11
Fri Sept 8 1.2 Geometry of complex numbers modulus, conjugate, basic inequalitites 1.2: 7(def), 16, 17 due: Mon Sept 18
Mon Sept 111.3, 1.4basic inequalitites, set descriptions, complex exponentials 1.3: 5(d), 7(e), 11, 13, 23; 1.4: 3(c), 12(b), 20(b) due: Mon Sept 18
Wed Sept 131.5 complex exponentials ctd, polar form, arg and Arg 1.5: 5(ae), 11, 16 due: Mon Sept 18
Fri Sept 15 geometry of multiplication, roots of unity
Mon Sept 181.6, (1.7)roots of a complex number, classification of sets. (If time, Riemann sphere. I won't test you on this.) 1.6: 2-8(a)(b), due: Mon Sept 25
Wed Sept 202.1Complex Functions 2.1: 3(abcd), 5(abcde), 6(abc), 10(a), 11(a) due: Mon Sept 25

7. Files