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
(omit) 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
(omit) 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
(omit) 6.7   Argument principle
(omit) 7.3   Moebius transformations
(omit) 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
Fri Sept 222.2 Mapping examples, limits of sequences
Mon Sept 252.3 Analyticity 2.2: 11(ace), 12; 2.3: 4(b), 11(f) due: Mon Oct 2
Wed Sept 272.4 The Cauchy-Riemann equations 2.4: 5, 8, 10, 12 due: Mon Oct 2
Fri Sept 292.5 The Cauchy-Riemann equations,ctd. The test next Friday will cover 1.1–1.6 and 2.1–2.5 2.5: 3(c)(d)(e), 6, 8(b), 9, 12 not due: I'll post solutions
Mon Oct 2 Harmonic functions
Wed Oct 4 Finding a harmonic conjugate using Green's theorem. Level curves of harmonic conjugates
Fri Oct 6 Test 1
Wed Oct 113.1,3.2Partial fractions, the complex exponential, trig functions3.1: 13(a)(d), 17,18,19; 3.2: 11, 13(a), 17(a)(b)(c), 20, 23 due: Mon Oct 16
Fri Oct 13 Properties of complex trig functions, mapping properties of complex exponential
Mon Oct 163.3 The complex logarithm, Cauchy-Riemann equations in polar co-ordinates 3.3: 1(b)(d), 4, 5(b), 9, 12, 15 due: Mon Oct 23
Wed Oct 18 3.5 Complex powers 3.5: 1(a)(d), 3(b), 5, 11, 15(a) due: Mon Oct 23
Fri Oct 20 When does $z^\alpha$ take on finitely many values? $z^\alpha w^\alpha$ is a branch of $(zw)^\alpha$. Quadratic formula, inverse trig functions.
Mon Oct 23 4.1, 4.2Complex integration. Warm-up: $\int_a^bf(t)dt$, Contours in the complex plane. 4.2: 5, 6(a)(b)(c), 7, 11(a)(b)(c), 14(a)(b) due: Mon Oct 30
Wed Oct 25 4.3 Complex integrals, antiderivatives and path independence4.3: 1(b)(c)(g)(h), 5 In addition: Derive formula (10) on p 135 for $\arctan(z)$ and use it to find the derivative due: Mon Oct 30
Fri Oct 27 Path independence ctd. Calculation of $\oint (z-z_0)^n dz$ for $n\in{\mathbb Z}$ in two different ways
Mon Oct 30 4.4b Cauchy's integral theorem 4.4: 10, 11, 13, 16, due Mon Nov 6 Also, make sure you understand 18.
Wed Nov 14.5 Cauchy integral formula 4.5: 1, 3(b)(e), 5, 9 due Mon Nov 6
Fri Nov 3 4.6 Bounds on analytic functions. Test next Friday covers material from last test to the end of this lecture. 4.6: 5, 7, 10, 11, 16, 17 not due
Mon Nov 6 Maximum modulus principle, intro to series
Wed Nov 8 5.1, 5.3 Power series, review for midterm
Fri Nov 10 Midterm
Wed Nov 15 5.2 Uniform convergence, power series converge unifomly in closed subdisks. Check out the summary sheet below. 5.1: 7(d)(e)(f), 8(c), 10, 11(b), 5.2: 5(a)(g), 11(a)(b), 5.3: 3(e)(f), 4, 5, due Wed Nov 22
Fri Nov 17Power series
Mon Nov 20Taylor's theorem, formula for coefs, term by term differentiation
Wed Nov 225.5, 5.6Examples of finding Taylor series, Laurent series 5.5: 3(a)(b)(c), 6, 7(b), 5.6: 1(a)(d), 13, 14, 6.1: 3(e), 3(f), due: Wed Nov 29
Fri Nov 246.1Classification of singularities, Residue theorem
Mon Nov 276.2Finding the residue6.2: 2,6 not due solutions are posted
Wed Nov 296.3Trig examples6.3: 2, 6, 9 not due solutions are posted
Fri Dec 16.4Integrals over $\mathbb R$6.4: 3, 7, 9, 12 not due solutions are posted

Request from grader: Could you please make sure that your name and student number are clearly legible on your homework, and that the name matches what is in the registrar's records. There are many students with the same surname in this class!

7. Files

8. Relevant Problems for Final Exam

I have compiled a list of problems that you should be able to do. They are taken from the three most recent final exams for this course and from the textbook. Most of the problems on the final exam will be similar to one of these. There are many more problems here than you could reasonably do, but you should go through and make sure you know how to approach each one.

Arithmetic operations, conjugate, modulus and argument, polar form.
Old exams: 2014WT1 1
Textbook: 1.1: You should be able to do all the problems for this section, although some (eg 29, 31 32) would not be suitable exam questions. 1.3: 3,4,5,7,11,12,13,17,22,23,25

Geometry of complex numbers, simple mapping examples, descriptions of sets
Old exams: 2016WT2 1(b)
Textbook: 1.2: 1-19, 1.6: 2-13

Exponential and trig functions, defintions, mapping properties
Old exams: 2016WT2 1(a)(d)
Textbook: 1.4: 1-13,17-21, 3.2: 1-23

Root finding: solve $z^n=a$, $a z^2 + b z + c = 0$ for $z$
Old exams: 2016WT2 2; 2015WT2 1(a)
Textbook: 1.5: 4,5,7-11,16,17

Complex functions, Cauchy Riemann equations, complex differentiable vs analytic,
Old exams: 2016WT2 1(c)
Textbook: 2.1: 1-13, 2.2: 7,11-14, 2.3: 3,4,7,9-13, 2.4: 1-15

Harmonic function, harmonic conjugates.
Old exams: 2014WT1 4(a)(b)
Textbook: 2.5: 1-15

Branches of the log: finding branches with prescribed properties, branches of $z^r$.
Old exams: 2015WT2 1(c)(e); 2014WT1 3, 11(f)
Textbook: 3.3: 1-5,8-17, 3.5: 1-17

Complex integration, definition and basic properties, antiderivatives, connection with integration around closed loops and path independence, Morera's theorem
Old exams: 2015WT2 2(b); 2014WT1 5 (note that integrand is Re$(z^2) + 2z^2$), 6, 11(d)
Textbook: 4.1: 1-5,7-11, 4.2: 6-14, 4.3: 1-8,12

Cauchy integral thoerem and Cauchy integral formula
Old exams: 2016WT2 3, 4(a)(b); 2015WT2 1(d); 2014WT1 7
Textbook: 4.4: 9-20, 4.5: 1-17 (the residue calculus gives an easier way to compute many of these)

Maximum modulus theorem, Liouville's theorem, mean value theorem
Old exams: 2015WT2 2(a); 2014WT1 11(g)
Textbook: 4.6: 1-19

Sequences and series, ratio test, comparison test. Power series, radius of convergence.
Old exams: 2016WT2 5, 6; 2014WT1 10, 11(j)
Textbook: 5.1: 1,2,3,4,5,6,7,8,9,10,11,14, 5.2: 1-7,11,13-16, 5.3: 1,2,3,4,5,6,7,8,9,10

Calculation of Laurent series, finding the residue
Old exams: 2016WT2 7; 2015WT2 1(b), 1(f), 3(a)(b); 2014WT1 9,11(h)
Textbook: 5.5:1-7, 5.6: 1,2,3,5,6,7,13,14

Classification of singularities (pole of order n, simple pole, removable, zero of order n)
Old exams: 2016WT2 7; 2014WT1 11(a)(b)(c)
Textbook: 6.1: 1-7

Integration of $p(e^{i \theta})/q(e^{i \theta}) d \theta$ for $\theta$ in $[0,2\pi]$
Old exams: 2015WT2 6
Textbook: 6.2: 1-8

Integration of $p(x)/q(x)$ over the real line by residues
Old exams: 2016WT2 8; 2015WT2 4; 2014WT1 8
Textbook: 6.3: 1-8

Integration of $e^{ikx}p(x)/q(x)$ over the real line by residues
Old exams: 2015WT2 5
Textbook: 6.4: 1,3,5,6,7