We will cover sections from Chapters 1–6. See the outline below.
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
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
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.
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:
|Homework (lowest score dropped):||10%|
Check back here for homework assignments and solutions, notes and links as the term progresses.
|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 11||1.3, 1.4||basic 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 13||1.5||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 18||1.6, (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 20||2.1||Complex Functions||2.1: 3(abcd), 5(abcde), 6(abc), 10(a), 11(a) due: Mon Sept 25|
|Fri Sept 22||2.2||Mapping examples, limits of sequences|
|Mon Sept 25||2.3||Analyticity||2.2: 11(ace), 12; 2.3: 4(b), 11(f) due: Mon Oct 2|
|Wed Sept 27||2.4||The Cauchy-Riemann equations||2.4: 5, 8, 10, 12 due: Mon Oct 2|
|Fri Sept 29||2.5||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 11||3.1,3.2||Partial fractions, the complex exponential, trig functions||3.1: 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 16||3.3||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 take on finitely many values? is a branch of . Quadratic formula, inverse trig functions.|
|Mon Oct 23||4.1, 4.2||Complex integration. Warm-up: , 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 independence||4.3: 1(b)(c)(g)(h), 5 In addition: Derive formula (10) on p 135 for and use it to find the derivative due: Mon Oct 30|
|Fri Oct 27||Path independence ctd. Calculation of for 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 1||4.5||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 17||Power series|
|Mon Nov 20||Taylor's theorem, formula for coefs, term by term differentiation|
|Wed Nov 22||5.5, 5.6||Examples 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 24||6.1||Classification of singularities, Residue theorem|
|Mon Nov 27||6.2||Finding the residue||6.2: 2,6 not due solutions are posted|
|Wed Nov 29||6.3||Trig examples||6.3: 2, 6, 9 not due solutions are posted|
|Fri Dec 1||6.4||Integrals over||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!
Here is a Julia set demo
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 , for
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 .
Old exams: 2015WT2 1(c)(e); 2014WT1 3, 11(f)
Textbook: 3.3: 1-5,8-17, 3.5: 1-17
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), 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 for in
Old exams: 2015WT2 6
Textbook: 6.2: 1-8
Integration of over the real line by residues
Old exams: 2016WT2 8; 2015WT2 4; 2014WT1 8
Textbook: 6.3: 1-8
Integration of over the real line by residues
Old exams: 2015WT2 5
Textbook: 6.4: 1,3,5,6,7