| Week | Monday | Wednesday | Friday | Reading and Assignments |
|---|---|---|---|---|
| 1/28 | Administrivia 1.1-1.2 introduction, field axioms |
1.1-1.3 the complex plane, algebra of complex numbers 1.4-1.6 vectors, modulus, conjugates, triangle inequality. 1.7-1.9 exponential form, argument, products, quotients |
1.10, 1.11 Euler's formula, powers and roots | Read all of Chapter 1 (sections 1-12) HW1 due Wednesday 2/6 solutions |
| 2/4 | 1.12 topology: neighborhoods, open sets, boundary, accumulation points 2.13 functions and mappings |
2.14 visualizing z → z2 2.15,2.16 limits |
2.17 stereographic projection (see this interactive app), limits at infinity 2.18 continuity end of add/drop, Friday 4pm |
Read Chapter 2 through 2.23 HW2 due Wednesday 2/13 solutions |
| 2/11 | More on stereographic projection 2.19, 2.20 derivatives |
2.21, 2.22 Cauchy-Riemann equations 2.23 conditions for differentiability Quiz 1: Here is the quiz and the solutions. |
Geometric interpretation of the derivative 2.24* Cauchy-Riemann in polar coordinates 2.25, 2.26 Analytic Functions and examples |
Read the rest of Chapter 2 HW3 due Wednesday 2/20 solutions |
| 2/18 | 2.26 analytic functions continued 2.27 harmonic functions |
2.28 uniquely determined analytic functions 2.29* reflection priniciple |
3.30 The exponential function 3.31,3.32 The logarithm 3.33 Branches of the logarithm |
Read Chapter 3 through 3.38 HW4 due Wednesday 2/27 solutions |
| 2/25 | 3.33, 3.34 derivative and identities of logarithms 3.35, 3.36 The power function, examples 3.37 sine and cosine 3.38 zeros and singularities of trig functions |
3.39 Hyperbolic functions 3.40 Inverse trig and hyperbolic functions Quiz 2: Here is the quiz and the solutions. |
4.41, 4.42 Derivatives and integrals of functions z=w(t) | Read the rest of chapter 3 and Chapter 4 through 4.46 HW5 due Wednesday 3/6 solutions |
| 3/4 | 4.43 Contours 4.44, 4.45 Contour integrals and examples |
4.46 Examples involving branch cuts 4.47 Upper bounds on moduli of contour integrals |
4.47 moduli of contour integrals (continued) 4.48, 4.49 Antiderivatives |
Read through chapter 4 section 4.52 HW6 due Friday 3/15 solutions |
| 3/11 | 4.48, 4.49 Antiderivatives (continued) 4.50 The Cauchy-Goursat theorem |
Midterm Here is the midterm and the solutions, as well as information about the grades. |
Some discussion of midterm 4.50, 4.51 The Cauchy-Goursat Theorem |
Read the rest of chapter 4 |
| 3/18 | Spring Break (probably not like this) |
|||
| 3/25 | 4.52 Simply connected domains 4.53 Multiply connected domains 4.54 The Cauchy integral formula |
4.54, 4.55 The Cauchy integral formula and extensions 4.56, 4.57 Consequences of the extension |
4.58 Liouville's theorem, Fundamental Theorem of Algebra 4.59 The maximum modulus principle Last day to drop or G/P/NC, Friday at 4pm. |
Finish chapter 4 HW7 due Wednesday 4/3 solutions |
| 4/1 | 5.60,5.61 Convergence of sequences and series 5.62, 5.63 Taylor series, some examples |
5.64 proof of Taylor's Theorem 5.65 series with negative powers |
5.66 Laurent series 5.67, 5.68 Laurent's theorem and examples |
Read Chapter 5 through 5.69 HW8 due Wednesday 4/10 solutions |
| 4/8 | 5.69 Absolute and uniform convergence 5.70 Continuity of sums 5.71 Integration and differentiation of series 5.72 Uniqueness |
5.71 Proof that Integration and differentiation of series works Quiz 3: Here is the quiz and the solutions. |
5.73 multiplication and division of series 6.74,6.75 Isolated singularities and residues |
Finish reading chapter 5, start chapter 6 HW9 due Wednesday 4/17 solutions |
| 4/15 | 6.76 The Cauchy Residue Theorem 6.77 Residue at infinity 6.78, 6.79 Poles, removable and essential singularities |
6.80, 6.81 The residue at a pole 6.82 Zeros of analytic functions |
6.83 Zeros and poles 6.84 behaviour of functions near singularities |
finish reading chapter 6 HW10 due Wednesday 4/24 solutions |
| 4/22 | 6.84 Riemann's theorem and Casorati-Weierstrass theorem Behavior of exp(1/z) near 0 |
7.85, 7.86 Residues and improper integrals | 7.87 improper integrals from Fourier analysis 7.88 Jordan's lemma |
read chapter 7 HW11 due Wednesday 5/1 solutions |
| 4/29 | 7.89, 7.90 Indented paths and branch points 7.91 Integration along a branch cut 7.92 Integrals involving sines and cosines |
7.93 The argument principle Quiz 4: Here is the quiz and the solutions. |
7.93 finish proof of Argument Priniciple 7.94 Rouché's Theorem and Fundamental Theorem of Algebra |
finish chapter 7 |
| 5/6 | 8.96 Linear maps 8.97, 8.98 z → 1/z (see this interactive app) 8.99, 8.100 Linear fractional transformations |
8.107, 8.108 z -> z2 and the square root 8.109 square roots of polynomials 8.110 Riemann surfaces |
read Chapter 8 |
|
| 5/13 | Final Cumulative Thursday, May 16, 11:15am-1:45pm in Engineering 145 see this page for more information. |