## Math 185, Fall 2017, instructor: Dmitry Tonkonog

Classes: 241 Cory, Tue Th 8-9:30. Textbook: Brown and Churchill, 9th ed.

Office hours: Mon 11-12:30, Tue 3:30-5 (813 Evans).

GSI office hours: MTWThF 9-11 (959 Evans, Chanwoo Oh).

### RRR week schedule

Office hours: Mon 11-12:30, Tue 3:30-5, Thu 9:30-11.

Review of practice final: Thu 08:30-09:30 in class (241 Cory). Note we are starting 30 mins late.

### Exams

[Practice MT 1]   [MT 1]   [Practice MT 2]   [MT 2]   [Practice Final]   [Final]

The readings and homework are according to Brown and Churchill, 9th ed. The corresponding hw numbers from 8th ed are provided.

 Date Topic Readings Hw Hw notes 8/24 Overview. Complex numbers. 1-6 8/29 Exponential form of a complex number. Open and closed sets, domains. Functions. The function z^2. 7-14 Hw1 out: 2.11, 3.1(b), 5.4, 5.5, 6.2, 6.10, 9.1(a), 11.7, 12.4(a,b) 8/31 The functions z^2 and z^1/2. Limits. Theorems on limits. 13-16 9/5 Complex infinity. Continuous functions. 17-18 Hw2: 18.3(b), 18.5, 18.7, 18.10(a,b,c), 18.13, 20.2(c,d), 20.8, 20.9 8th ed: 20.2(c,d)->20.1(c,d), no other difference 9/7 Complex derivative. Differentiation rules. Cauchy-Riemann equations. 19-21 9/12 Cauchy-Riemann equations, sufficient conditions for differentiability. Analytic functions, basic properties. 22,23,25,26 Hw3: 24.1(b,c) 24.2(c,d) 24.3(c) 29.4 8th: shift sec number by -1 9/14 Reflection principle, exp and log. 29-33 Practice MT 1 9/19 Power, trig, hyperbolic functions. Complex integrals on the real line. 34-39,41-42 Hw 4: 30.1(a), 30.11, 33.10(a), 36.2(c), 38.8(a), 39.17, 42.2(bcd) 8th ed: 29.1(a), 29.11, 31.9(a), 33.2(c), 34.8(a), 35.16, 38.2(abc) 9/21 Midterm 1, 08:30-09:30, 241 Cory 9/26 Recap on derivatives. Arcs, countours, length. Contour integral. 41,43,44 9/28 Recap on substitution of variables. Contour integral, invariance under reparametrisation, examples. 44-46 Hw 4 due 10/3 Bounds on moduli of integrals. Antiderivatives and computations using them. 47-48 Hw5: 42.3, 43.1(b), 46.1(abc), 46.6, 46.10, 46.13, 47.1(b) 47.3 8ed: 38.3, 39.1(b), 42.1(abc), 42.7, 42.8, 42.10, 43.1, 43.3. Note: prob 46.13(9ed) resp. 42.10(8ed) are worded slightly differently. Please compute the integral from that problem without referring to any exercises/computations that were not in class/homework. 10/5 The theorem on antiderivatives. Cauchy theorem. Cauchy-Goursat theorem (proof beginning). 49-51 10/10 Proof of Cauchy-Goursat theorem. 51 Hw6: 49.2(bc), 49.4, 49.5, 53.1(cf), 53.2(a), 53.4, 53.6, 53.7 8ed: Sec 49->45, 53->49, same numbers. 10/12 Comparing contour integrals using Cauchy-Goursat. Cauchy integral formula, proof. Cauchy integral formula for higher derivatives, statement. 52-55 10/17 Cauchy fla for higher derivatives, Liouville thm, maximum modulus principle. 56-59 Hw7: 57.1(abcde), 57.2(ab), 57.3, 57.4, 57.5, 57.6, 57.7 8ed: Sec 57->52, same numbers. 10/19 Max modulus principle. Convergence of sequences and series. 59-61 10/24 Taylor series. Excursion: ideal fluids via complex analysis. 62-64 Hw8: 57.10, 59.2, 59.5, 59.6, 59.8, 65.1, 65.3, 65.6 8ed: 52.10, 54.3, 54.6, 54.7, 54.9, 60.1, 60.3, 60.10 and additionally in 60.3 replace f(z)=z/(z^4+9) by f(z)=z/(z^4+4) 10/26 Laurent series. 64-68 10/31 Uniform convergence. Differentiation and integration of series term-by-term. 69-71 Hw9: 68.4, 68.6, 68.7, 72.1, 72.3, 72.5, 73.1, 73.4 8th ed: 62.4, 62.6, 62.8, 66.1, 66.3, 66.5, 67.1, 67.3 11/2 Uniqueness of expansions. Multiplication and division of series. Residues. 72-75 Practice MT 2 11/7 Residues, contd. Residue at infinity. Three types of isolated singularities. 76-79 Hw10: 73.9, 77.1(abcd), 77.2(abcd), 77.5, 79.1(abcd), 79.2(abc) 8ed: 67.8, 71.1(abcd), 71.2(abcd), 71.4, 72.1(abcd), 72.2(abc) 11/9 Zeroes and poles. Improper integrals. 80-85 11/14 Computing improper integrals using residues 85-88 Hw 11: 81.1(cd) 81.2(bc) 81.4(ab) 81.5(ab) 81.6 81.7(ab) 83.3(b) 83.4(ab) 8th ed: 74.1(bc) 74.2(bc) 74.3(ab) 74.4(ab) 74.5 74.6(ac) 76.2(b) 76.3(ab) 11/16 Midterm 2, 08:30-09:30, 241 Cory 11/21 Definite integrals involving sines and cosines. Winding number, informally. 92,93 Hw 12: 86.2 86.3 86.5 86.10 88.2 88.4 92.5 92.6 8ed: 79.2 79.3 79.5 79.9 81.2 81.5 85.6 85.7 11/23 No class, academic holiday 11/28 Argument principle. 93 Hw12 due 11/30 Rouche theorem. Review. 94 12/5 RRR, no class 12/7 RRR, practice final review, starts 08:30 12/14 Final, 7-10pm

* Homework: 25%

* First midterm: 25%

* Second midterm: 25%

* Final exam: 25%

* Two worst homework scores will be dropped.

* The lowest midterm score can be dropped and replaced by the final exam score.

More precisely, each of the above four grades will first be curved into a number on a consistent scale. Your lowest midterm grade will be replaced by the curved final exam grade if it is higher. Finally, the four curved grades are added up and converted into a final course grade.

### Midterms and exams

Midterms will be in class. Notes and calculators are not allowed.

In the case of a fire alarm during either of the midterms or the final exam, leave your exams in the room, face down, before evacuating. Under no circumstances should you take the exam with you.