Complex Variables: Math 421

Meeting : MWF 9:05    LGRT 117

Instructor : Luc Rey-Bellet

Office :  1423 J LGRT
Phone :  545-6020
E-Mail :  lr7q@math.umass.edu
Office Hours :   Monday 1:30--3:00,   Wednesday 1:30--3:00,   or by appointment.

Text: Complex Variables and Applications, 6-th Edition, by James Ward Brown and Ruel V. Churchill, McGraw-Hill.

Syllabus:

• 1) Complex Numbers: algebraic and geometric properties, polar form, powers and roots.

• 2) Analytic functions: Differentiability and Cauchy-Riemann equations, Harmonic functions, examples.

• 3) Elementary functions of a complex variable: exponential and trigonometric functions, logarithms.

• 4) Path integrals: contour integration and Cauchy's integral formula; Liouville's theorem, Maximum modulus theorem, the Fundamental Theorem of Algebra.

• 5) Series: Taylor and Laurant expansions, convergence, term-by-term operations with infinite series.

• 6) Isolated singularities and residues. Essential singularities and poles. Evaluation of Improper integrals via residues.

• 7) (If time permits) Mappings by elementary functions and linear fractional transformations; conformal mappings.

• 50% Exam1 and Exam2 (25% each).
• 30% Final Exam
• 20% Homework, Attendance and participation

Exams:

• Exam1:   Wednesday March 12   9:00-10:00 am     Room: Classroom

• Exam2:   Wednesday April 30      6:00-7:30 pm       Room: LGRT 117         Practice I     Practice II

• Final:     Monday May 19          10:30-12:30         Room: LGRT 321         Practice I     Practice II

No calculators will be allowed in the exams.

Homework : Homework will be assigned weekly and will be due every Wednesday, unless mentionned otherwise. No late homework will be collected. The three lowest grades will be dropped.

Week 1 (due on 2/5):
Section 2, pp.5 : 1(d,e,f), 3
Section 4, pp.11 : 2(c,d), 4, 9, 10(b,c), 12, 18
Section 6, pp.17 : 1(a), 2(d), 3, 4 , 5

Week 2 (due on 2/12):
Section 6 pp. 17: 6, 9, 10, 13
Section 7 pp. 22: 2 (a,d), 5, 7, 8
Section 8 pp. 23: Read It!
Section 14 pp 42: 4
Section 16 pp 47: 1 (b),(d), 4, 8(b), 9

Week 3 (due on 2/19):
Section 19 pp 54: 1 (c,d), 2(b,d), 3(a,b), 10
Section 22 pp 62: 1 (a,c), 2(b), 3, 7(a,b), 10(b,c)

Week 4 (due on 2/26):
Section 23 pp 67: 1, 4, 7, 8
Section 24 pp 71: 2, 5(a), 7, 13, 17

Week 5 (due on 3/5):
Section 27 pp 79: 1, 2, 3, 5 (b), 7
Section 28 pp 84: 1, 2(c), 3, 5, 6

Week 6 (due on 3/12):

TEST 1

Week 7 (due on 3/19):

SPRING BREAK

Week 8 (due on 3/26):
Section 31 pp 92: 1, 3, 5, 9
Section 33 pp 102: 1, 3, 5, 6, 8, 9, 10, 13

Week 9 (due on 4/2):
Section 38 pp 119: 2(a,b), 3, 4(all), 5(all) 6, 7, 8, 12

Week 10 (due on 4/9):
Section 40 pp 128: 1(all), 2, 3, 4, 5, 9, 10
Section 42 pp 136: 1,2,3

Week 11 (due on 4/16):
Section 42 pp 136: 4, 5, 6, 7, 8
Section 43 pp 142: 1, 4, 5
Section 45 pp 149: 1, 2, 3, 4, 5, 6, 10(a,b), 11

Week 12 (due on 4/23):
Section 47 pp 156: 1, 2, 3, 4, 5, 6, 7
Section 55 pp 188: 1(all), 2, 3, 4, 5, 6, 7

Week 13 (due on 4/30):
TEST 2

Week 14 due on MONDAY 5/12 :
Section 57 pp 197: 1(b), 2(b), 3(a,b), 4, 5, 6(b), 7(a)
Section 60 pp 208: 1, 2, 3, 5, 6,
Section 61 pp 214: 2, 9
Section 62 pp 218: 1, 2