UNIVERSITY OF WAIKATO

Department of Mathematics

MATH311-07A – Advanced Calculus - 2007

Complex part web page
The test was handed back on Thursday - your script is with the Maths Secretary if you didn't collect it. (solutions are below)

The problem with the fundamental theorem of integral calculus for the complex case is nicely illustrated by the example of f(z)=1/z on the unit disk with the origin z=0 removed. Finding an F(z) with F'(z)=f(z) everywhere on this set is impossible. The theorem is still true!

Office hours during the examination preparation period are Monday-Wednesday , 18th - 20th June, 3-5pm in G3.22. Also Wednesday 13th June 3-5pm. The final examination is scheduled for 2.15 pm, Thursday 21st June. You will be asked to do 5 questions out of 8 with at least 2 from section A (Professor Kalnins material) and at least 2 from section B (my material).

Web Site:

The URL for this is http://www.math.waikato.ac.nz/~kab: click on the link pointing to this page.

Lecturer

Timetable

  1. Tuesday 1.10 p.m - 2.00 p.m., G3.33
  2. Wednesday 9.00 a.m. - 9.50 a.m., G3.33
  3. Thursday 11.00 a.m. - 11.50 a.m., J2.18
  4. Friday 3.10 p.m. - 4.00 p.m., G3.33

Prescribed Text:

  • Complex Variables by M. R. Speigel, (McGraw-Hill).

    On-line lecture notes for the Complex section:

    The text for the on-line lecture notes, Lecture Notes for a Course in Complex Calculus by Kevin A. Broughan Copyright (C) 1998-2004, for the complex part may be obtained by clicking on the links below:

    Exercises and Readings set in the complex part, 2007:

    1. Complex numbers: read lecture notes (LN) p1-6, Spiegel (S) p16 #20,23(a),29, 32,48. Do p25 #54(e), 71(a),(b), 81(a),96(a).
    2. Cauchy Riemann equations: read LN 12, 13, 16, 18, 20-22, S p72 #5-8, Do page 86 46(a), 47(a)(b), 48, 50(a)(b),55.
    3. Elementary functions: Read LN p26-30, S p44 # 8-12, Do page 58 # 58, 61, 62(a).
    4. Multiple valued functions: Read LN p30-35, S p46 # 13-16, Do p59 # 55, 74, 79(b), 85.
    5. Complex integral: Read LN p36-42 S p98 #1,2, Do S page 112 # 33(a), 36(a)(b), 37(a), p114 #60(a), 62.
    6. Cauchy's integral formula: Read LN p43-50, S p122 #5, Do p134 #30-34.
    7. Cauchy's integral formula for the n'th derivative: Read LN p51-56, Do S p135 #48,49,50(b),51(a)(b),52,54.
    8. Singularities I: Read LN p59-61,S p78 #21-23, #25,26, Do p89 #82(a)(b), 83, 85(a).
    9. Radius of convergence: read LN p67, 71,73-75, S p142-143,153 #22-24. Do S p164 # 59(a)(c), 60,65, #78(a)(c)(d), 79(a)(b)(c)(e),88.
    10. Taylor series: Read LN p75,76,78-80,82, S p154 #23,24. Do p165 #80, 81,83(a).
    11. Laurent series: Read LN p82-87, S p155 #25-27. Do S p166 #91, 92.
    12. Singularities II: Do S p167 #96(a)-(d), 98.
    13. (Last entry) Contour Integration: Read LN p95-118, S p179 #9-23 (selection). Do p195 #39(a)(c)(e),44-46, 49,50,54,58,64(a). This includes trigonometric integrals, integrals obtained using a semicircular contour, Jordans lemma, rectangular doubly infinite and singly infinite contours, integrals with poles on the contour, integrals with multiply valued functions and, lastly, an integral with a sector of a circle contour.

    Assignments and Test for the complex part, 2007:

    1. Complex Assignment 1, solutions.
    2. Complex Assignment 2, solutions.
    3. Complex Assignment 3, solutions.
    4. Complex test, solutions.

    Assignments and Test for the complex part, 2004:

    1. Complex Assignment 1, solutions.
    2. Complex Assignment 2, solutions.
    3. Complex Assignment 3, solutions.
    4. Complex Assignment 4, solutions.
    5. Complex test, solutions, more solutions.

    Assignments, tests and solutions (Complex Section):

    From 2003:

    1. Complex Assignment 1, Solutions.
    2. Complex Assignment 2,Solutions.
    3. Complex Assignment 3, Solutions
    4. Complex test, solutions, more solutions, more solutions.
    5. Complex Assignment 4,
    6. Complex Assignment 5,

    From 2002:

    1. Complex Assignment 1, solutions.
    2. Complex Assignment 2,. solutions.
    3. Complex Assignment 3,. solutions.
    4. Complex Assignment 4,. solutions.
    5. Complex Assignment 5,. solutions.
    6. Complex Test, solutions.

    Assessment

    General Rules for the complex part:

    The internal assessment will consist of one formal one hour test, worth 15% of the final mark, and three regular marked assignments worth a total of 10%. Assignments should be handed in through the slot marked 311A under the Mathematics Office reception counter (G3.19).

    There will be a three hour Final Examination for this paper covering both parts.
    The (Internal Assessment):(Final Examination) ratio will be 1/2:1/2.

    Dates and values of Tests, Assignments and the Final Examination

    1. Test 2 (Complex Section):Friday 25th May, 3-4pm, value 15%.
    2. Complex assignment 1: out Friday 27th April, back Friday 4th May, value 3%.
    3. Complex assignment 2: out Friday 4th May, back Friday 11th May, value 3%.
    4. Complex assignment 3: out Friday 11th May, back Friday 18th May, value 4%.
    5. Final Examination: in June after the study break, at a date to be announced, value 50%.

    Associate Professor Kevin Broughan

    31st May 2007