MATH 366, Honours Complex Analysis, Fall 2014

Course web page: http://www.math.mcgill.ca/jakobson/courses/math366.html

Lectures: Tue, Thurs 10:00-11:30, Burnside 1205
Instructor: D. Jakobson
Office: BH1220, Office Hours: Tuesday, 11:30-12:30; Wednesday, 9:20-10:20; or by appointment
Tel: 398-3828
E-mail: jakobson AT math.mcgill.ca
Web Page: www.math.mcgill.ca/jakobson
Prerequisite: Math 248. Co-requisite: Math 354
Marker: Xianchao Wu
Email: xianchao.wu AT mail.mcgill.ca
Text
  • Required: E. Stein and R. Shakarchi. Complex analysis. Princeton Lectures in Analysis II. Princeton Univ. Press.
    • Syllabus: Functions of a complex variable, Cauchy-Riemann equations, Cauchy's theorem and its consequences. Uniform convergence on compacta. Taylor and Laurent series, open mapping theorem, Rouche's theorem and the argument principle. Calculus of residues. Fractional linear transformations and conformal mappings.

    Tests
  • Take home midterm.
  • Solutions are posted on Mycourses.
  • There will be a three hour final exam. It will be held on December 9, 14:00-17:00, in Burnside 1205.
    • Grading: Your final mark will be the largest of the following: [20% Assignments + 30% Midterm + 50% Final]; OR [20% Assignments + 80% Final].
    WebCT: Your scores on assignments, midterm, final, and your final mark will be posted on WebCT
    Supplemental There will be a supplemental exam, counting for 100% of the supplemental grade. No additional work will be accepted for D, F or J.

    Handouts

  • Notes on Introductory Point-Set Topology by Allen Hatcher.
  • Point set topology and metric spaces handout (for math 466): postscript and pdf.
  • Differentiation under the integral sign handout (for math 466) (postscript, pdf)
  • Pages about simply-connected domains in Wikipedia and in Wolfram Mathworld
  • Pages about path homotopy (continuous deformation) in Wikipedia
  • Conservative vector fields, notes by L. Laayouni for Math 264. Note that conservative vector fields in the plane are the same thing as exact differentials, and a necessary condition for the vector field to be conservative is that the corresponding differential is closed.
  • Line Integrals, notes by L. Laayouni for Math 264.
  • Green's theorem, notes by L. Laayouni for Math 264.

    • Assignments

  • Homework will be assigned in class and will be due by midnight by the specified deadline.
  • Assignment 1, due Friday, Sept. 19. Do any 10 of the following problems: Stein and Shakarchi, Chapter 1: # 3, 7, 8, 9, 10, 11, 14, 16, 21, 22, 23, 25. You may do more than 10 problems for extra credit (if you miss some problems later, you can get credit for extra problems in Assignment 1.
  • Solutions to assignment 1.
  • Assignment 2, due October 16. Do any 8 of the following problems: Stein and Shakarchi, Chapter 2: Exercises # 1, 3, 6, 8, 9, 12, 13. Problems # 1, 2, 4, 5.
  • Solutions to assignment 2: version 1 and version 2.
  • Assignment 3, due date to be announced. Solutions posted on MyCourses.
  • Assignment 4, due date TBA: do the following problems: Stein and Shakarchi, Chapter 3: Do any 4 of exercises # 4, 5, 6, 8, 10, 12. Chapter 3: # 14, 17, 19, 20. Problem 1 (extra credit), 2, 3, 5 (extra credit).
  • Assignment 4, due date to be announced. Do the following problems: Stein and Shakarchi, Chapter 4: Exercise # 5, 7, 8. Do any two of the following problems: Stein and Shakarchi, Chapter 4, Exercise # 9, 10, 11, 12. Extra credit: Stein and Shakarchi, Chapter 4: Problems 2, 3. Do any 2 of the following problems: Stein and Shakarchi, Chapter 5, Exercise # 2, 3, 5, 6. Do the following problems: Stein and Shakarchi, Chapter 5, Exercise # 8, 11, 15. Extra credit: Stein and Shakarchi, Chapter 5, Exercise # 15, Problems 2, 3, 4.

    • Web Links

    Complex Analysis:
  • McGill Analysis Seminar
  • Function viewers: an applet
  • Stereographic projection.
  • Spirographs: applet 1 and applet 2.
  • A nice links page.
  • Douglas Arnold's Graphics for Complex Analysis: java or non-java.
  • Curt McMullen's gallery.
  • Minnesota Geometry Center graphics page.
  • A web page on quaternions etc
  • Descriptions of the Mandelbrot and Julia sets.
  • Java applets for drawing the Mandelbrot set: applet 1 and applet 2.
  • Java applets for drawing Julia sets: applet 1, applet 2 and applet 3.
  • Computer code for drawing the Mandelbrot and Julia sets: Java You can also download programs from some of the sites above and from this list.
    • Functional Intro to Analysis and Topology (bottom of the page), this is my old math 354 page
    Calculus, Algebra, Geometry and Differential Equations
    HELPDESK and their email: helpdesk@math.mcgill.ca
    NOTICE: McGill University values academic integrity. Therefore, all students must understand the meaning and consequences of cheating, plagiarism and other academic offences under the Code of Student Conduct and Disciplinary Procedures (see McGill web page on Academic Integrity for more information).
    NOTICE: In accord with McGill University's Charter of Student Rights, students in this course have the right to submit in English or in French any work that is to be graded.
    NOTICE: In the event of extraordinary circumstances beyond the University's control, the content and/or evaluation scheme in this course is subject to change.