189-466A, Complex Analysis, Fall 2002

  • Lecture: MWF 11:30-12:30, Burnside 920

    • Instructor

  • Dmitry Jakobson
  • Office: BURN 1212. Phone: 398-3828. Email: jakobson@math
  • Office hours: TW, 10:00-11:00

    • Marker (if approved)

  • Sidney Trudeau
  • Email: trudeau@math.mcgill.ca
  • Office: BURN 1023

    • Text

  • Required: R. Greene and S. Krantz, Function Theory of One Complex Variable
  • Recommended: Ahlfors, Complex Analysis.
  • Both texts are on reserve in the library.

    • Tests

  • Take-home Midterm: Given out in class on Monday, October 21. Due in class on Friday, November 1, in class: ps format, pdf format.
  • In-class Midterm: Wednesday, October 30, Burnside 1234, 5pm-6pm.
  • Take-home Final (due on Friday, December 20, by 2pm): ps, pdf).
  • In-Class Final: Friday, December 20, Leacock 31, 2pm-5pm
  • Practice problems on linear-fractional transformations (for the final): ps, pdf.

    • Grading

  • Midterm: 25%
  • Final: 50%
  • Homeworks: 25%

    • Misc

  • There will be a supplemental exam, counting for 100% of the supplemental grade. No additional work will be accepted for D, F or J.
  • Homework will be assigned in class and will be due by 5pm by the specified deadline. There will be a 50% deduction for late homework.
  • Related courses: Math 566B (Advanced Complex Analysis) - taught by Prof. Koosis next semester.

    • Various math announcements

  • PUTNAM organizational meeting - Monday, September 9, at 4PM in 1024 Burnside. You can also contact Jim Loveys (loveys@math) or Wilbur Jonsson (jonsson@math).

    Homeworks, handouts

  • Point set toplogy and metric spaces handout: postscript and pdf.
  • Differentiation under the integral sign handout (postscript, pdf)
  • Problem Set 1 (due Monday, September 16 by 5pm): Greene and Krantz, Chapter 1, # 6; 10; 16 & 17 (one problem); 32; 41; 44; 49; 51; 53. Extra credit: Greene and Krantz, Chapter 1, # 57; Also, prove that the convex hull of the roots of a polynomial P(z) contains the roots of its derivative P'(z). (Hint: prove that a straight line cannot separate a root of P'(z) from all the roots of P(z)). Remark: the problems are identical in the 1st and the 2nd editions. Chapter 1: page 1, page 2, page 3, page 4
  • There is a typo in Problem 51: in the last sentence the expression ``dh,dy'' should actually be ``dh/dy.''
  • Problem Set 2 (due Monday, September 30, by 5pm): Greene and Krantz, Chapter 2, # 5; 11; 18bdf; 40. Chapter 3, # 11bcdfi; 14; 16; 21. Extra credit (open deadline): Consider all rearrangements of a conditionally convergent series of complex numbers, such that the new series converges. What sets of limits can one get this way? Reminder: by Gauss's theorem, you can get any real number as a sum of a rearrangement of a conditionally convergent series of real numbers. Chapter 2: page 1, page 2, page 3, page 4, page 5. Chapter 3: page 1, page 2, page 3, page 4, page 5.
  • Problem Set 3 (due date TBA): Greene and Krantz, Chapter 3, # 33; 35; 37; 39; 44; 45. Additional problems: postscript, pdf.
  • Problem Set 4 (due date TBA): Greene and Krantz, Chapter 4, # 5bdef; Chapter 5, # 5; 6; 9; 10acde; 11; 12; 14; 18. Extra credit: let f(z) be an entire function such that for every complex number z, one of the derivatives of f at z vanishes; prove that f is a polynomial.
  • Problem Set 5 (due date TBA): Greene and Krantz, Chapter 4, # 8; 13adf; 24; 35begh; 36bdgi; 49 & 53 (one problem); 57 & 60 (one problem); 63 & 65 (one problem); Prove that the function f(z)=2-2*3z+3*4z^2+...+(-1)^n(n+1)(n+2)z^n has no zeros in in D(0,R) (where R is a real number between zero and 1), for large enough n. Extra credit: Greene and Krantz, Chapter 4, # 68. Find all entire functions f(z) satisfying f(0)=1, f(2z)=f(3z) for all complex z.
  • Problem Set 6 (due date - December 4): Greene and Krantz, Chapter 6, # 3; 7; 10; 12; 17; 21; 23; 24. Additional problems: postscript, pdf.
  • I will be away from Nov. 24 - Nov. 27; I will be back on Nov. 28. There will be two extra lectures on Nov. 18 (5:30-6:30pm), and on Nov. 20 (5pm-6pm), both in Burnisde 1120.

    • Web Links

  • McGill Analysis Seminar
  • Function viewers: an applet and a program together with some files for it. math466.html
  • 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.
  • Oded Schramm's gallery (postscript files).
  • 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, applet 2 and applet 3.
  • Java applets for drawing Julia sets: applet 1, applet 2 and applet 3.
  • Computer code for drawing the Mandelbrot and Julia sets: Java and two www pages of Mathematica code: page 1 and page 2. You can also download programs from some of the sites above and from this list.