College of the Holy Cross
Department of Mathematics and Computer Science

• Syllabus (pdf version)
  
  
• Assignments:
  
  
  • #1 Due Tuesday, January 27. Read Sections 1.1 and 1.2.
    • Sections 1.1: 2ace, 3defg, 5di, 7, 10, 12a.
      
      
    • Check the two links below to pages written by David E. Joyce of Clark
      
      • Complex Numbers
      • Mandelbrot and Julia Set Explorer
        
        
    • Handout 1: Complex Numbers (pdf version)
      
      
    • Assignment 1 Solutions (pdf version)
      
      
  • #2 Due Tuesday, February 3. Read Sections 1.2, 1.3 and 1.4.
    • Sections 1.2: 11 - 14, 23 - 26, 28, 29.
    • Sections 1.3: 2, 4, 5, 6, 10, 11 (for 2,4,5, and 6).
    • Extra credit: Problems 2 - 6 on Handout 2.
      
      
    • Handout 2:Basic Point-set Topology in C (pdf version)
      
      
    • Assignment 2 Solutions (pdf version)
      
      
  • #3 Due Tuesday, February 10. Read Sections 1.4 and 1.5.
    • Sections 1.4: 2, 3, 6, 11, 14 - 16, 22, 23, 31, 33.
    • Sections 1.5: 4, 8, 12, 15a, 16 (cosine only), 18 (cosine only), 23, 24.
    • Exercises 1 - 4 on Handout 3.
      
      
    • Extra credit: Receive 10 extra credit points each for going to two of the job interview presentations. In order to receive credit, you must speak to me about the presentation or send me your comments by e-mail. The presentations are scheduled for Thursday, February 12, Friday, February 13, Tuesday, February 17, and Wednesday, February 18, all at 4 PM. (You are of course encouraged to go to more than two talks. Credit will only be given for two talks, however.)
      
      
    • Check out the link to the complex function viewer by Prof. Sérgio Santa Cruz, Paulo Soares Fonseca Departamento de Matemática Universidade Federal de Pernambuco:
      • Bombelli
        
        
    • Handout 3: The Riemann Sphere and Stereographic Projection (pdf version)
      
      
    • Assignment 3 Solutions (pdf version)
      
      
  • #4 Due Tuesday, February 17. Read Section 1.6.
    • Sections 1.6: 2, 3, 4, 6, 8, 9, 12, 14, 15, 16
    • Exercises 1 - 4 on Handout 4.
      
      
    • Extra credit: Receive 10 extra credit points each for going to two of the job interview presentations. In order to receive credit, you must speak to me about the presentation or send me your comments by e-mail. The presentations are scheduled for Thursday, February 12, Friday, February 13, Tuesday, and February 17. (You are of course encouraged to go to more than two talks. Credit will only be given for two talks, however.)
      
      
    • Assignment 4 Solutions (pdf version)
      
      
    • Handout 5: Contour Integrals (pdf version). In-class discussion for February 12.
      
      
  • #5 Due Tuesday, February 24. Read Sections 2.1 (excluding 2.1.1) and 2.2.
    • Section 2.1: 1acefg, 14, 16, 17, 20, 21, 22, 24, 25.
    • Exercises 1 - 6 on Handout 6.
      
      
    • Handout 6: Complex differentiation (pdf version)
      
      
    • Assignment 5 Solutions (pdf version)
      
      
  • #6 Due Tuesday, March 2. Read Sections 2.2 and 2.3 (excluding 2.3.1)..
    • Section 2.2: 2, 4, 6, 8, 12, 14, 15, 22.
    • Section 2.3: 1, 2, 4, 5, 6, 9, 10, 14, 15, 16.
      
      
    • Assignment 6 Solutions (pdf version)
      
      
  • #7 Due Tuesday, March 16. Read Section 2.4.
    • Sections 2.4: 1, 3, 4, 6, 8, 9, 12, 14, 18, 22, 24, 28.
      
      
    • Handout 7: Analytic functions, Cauchy integral formula, and power series (pdf version). In-class, Thursday, March 4. Problems 3, 4, and 5 may be done for extra credit.
      
      
    • Assignment 7 Solutions (pdf version)
      
      
    • Extra Credit Review Problems. Do up to 10 of the following problems from the text for extra credit:
      • Section 1.2: 4, 8, 18, 36
      • Section 1.3: 12
      • Section 1.4: 4, 8, 10, 12, 18, 20
      • Section 1.5: 6, 14, 20
      • Section 1.6: 10, 13
      • Section 2.2: 10, 16, 18
      • Section 2.3: 8, 12, 18a, 21a,c
      
      
  • Take home midterm exam will be distributed in class, Thursday, March 18 and is due Thursday, March 25 in class. 1/5 of the midterm grade will an oral portion.
    
    
    • Handout 8: Oral Portion of the Mid-term Exam (pdf version). Information for the oral portion of the mid-term.
      
      
    • Handout 9: Take home portion of the mid-term exam (pdf version). Due in-class, Thursday, March 25.
      
      
    • Take home solutions (pdf version)
      
      
  • #8 Due Tuesday, April 13. Read Sections 2.5, 2.6, 3.1, 3.2.
    • Section 2.5: 2, 8, 14, 22, 23a.
    • Section 2.6: 2, 5, 10, 22a, 26a.
    • Section 3.1: 6, 7, 13, 22.
      
      
    • Assignment 8 Solutions (pdf version)
      
      
    • Handout 10: Groups of Linear Fractional Transformations (pdf version)
      
      
    • Handout 10 Solutions (pdf version)
      
      
    • Extra Credit: Receive 10 extra credit points either for attending the department colloquium: ``The Geometries of Escher'' on Monday, April 5, at 4 PM (O'neil 112) or by reading the article ``Artful Mathematics: The Heritage of M.C. Escher'' (Escher). and commenting on the use of complex analysis in the article. In either case you have to send me an e-mail with your comments to receive credit.
      
      
  • #9 Due Tuesday, April 20. Read handout up through page 222.
    
    
    • Handout 11: Not too complicated Julia sets (pdf version)
      
      
    • Handout 11 Solutions (pdf version)
      
      
  • #10 Due Thursday, April 29. Read sections 14.2 and 14.3 of the handout.
    
    
    • Problems 7, 9, 10, 11, 12, 15, 16, 17, and 18 of the handout.
      
    • Read and work through the The Mandelbrot Set Explorer on Bob Devaney's web page at Boston University.
      
      
    • Assignment 10 Solutions (pdf version)
      
      
  • Take Home Final ExamThe written portion is due 7 days after you pick it up or Friday, May 14, whichever comes first. This covers textbook material covered since the last test. The oral portion of the midterm must be scheduled between Wednesday, May 5 and Thursday, May 13. It covers material on complex dynamics from the handout, Chapter 14 of Fractal Geometry: Mathematical Foundations and Applications, by Kenneth J. Falconer, class notes, and from the Mandelbrot Set Explorer on Bob Devaney's website.
    
    
    • Handout 12: Take-home portion of the Final Exam (pdf version).
      
      
• Links of interest:
  
  
  • Resources on definitions, terminology, and concepts:
    
    
    • Complex Numbers, from David E. Joyce of Clark University. This is a nice concise on-line introduction to complex numbers.
      
      
    • Complex Analysis--Maple Powertools, from Maplesoft, which produces the software package Maple. This is a complete web course on complex analsys. It contains downloadable Maple worksheets on a variety of topics.
      
      
    • Complex Analysis--Mathworld, from Wolfram Research, which produces the software package Mathematica. A good on-line source with short entries on a wide range of mathematical topics. This link is to the main page for complex analysis.
      
      
  • Complex analysis software and demos:
    
    
    • Bombelli, a java complex function viewer from Prof. Sérgio Santa Cruz, Paulo Soares Fonseca Departamento de Matemática Universidade Federal de Pernambuco Brazil.
      
      
  • Resources on Fractals and dynamics:
    
    
    • Fractals--Mathworld, from Wolfram Research. This link is to the main page for complex analysis.
      
      
    • Dynamical Systems--Mathworld, from Wolfram Research. This link is to the main page for dynamical systems.
      
      
  • Fractals and dynamics software and demos:
    
    
    • Mandelbrot and Julia Set Explorer, from David E. Joyce of Clark University. A terrific, easy to use interactive web page for exploring these sets.
      
      
    • The Mandelbrot Set, from Peter Alfeld of the University of Utah. A more involved interactive page.
      
      
    • The Dynamical Systems and Techonology Project, from Robert L. Devaney of Boston University. Check out the Mandelbrot Set Explorer link.
      
      
    • Mandelbrot/Julia Set Browser, from Michael G. Henderson of Los Alamos National Laboratory.
      
      
  • Minimal Surfaces (an application of harmonic functions):
    
    
    • Minimal Surfaces--Mathworld., from Wolfram Research. This link is to the page for minimal surfaces and contains a basic introduction.
      
      
    • Minimal Surfaces, from Matthias Weber of the Indiana University, Bloomington. This contains a dazzling gallery of ray-traced images of minimal surfaces.
      
      
    • GANG. Home page of the Geometry, Analysis, Numerics Graphics group at the University of Massachusetss, Amherst.
      
      
    • Minimal Surface Library, from Martin Steffens and Christian Teitzel of GRAPE, University of Bonn.