MAT 542 Complex Analysis I, Spring 2007.

This is an introductory course in complex analysis. Although we will start from the foundations and try to make the course self-contained, some basic knowledge appropriate background from real analysis, topology and algebra will be useful.
The final exam is now graded!
The grades are available in the Blackboard. (I put in the exam scores, letter grades for the exam, and the course grades. The course grades will be in Solar shortly). The exam letter grade is supposed to give you an idea of how you did. B roughly corresponds to four problems solved. If your grade is C or below, I feel you are not quite prepared for the Comps yet. If you're unable to use the Blackboard system, you can email me for your grades.

Sorry the grading took me a while.

A few comments on the test:

in #1, I meant to ask for a Laurent series centered at 0 valid around 2, but the words "centered at 0" were missing from the question, so I accepted any other expansions provided they were correct (Taylor series centered at 2, Laurent centered at 1, etc).

#4 was a very easy application of the open mapping, but a lot of people were tricked into harmonic functions arguments (with varying degrees of success)

in #7, most people tried to use rotation to reduce to the usual reflection principle (which is correct), but few people got the right answer. The answer, however, is easy to guess as points symmetric wrt imaginary axis should go to symmetric points -- so is f(x+iy)= u+iv, then f(-x+iy)= -u+iv.

  • References :
    Serge Lang, Complex Analysis
    John B. Conway, Functions of One Complex Variable I
    Lars Ahlfors, Complex Analysis
    The first two of these books are available in the campus bookstore, and all three are placed on reserve in the library. You are not required to have any of these texts, but we will loosely follow Lang at least part of the time.
  • Exams: there will be a midterm exam and a final exam (TBA).
    The midterm will be on Mar 15, Thursday, 9:40-11:10.
    Final exam info: the exam will be given in class on May 3, Thursday (the last class of semester), 9:00-11:10.
  • Homework: weekly assignments will be posted on this page. Homework will constitute a significant part of your course grade.

    Important: Please write up your solutions neatly, be sure to sign it and staple all pages. Illegible homework will not be graded. You are welcome to collaborate with others and even to consult books, but your solutions should be written up in your own words, and all your collaborators and sources should be listed.

    Homework 1, due Feb 6.
    Homework 2, due Feb 13.
    Homework 3, due Feb 20.
    Homework 4, due Feb 27.
    Homework 5, due Mar 6.
    Homework 6, due Mar 13.
    Homework 7, due Mar 27.
    Correction (3/22/07) Question 6 (i) on Homework 7 was incorrect as stated. To make the statement correct, we have to understand "accumulation points" in a rather special sense. The question is now corrected.
    Homework 8, due Apr 17.
    Homework 9, due Apr 24.
    Homework 10, due May 1.

  • Syllabus: we will follow the basic outline from the graduate core course requirements (see below), not necessarily in the same order, with additional topics as time permits. (In particular, I'd like to get a glimpse into complex dynamics and discuss the definition and basic properties of the Julia set).
    1. The field of complex numbers, geometric representation of complex numbers
    2. Analytic functions
      • Definition, Cauchy-Riemann equations
      • Elementary theory of power series, uniform convergence
      • Elementary functions: rational, exponential and trigonometric functions
      • The logarithm
    3. Analytic functions as mappings
      • Conformality
      • Linear fractional transformations
      • Elementary conformal mappings
    4. Complex integration
      • Line integrals and Cauchy's theorem for disk and rectangle
      • Cauchy's integral formula
      • Cauchy's inequalities
      • Morera's theorem, Liouville's theorem and fundamental theorem of algebra
      • The general form of Cauchy's theorem
    5. Local properties of analytic functions
      • Removable singularities, Taylor's theorem
      • Zeros and poles, classification of isolated singularities
      • The local mapping theorem
      • The maximum modulus principle, Schwarz's lemma
    6. The calculus of residues
      • The residue theorem
      • The argument principle
      • Rouche's theorem
      • Evaluation of definite integrals
    7. Power series
      • Weierstrass theorem
      • The Taylor and Laurent series
      • Partial fractions and infinite products
      • Normal families
    8. The Riemann mapping theorem
    9. Harmonic functions
      • The mean-value property
      • Harnack's inequality
      • The Dirichlet problem
    10. Picard's theorem

    Students with Disabilities: If you have a physical, psychological, medical, or learning disability that may impact on your ability to carry out assigned course work, you are strongly urged to contact the staff in the Disabled Student Services (DSS) office: Room 133 in the Humanities Building; 632-6748v/TDD. The DSS office will review your concerns and determine, with you, what accommodations are necessary and appropriate. A written DSS recommendation should be brought to your lecturer who will make a decision on what special arrangements will be made. All information and documentation of disability is confidential. Arrangements should be made early in the semester so that your needs can be accommodated.