Final Exam: Friday 12/17/04    10:30-12:45    LT005
See me on Monday or Tuesday if you need to take the Make-up on Thursday.

Office Hours:                Office Hours during Exam Week:
    M 11:00-12:30            M    9:30-11:30
    T   2:00-3:30                T    10:30-12:30 (Math Lab)
   Th 3:30-5:00                 W    1:30-3:15
    F  2:00-3:15                 Th    2:00-4:00

Assignments and announcements:

Thursday 12/9:    Review for Examination

Tuesday 12/ 7:    We discussed uniform convergence and proved that uniform limits of
        continuous functions are continuous and that uniform limits commute with integration.
    We showed that power series converge uniformly on close disks inside their circle of convergence.
        From this we concluded that power series are analytic inside their circle of convergence.
    Please be prepared to present a problem on Thursday if you have not presented at least three.

Thursday 12/2:    We began our disucssion of power series expansion of analytic functions
        by stating the main results and computing several examples.
    In Tuesday's class we will discuss the proofs of these results.
   The homework will be due on Thursday of next week, not Tuesday.

Tuesday 11/30:    After the home presentations, I did a review of the basic theory
        of sequences and series and its generalization to complex sequences and series.
        Thie material is covered in section 3.1
    On Thursday, we will discuss uniform convergence of of analytic functions, power series
        and Taylor's Theorem for analytic functions. This material is covered in
        sections 3.1 and 3.2
    For next Tuesday, please prepare 3.1 # 1, 2, 4, 6, 7, 10, 14

Thursday 11/18:  We discussed the proofs of some of the Topological Exercises
        in some detail.
    We proved the Global Maximum Modulus Principle.
   This class will not meet next week.
  For  Tuesday 11/30, please prepare 2.5 # 1, 2, 3, 5, 15, 18
        and the last four propostions from the Topological  Exercises.  You may use
        the first four propositions in proving the last four.

Tuesday 11/16:    After reviewing the homework, we began a discussion of the
        Maximum Modulus Principle.
    Please prepare 2.5 #1, 2, 3, 5, 15, 18 for Tuesday 11/30.
    This class will not meet on Tuesday 11/23.

Thursday 11/11:    We derived the Cauchy Inequalities, Liouville's theorem,
        the Fundamental Theorem of Algebra and Morera's Theorem from
        the Cauchy Integral Formulas.
    I gave the abstract topological definition of of the boundary of a set and several
        related ideas.  I also stated several propositions about thiese ideas without proof.
    For Tuesday's homework assignments, you may submit solutions to 2.4 # 3, 6, 16
        or proofs of the propositions about the boundarty etc.  The proof of a single propostion
        counts as a single problem.
    Click to download the definitions and propositions as a Word document.

Tuesday 11/9:    After presenting the homework, we discussed the derivation of the
        Cauchy Integral Formulas for derivatives.
    On Thursday, we shall discussthe Cauchy Inequalities, Liouville's Theorem,
        the Fundamental Theorem of Algebra, and other results that follow from
        the Cauchy Integral Formulas. if time permits, we shall discuss the Maximum Modulus Principle,
    For next Tuesday,please prepare 2.4 # 3, 6, 16.

Thursday 11/04:    In today's class, we derived the Cauchy Integral Formula for f(z),
        worked a few example like like Example 2.4.14 in the text and finished by stating,
        but not deriving the Cauchy Integral Formula for the derivatives of f(z).
    Since we did not get as far I expected, I am changing the homework asssignment.
        For next Tuesday, please prepare 2.3 # 5, 8 and 2.4 # 2, 5, 13, 15, 18, 19.
        You may submit any of the other problems that I assigned on Tuesday, but they require
        theorems that we have not discussed yet.

Tuesday 11/2:    After reviewing the homework, we discussed the proof of the homotopy
        forms of Cauchy's theorem.
    On Thursday, we will discuss Cauchy's Integral Formula and it consequences.  This material
        is covered in section 2.4 of the text.
    Please prepare 2.3 # 5, 8 and 2.4 # 2, 3, 5, 12, 15, 16, 18, 19
 

Thursday 10/28:    We proved Cauchy's Theorem for an open disk without contiunity
        assumptions on f'(z).
    We discussed the defintion of homotopy and several related examples and propositions.
        We stated but, did not prove the homotopy version of Cauchy's Theorem.  You may use
        this theorem without proof in doing  this weeks's homework.
For next Tuesday, please prepare 2.2 # 1, 2, 3, 5, 11 and 2.3 # 4, 7d, 9b.
 

Tuesday 10/26:    We discussed some examples of the use of Cauchy's theorem and
        we proved Cauchy's theorem for rectangles.
    For next Tuesday, please prepare 2.2 # 1, 2, 3, 5, 11 and 2.3 # 4, 7d, 9b.
        Be sure that you write out all of the explanations necessary to justify your solutions to
        these problems.  You may omit routine algebraic calculations, but if you claim that two curves
        are homotopic, write out the homotopy.

Thursday 10/21: We finished the proof that path independence in a domain is equivalent to
        the existence of and an antiderivative.
    No assignment overt the break.

Tuesday 10/19:    After disccuusing the rersults of the midterm examination, we
        began to prove that path independence is equivalent to the existence of
        an antiderivative. (Thm 2.1.9)
    For Thursday, please review  problems 2.1 # 2, 5, 7, 8, 9, 10, 12, 14.  Be sure
        that you can solve all of them.  Be prepared to ask questions if you cannot.

Tuesday 10/12:    We presented the homework and reviewed for the test on Thursday.
    Click her to download the review sheet.

Thursday 10/7:    We introducted the comple line integral.  We discussed all of
        section 2.1, except for Theorem 2.1.9
    Please prepare problems 2, 5, 7, 8, 9, 10, (12, 14) for Tuesday.
    We decided to have the midterm examination on Thursday 10/14.

Tuesday 10/5:    We spent the entire class discussing the homework.

Thursday 9/30:    We discussed, the Inverse Function Theorem for Analytic functions,
        harmoinic functions and the harmonic conjugate, and the deriveative of the expoenetial,
        logarithm and trigonometric fucntions. This material will be found in sections 1.5 and 1.6
     For next Tuesday, please prepare 1.5 # 22, 27, 28 and 1.6 # 1, 2, 3, 4, 6, 7.8, 12, 14

Tuesday 9/28:    After reviewing the homework, we disucssed the difference between having a
        derivative as a real function from R^2 to R^2 and having a complex derivative.
    On Thursday, will discuss harmonic functions and we th derivatives of the remaing elementary
        functions.  This material will be found in sections 1.5 and 1.6
    On Thursday, we will also vote on the date of the midterm.
    For next Tuesday, please prepare 1.5 # 22, 27, 28 and 1.6 # 1, 2, 3, 4, 6, 7.

Thursday 9/23:    We began discussion of the complex derivative and analytic functions.  We got
        approximately to pp. 66-68 in section 1.5.
    For next Tuesday, please prepare 1.5 # 2, 3, 6, 7, 9, 10, 15, 16.  Please hand in problem chosen
        from the even numbered problems.  I will collect the homework at the beginningof the class, so be
        sure that you keep extra copies if you want to use your notes when you present at the board.

Tuesday 9/21:    After going over the homework, we discussed the definition of path-connected domains.
    On Thursday, we wil begin our discussion of  complex differentiation.
    For next Tuesday, please prepare 1.5 # 2, 3, 6, 7, 9, 10, 15, 16.
 

Thursday 9/16: We discussed limits and continuity in the complex plane, and open and closed sets.
        This material is covered is section 1.4 pp. 41-46.  We did not discuss connectedness, path connectedness
        or compactness.
    For next Tuesday, please prepare 1.3 # 15, 17, 22, 30c, 31, 34  and 1.4 # 2, 3, 4, 7, 8, 13, 17, 20, 21

Tuesday 9/14:  After going over the homework, we finished the discussion of comple trigonometric functions.
    Please prepare problems 1.3 # 15, 17, 22, 30c, 31, 34
    In the next clas we wil discuss some but not all of the material in section 1.4 Try to readpp.41-44 and
        the discussion of the Riemann sphere on pp. 54-55.

Thursday  9/9:    I returned the homework from Tuesday.
    We discussed the complex exponential and logarithm functions., but not the complex trig functions from
        Section 1.3.
    For next Tuesday, please read section 1.3 and prepare Problems 2a, 6, 9, 11, 14, 21, 23 from section 1.3

Tuesday 9/7:    After reviewing the homework, we discussed the matrix representation of complex numbers.
        Please review th properties of this representation as described in problem 1.2 # 20.
    In Thursday's class,  we will discuss the complex versions of the elementary functions of Calculus.
        This material is covered in section 1.3 of the text.

Thursday 9/2:    We discussed the geometric significance of complex multiplication and derived D'Moivre's
        Theorem and the existence of nth roots.
    For Tuesday' class please read Section 1.2 prepare problems 1, (3, 4), (7, 8), 9, (11, 12), 15, 23, 26, 29.
    Remember that you must hand in written solutions to two of the problems that I assigned this week
        at the beginning of Tuesday's class.  I will also expect three of you to present problems at the board.

Tues  8/31:  We discussed the definitons of complex numbers and their basic algebraic properties
    For next class, please look as sections 1.1 and 1.2.
    Home problems: Sect 1.1 # (1,2), 4, 5, 7(Proof or counterexample),12a, 16(formal induction proof),19.