Important announcement about the textbook. There is only one textbook for this course: "Complex Variables" by Levinson and Redheffer. This book is out of print, and you have two options. You can either purchase a photocopied version from UpClose Printing on 6th street, or at considerably greater expense, buy a used copy from either IUB or TIS. Both these bookstores refer to the UpClose notes, but leave the misleading impression that these are additional to the book, rather than an alternative.
F 1/21 -- I was out of town. Guest lecture by Prof. J. Wetzel.
M 1/24 -- Return from trip. Announcement of new homework
policy to handle grad student credit choices. We've pretty much
covered the first three sections of of the book.
W 1/26 -- First "normal" day. Example 1.2.3 and the beginning
of topology. New course organization handed out, Revised Course
Organization. First homework assignment distributed [see
corrections, 1/28 and solutions 1/31]:
First Homework.
F 1/28 -- Corrections to the First homework announced:
1. In #2, the problem is only feasible if you correct a typo!
for 2+3i in the denominator, read 2-3i.
2. In #7, do only the first *two* transformations; the next two are
deferred to the next homework.
3. In #9, as a hint, consider induction, let f_n(z) = z^n + z^{-n}
and consider f_1*f_n. (Here, "_" denotes subscripts and "^" denotes
superscripts and "*" denotes multiplication.). Otherwise, we finished
the first chapter. I also passed out a handout of additional
computations, featuring the quadratic formula, the computation of trig
functions at 2*Pi/5 and the image of vertical and horizontal lines
under w = z^2.
M 1/31 -- Material covered; § 2.1. Handouts:
Second
Homework,
First
Homework Solutions,
and a copy from an old trigonometry
book, giving the value of sine at multiples of 3°.
Being differentiable is harder for a complex function than for a real
function, but the implications are stronger too.
W 2/2 -- Discussion of complex exponentials, the sine
and the cosine. Homework one returned. How to solve it
handout. Additional remarks on homework 1 handout.
F 2/4 -- More of the same, with a one-shot use of the symbols
E(z), C(z), S(z), so we can look at complex functions in terms of
the familiar real functions without having our heads spin. Also,
an introduction to the mysterious complex logarithm, and the
(consequentially) multiple valued za (when a is not an
integer.) Handout on polynomials, including a review.
M 2/7 -- Handouts:
Third
Homework,
Second
Homework Solutions.
Discussion of homework, and also more on the log, on complex powers,
and on the inverse sine.
W 2/9 -- Cauchy-Riemann implies harmonic. How to compute a
harmonic conjugate. The completion of &S 2. Handout on "Dramatis
personae to date".
F 2/11 -- Homework two returned. We agree on an evening date for
the first hour exam: M 2/28, to cover through homework four. We begin
complex integration. Phone and e-mail list distributed; and "today's
tedious calculation" handout.
M 2/14 -- Handouts:
Fourth
Homework,
Third
Homework Solutions.
Some discussion of the homework, and more on complex integration,
plus estimates of the integrals.
W 2/16 -- Homework three returned. Handout on Green's theorem.
First proof of Cauchy-Goursat, and we begin to see the light. Minor
typos reported on hw 3 solutions and hw4; to be specific, the second
integral in 9 should be over Cm.
F 2/18 -- The room for the exam is announced: 145 Altgeld.
This is the big day when we prove the Cauchy Integral formula, and
start talking about its amazing consequences. In particular, if f is
analytic at z, then all its derivatives exist and are analytic, too.
M 2/21 -- Handouts:
Fifth
Homework,
Four
Homework Solutions. Handwritten notes on a few useful
calculations. Note that the fifth homework is due on 3/1,
not 2/28, as might have been expected. Review of Cauchy, for those who
weren't in class 2/18, plus Liouville's Theorem and the beginning of
the Fundamental Theorem of Algebra.
W 2/23 -- Homework four returned, with a sheet of corrections
and additional remarks. More of FTA, with a crucial lemma on polynomial growth,
a few examples, Morera's Theorem, and the beginning of Taylor Series.
F 2/25 -- Handout (the lemma, some geometric series algebra), and
a detailed proof of the uniform convergence of Taylor series on
appropriate disks.
M 2/28 -- Review session for first exam, which will be
held in the evening, starting at 7:00 pm tonight in 145 Altgeld.
M 2/28 (evening) --
First Test.
W 3/1 -- Brief discussion of class performance on the exam.
Behavior of analytic functions near zeros. Jump ahead to the
reflection principle. Averaging property of harmonic functions.
The Maximum Principle
F 3/3 -- Handouts:
Sixth
Homework,
Fifth
Homework Solutions. Homework five discussed. More on the
Maximum Principle and its ramifications. L'Hopital's Rule. The
Schwarz Lemma. Beginning of isolated singularities and broad overview.
M 3/6 --Handout: More on polynomials, a continuation
of the handout from 2/4/00. A more careful and detailed account of
the classification into removable singularities, poles of order m
and essential singularities. The Riemann Theorem (bounded near an
isolated singularity implies that it's removable.)
W 3/8 -- Handout:
Seventh
Homework. The Weierstrass-Caseroti Theorem (essential
singularity at alpha means that the image of every punctured
neighborhood of alpha is dense in C. Homework six made due on 3/9
either in my mailbox or outside my office.
F 3/10 -- Informal meeting to answer questions. Material covered
was the proof of the Cauchy-Goursat Theorem from § 3.4, which was
omitted in the usual sequence. Distribution of Sixth
Homework Solutions to those who didn't get it on Thursday
M 3/20 -- Distribution of Sixth
Homework Comments and Corrections
The beginning of the discussion of Laurent
series -- any analytic function f with an isolated singularity
at alpha can be written as the sum of a function analytic at
alpha and a function analytic everywhere in C, except alpha.
W 3/22 -- Visit from Zsuzsanna Fagyal and historical notes
about some of the mathematicians mentioned in the course.
More details on the Laurent series, and how it
can be used to classify isolated singular points.
F 3/24 -- Handouts: Seventh
Homework Solutions, Eighth
Homework. Discussion of the homework.
Beginning of section four, and the generalization from star-shaped
regions to simply connected regions.
M 3/27 -- Distribution of Seventh
Homework Comments and Corrections. Applications of the broader
version of Cauchy's formula and the Residue Theorem.
W 3/29 -- Examples of the residue theorem in action; an
introduction to the "Protractor Contour"; the large semi-circle in
the upper half plane with the diameter on the real axis from -R to R.
F 3/31 -- Handout: Eighth
Homework Solutions. Discussion of the homework and more examples
of the residue theorem in action, Jordan's Lemma, and the way to
integrate real rational functions p(x)/q(x) from -infinity to
infinity, when q is positive and has degree two or more greater
than the degree of p.
M 4/3 -- Review for the first part of Monday's class,
then §4.4.
Second Exam, at 7 pm, in the same room as before. Here is
a copy of the second test: Second Test.
W 4/5 -- Handout for homework nine:
Ninth
Homework. Review of exam and integration around branch cuts.
F 4/7 -- More integration around branch cuts and Rouche's Theorem
and the N-P Theorem.
M 4/10 -- Ninth
Homework Solutions and Tenth
Homework. Discussion of homework and more on § 4.6.
Beginnings of the discussion of mappings.
W 4/12 -- Handout: More on
Homework Nine and more on bilinear transformations.
F 4/14 -- Yet more on bilinear transformations.
M 4/17 -- Tenth
Homework Solutions and Eleventh
Homework Discussion of homework and bilinear transformations.
W 4/19 -- Finally, an end to bilinear transformations! And
a very brief discussion of the Dirichlet problem.
F 4/21 -- Careful review of Homework 10, and the beginning
of the discussion of local mappings.
M 4/24 -- Eleventh
Homework Solutions More on local and global mappings.
W 4/26 -- no class (In favor of Test Two at night.)
F 4/28 -- Last new material of the course, a completion of
some global mapping theorems.
M 5/1 -- Class at 10 will be review. Third Exam, at 7pm as usual.
W 5/3 -- Return of third exam.
M 5/8 -- Final Exam, in class, 8am - 11am.