Math H185: Honors introduction to complex analysis
UC Berkeley, Spring 2012
Office phone: 510-642-4329.
Office: 923 Evans.
Tentative office hours: Wednesday 2:00-4:00.
The textbook for this course is Ahlfors, Complex analysis, 3rd
edition, Addison-Wesley. A custom edition is available in the Cal
store at a reduced price (but still rather expensive, sorry). It's a
nice book, but you can maybe survive without it if you copy the
homework assignments from someone. Lectures will generally correspond
to parts of the book (indicated on the syllabus), but I will sometimes
do things differently. I may also post some notes here.
Homework is due every Tuesday (except for the first two Tuesdays and
the weeks of the midterms) at the beginning of class. You can either
bring it to class or slide it under my office door. (If it doesn't
fit under the door, please be more concise!) Homework assignments
will be posted below at least a week before they are due. No late
homeworks will be accepted for any reason, so that we can go over the
homework right after it is handed in (which is when people are most
eager to see solutions to troublesome problems). However it is OK if
you miss the deadline once or twice, because your lowest two homework
scores will be dropped.
When preparing your homework, please keep the following in mind:
1) You are encouraged to discuss the homework problems with your
classmates. Perhaps the
best way to learn is to think hard about a problem on your own until
you get really stuck or solve it, then ask someone else how they
thought about it. However, when it comes time to write down your
solutions to hand in, you must do this by yourself, in your own
words, without looking at someone else's paper.
2) All answers should be written in complete, grammatically correct
English sentences which explain the logic of what you are doing,
with mathematical symbols and equations interspersed as appropriate.
Results of calculations and answers to true/false questions
etc. should always be justified. Proofs should be complete and
detailed. Avoid phrases such as
"it is easy to see that"; often what follows such a phrase is actually
a tricky point that needs justifiction, or even false. You can of
course cite theorems that we have already proved in class or from the
Now here are the assignments. Below, "x.y" means exercise y on page x of Ahlfors.
There will be in-class midterms on Thursday 3/1 and Thursday 4/5. The
final exam is on Wednesday 5/9.
- HW#1, due 1/31: 6.1, 9.4, 15.2, 16.2, 17.5, 20.1, 28.4, 28.5, 37.3. Note that in 28.5, "simultaneously" means "if and only if". Most of these problems are elementary because we have not introduced much material yet. They are not always easy though; there are some pretty tough problems in chapter 1. Selected solutions
- HW#2, due 2/7: 28.3 (just use integration), 32.2, 32.3 (give a formula for the polynomial), 41.7, 41.8, 41.9, 47.6, 47.9 (use log to define angle). Also, prove that there does not exist a continuous logarithm function defined on the set of all nonzero complex numbers. Some of these are a bit tricky; just do what you can. Complete solutions
- HW#3, due 2/14: 78.4, 80.2, 80.3, 83.4, 84.1, 88.1, 88.5. (This last problem is challenging. Hint: think about eigenvectors and eigenvalues, and make sure you know what a unitary matrix is.) Selected solutions
- HW#4, due 2/21. Page 96 problems 1,2, page 108 problems 1-6. Selected solutions
- HW#5, due 2/28. Page 120 problems 1,2,3; page 123 problems 1, 2, 3. Also, prove that the winding number of a loop around a is invariant under homotopy through loops that do not intersect a. (Give two proofs: one using continuous loops and homotopies and the "lifting lemma" definition of winding number, and one using smooth loops and homotopies and the integral formula for winding number.)
- No homework due 3/6 because of the first midterm.
Midterm 1 solutions
- HW#6, due 3/13: 32.1 (yes that's way back in chapter 2), 130.1, 130.2, 130.3 (just do e^z^), 130.4, 130.5. Additional problems: (1) Show that every holomorphic bijection from the Riemann sphere to itself is a linear fractional transformation. (2) [Extra credit] Find all z such that the sum of z^n/n converges. Selected solutions
- HW#7, due 3/20: 133.1, 133.3, 136.1, 136.3 (just the part referring to 136.1), 136.5 (you can use what we proved in class), 154.1, 154.2.
- HW#8, due 4/3: 161.1, 161.3 (lots of calculations). Selected solutions
- No homework due 4/10 because of the second midterm.
Midterm 2 solutions
- HW#9, due 4/17: 171.5, 184.3, 184.5, 186.1, 186.2 (you can be non-rigorous about the topological aspects of this), 186.4, 190.1.
- HW#10, due 4/24. pdf. Since we did not introduce much new material this week, this homework explores some interesting supplementary topics which review previous material.
- HW#11, due 5/1. pdf.
There will be no makeup exams. However you can miss
one midterm without penalty, as explained in the grading policy below.
There is no regrading unless there is an egregious error
such as adding up the points incorrectly. Every effort is made to
grade all exams according to the same standards, so regrading one
student's exam would be unfair to everyone else.
grade will be determined as follows: homework 25%, midterms 25% each,
final 50%, lowest exam score -25%. All grades will be curved to a
uniform scale before being averaged.
The following is the core syllabus; I hope to discuss some
additional topics as time permits. Math 104 or equivalent is a
prerequisite; I will briefly review some of this material as needed,
but assume that you are generally comfortable with basic real
- Complex numbers (Ahlfors chapter 1)
- Holomorphic functions, power series, exponential and trigonometric
(Ahlfors chapter 2)
- Conformal maps (parts of Ahlfors 3.2-3.4)
- Complex integration and the fundamental theorems of complex analysis
- Residues and evaluation of definite integrals (Ahlfors 4.5)
- Laurent series (Ahlfors 5.1)
Review sheet for the final exam
- (1/17) Rapid review of complex numbers. See Ahlfors chapter 1.
- The Riemann sphere. (See end of Ahlfors chapter 1.)
- Review from real analysis of the notions of differentiable function, continuously differentiable function, infinitely differentiable function, and real analytic function, and why each of these conditions is strictly stronger than the previous one.
- Review of derivatives of multivariable functions in real analysis. (These last two items aren't covered so much in Ahlfors but I will post some notes about them here.)
- (1/24) Definition of complex differentiable (holomorphic) functions, and basic properties. (See the first few pages of Ahlfors chapter 2. Note that page 27 is somewhat impenetrable; don't worry about it.)
- (1/26) (Guest lecture by Prof. John Lott) Power series and the exponential function.
- (1/31) More about the exponential function. Logarithms. Started on harmonic functions. This is from various parts of Ahlfors chapter 2. Also, here are my notes for the first few lectures, which sometimes use a different approach than Ahlfors.
- (2/2) Harmonic functions and harmonic conjugates. Conformal maps.
- (2/7) Linear fractional transformations. Digression on the complex projective line and what it means to be holomorphic at infinity.
- (2/9) More about LFT's. A bit about Riemann surfaces from the end of Ahlfors chapter 3.
- (2/14) Review of line integrals. See the beginning of Ahlfors chapter 4. (Ahflors defines the integral of fdz first, while I defined the integral of fdx + gdy first because that should be familiar from Math 53, except that now f and g are complex valued.)
- (2/16) Cauchy's theorem; see Ahlfors chapter 4.1.
- (2/21) Winding number and Cauchy's integral formula; see
Ahlfors chapter 4.2.
- (2/23) Differentiation under the integral sign and Cauchy's
integral formula for higher derivatives.
- (2/28) Review for the first midterm.
- (3/1) Midterm #1
- (3/6) Taylor series, poles. See Ahlfors chapter 4.3.
- (3/8) All about counting zeroes (and poles) of holomorphic (meromorphic) fuctions. See Ahlfors sections 4.3.3 and 4.5.2.
- (3/13) Open mapping theorem, maximum principle, Schwarz lemma.
See Ahlfors sections 4.3.3 and 4.3.4.
- (3/15) Tried to explain the "big picture" behind Ahlfors section 4.4, in particular homology.
- (3/20) "Fun" stuff from Ahlfors section 4.5.
- (3/22) Harmonic functions. Ahlfors section 4.6.
- (3/27) No class (spring break)
- (3/29) No class (spring break)
- (4/3) Review for the midterm.
- (4/5) Midterm #2
- (4/10) Laurent series. Ahlfors section 5.1.
- (4/12, 4/17) Infinite partial fractions. Ahlfors section 5.2.
- (4/19, 4/24) Introduction to elliptic functions. Ahlfors chapter 7.
- (4/26) TBA
- (5/1) Optional review session (RRR week)
- (5/3) No class (RRR week)
- (5/9) Final exam, 11:30-2:30, location TBA