Math 185: Introduction to complex analysis
UC Berkeley, Fall 2012
Office phone: 510-642-4329.
Office: 923 Evans.
Tentative office hours: Tuesday
2:00-3:30; Thursday 10:00-11:00.
The textbook for this course is Theodore Gamelin, Complex
analysis. This can be ordered online, and a few copies are also
available at the campus bookstore. Please be sure to download the
will sometimes do things differently from the book, or in a
different order. In any case, for every lecture there are corresponding
sections of the book to read, indicated on the syllabus. I may also
occasionally post supplemental notes here. I expect
that the course
will cover most of the first part of the book, some of the second
part, and maybe a couple of topics from the third part.
There are many other complex analysis books out there, with
different styles and emphasis, which you
might find useful. For example, last semester for H185 I
used Complex analysis by Ahlfors. This book is more
difficult than the one we are using, and also a bit old-fashioned,
but very rewarding. Some other nice books at an advanced undergraduate
to beginning graduate level are Complex function theory by
Sarason, Complex analysis by Lang, Functions of one complex variable I by Conway and
Complex analysis by Stein-Shakarchi.
Homework is due on most Tuesdays 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. In addition, you must
acknowledge any collaboration.
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 this means "I don't feel like
explaining that", and what follows 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 the book.
There will be in-class midterms on Thursday 10/4 and Thursday 11/8. The
final exam is on Friday 12/14.
- HW#1, due 9/4.
- HW#2, due 9/11.
- HW#3, due 9/18.
- HW#4, due 9/25.
- HW#5, due 10/2: 106.1,2,4,5; 109.2; 110.5; 116.1(a)-(g).
- HW#6, due 10/16.
- HW#7, due 10/23: 143.2,5; 147.3; 148.7,8; 149.12,13.
- HW#8, due 10/30: 153.3, 157.1(a,c,e,g,i), 158.8, 170.1, 170.2,
- HW#9, due 11/6: 176.3, 181.1(a,c,e), 198.1(a,c,e,g,i), 198.3(a,c,e).
- HW#10, due 11/20: 202.4,8; 205.2,3; 207.1,2; 211.2,4.
- HW#11, due 11/27: 228.3,4; 230.2,4; 235.4; 245.3,7.
- HW#12 consists of the following review
problems and will not be graded.
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%. The homework score will not be
dropped. All grades will be curved to a
uniform scale before being averaged.
The following is the core syllabus, listed in the order in which it
is presented in the book, which is not always the order in which I
will cover it. I may 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, exponential and trigonometric functions (chapter I)
- Holomorphic functions, a.k.a. analytic functions,
a.k.a. conformal maps (chapter II)
- Harmonic functions, review of line integrals (chapter III)
- Complex integration (chapter IV)
- Power series (chapter V)
- Isolated singularities and Laurent series (chapter VI)
- Residues and evaluation of definite integrals (chapter VII)
- Topological considerations: winding numbers, counting
zeroes, simply connected domains (chapter VIII)
- The Schwarz lemma and conformal self-maps of the unit disk
- (Thursday 8/23) Review of complex numbers. See sections I.1 and I.2 of the book. Also, I proved that there is no continuous square root function defined for every complex number. See pages 4-5 of these notes.
- (Tuesday 8/28) Limits of complex numbers. Stereographic
projection and the extended complex plane a.k.a. Riemann
sphere. Convergence and absolute convergence of series. Introduction
to the exponential function. The principal branch of the logarithm
function. Most of this is in Chapter I of the book. (However I
defined the exponential function by the power series, while the
book's definition is something which I would rather regard as
theorem; the book's definition follows from the power series
definition using the power series for sine and cosine, and I will
give a nicer proof a little later.) Also see the above notes.
- (Thursday 8/30) Review of four increasingly strong notions of
regularity from real analysis: differentiable, C^1, C^infinity, and
real analytic. Definition of complex derivatives and basic
examples. Statement that the
above four notions of regularity are equivalent in the complex
case. Complex derivative rules including the chain rule. Started
discussing the Cauchy-Riemann equations, will clarify next time. See
the above notes and sections II.2 and II.3 of the book. (Note that the
book says "analytic" where I say "holomorphic".)
- (Tuesday 9/4) Real differentiability of multivariable functions
and Jacobians. More about the Cauchy-Riemann equations. Harmonic
functions and harmonic conjugates. This material is in section II of the book. See also pages 11-13 and 21-22 of
the above notes.
- (Thursday 9/6) Complex differentiation notation (see equations (2.5) and (2.6) in my notes and pages
124-125 of the book). Review of the inverse function theorem and application to local inverses of holomorphic functions (see section II.4 of the book). Chain rule II (see page 58 in the book and page 14 of my notes). Application to properties of the exponential function (see section 2.4 of my notes).
- (9/11-9/13) Conformal maps and linear fractional transformations. See Sections II.6 and II.7 of the book.
- (Tuesday 9/18) Review of line integrals and Green's theorem. See sections III.1 and III.2 of the book.
- (Thursday 9/20) Closed 1-forms and exact 1-forms. Line integrals involving dz. See sections III.2 and IV.1-3 of the book.
- (Tuesday 9/25) Cauchy's integral formula. See section IV.4 of
the book. The general rule for differentiation under the integral
- (Thursday 9/27) Examples of Cauchy's integral formula. Relation
to the mean value property of harmonic functions (the latter is explained
in chapter III).
- (Tuesday 10/2) Review for the midterm.
- (Thursday 10/4) Midterm #1: covers the lectures through 9/25.
- (Tuesday 10/9) More about the maximum principle. See section
III.5. Proof of the
"fundamental theorem of algebra" using the maximum
principle. Liouville's theorem, and another proof of the fundamental
theorem of algebra. See section IV.5.
- (Thursday 10/11) Introduction to power series. See the first four sections of chapter V.
- (Tuesday 10/16) Proof that a power
series defines a holomorphic function inside its radius of
convergence which can be differentiated
term by term, see section V.3. Proof
that "the radius of convergence is the distance to the nearest
singularity", see section V.4. Manipulation of power series, see section V.6.
- (Thursday 10/18) Guest lecture by Owen Gwilliam: More about
manipulation of power series. Orders of zeroes of holomorphic
functions, see section V.7.
- (Tuesday 10/23) Laurent series, see section VI.1.
- (Thursday 10/25) Classification of isolated
singularities. Partial fractions. See sections VI.2 and VI.4.
- (Tuesday 10/30) The residue theorem, see section VII.1 and a
bit of the following sections in Chapter VII. Section VII.1 may be
covered on the midterm, but material beyond
section VII.1 will not be covered on the midterm.
- (Thursday 11/1) Fun with residue calculus from chapter VII.
- (Tuesday 11/6) Review for the midterm.
- (Thursday 11/8) Midterm #2: covers the lectures through 10/30,
with emphasis on the lectures after 9/25.
- (Tuesday 11/13) More fun with the residue calculus from chapter
VII. (Note that we are skipping sections VII.6 and VII.8.) The
argument principle from section VIII.1.
- (Thursday 11/15) More about the argument principle and winding
numbers. Rouche's theorem. See sections VIII.2,4,6.
- (Tuesday 11/20) Simply connected domains. See section VIII.8.
- (Thursday 11/22) No class (Thanksgiving Day holiday)
- (Tuesday 11/27) More about winding numbers. General version of Cauchy's theorem. (This appears in the book as exercise 245.6, but a nicer way to state the assumption on the domain is that the domain is simply connected.) See sections VIII.6-8.
- (Thursday 11/29) The open mapping theorem, section VIII.4. The Schwarz lemma, section IX.1,2.
- (Tuesday 12/4) Optional review session (RRR week)
- (Thursday 12/6) No class (RRR week)
- (Friday 12/14) Final exam (8-11am, Room 1 LeConte): covers the
whole course, with a bit more emphasis on the later material.