Office phone: 510-642-4329.

Office: 923 Evans.

Tentative office hours: Tuesday 2:00-3:30; Thursday 10:00-11:00.

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.

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
book.

Now here are the assignments. Below, "x.y" means exercise y on page x of the book.

- 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, 176.1(a,c,e,g,i).
- 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.

The course
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.

- 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 (chapter IX)

- (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 sign.
- (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.