Math 132: Complex Analysis for Applications, Spring 2011
Math 132: Complex Analysis for Applications, Spring 2011
For information on this course, the instructor and TA, homework, exams, grading, holidays etc.
please read the Course Organization Handout.
More information can be found in the
course description
and at this page.
The TA's website is located here.
Announcements
- June 9 2011: The final exam grades and the final grades for the course have been uploaded to
MyUCLA. The solutions to the final exam can be found here for
version 1 and
version 2. Regardless of your result, I hope you
liked the course and learned lots of new cool mathematics. Enjoy your vacation and good luck in your
future endeavors.
- June 2 2011: There was a minor error in question 8 of this week's homework: z=0 should have been
taken out of the complex plane as well as part of the slit, since arg(z) is not defined for z=0.
Here is an updated version of homework week 10.
- June 1 2011: The location of the final exam is Boelter 2444. I have updated this in the
Course Organization Handout.
- May 27 2011: Question 10 in homework 9 turned out to be a tough challenge for many of you, so I
have written out the solution here.
- May 25 2011: There was a typo in question 3c of this week's homework. Here is an updated version
: Homework week 9.
- May 18 2011: Some extra references for more practice material. These books have plenty of
exercises (especially Schaum's outlines; which also provides solutions to many of its exercises) and
are available in the library:
Function Theory of One Complex Variable by Greene and Krantz; and
Schaum's outlines: complex variables: with an introduction to conformal mapping and its
applications.
Here is a website
with some more questions and solutions. Also have a look at the old homework an exam questions from
the Math 132 courses by prof.
Tao and prof. Sohn.
- May 16 2011: Here are the solutions to this morning's midterm:
version 1 and
version 2.
- May 13 2011: The midterm next Monday will be held 9:00-9:50 am in Boelter 5249. Be ten minutes
early and bring your Bruin ID card. The
Course Organization Handout has the updated location.
- May 6 2011: By request and after discussion in class this morning I have again changed my office
hours. The new dates and times are Monday 3:00-4:00 pm, Wednesday 10:45-11:45 am, and Thursday
1:00-2:00 pm. The Course Organization Handout has been
updated as well.
- May 4 2011: If you want more practice on certain subjects, here is a website with a nice
collection of exercises,
from the University of Oulu Finland. I think many of you will especially benefit from more practice
with line integrals (Chapter V on that website). Try to resist the urge to look up the solutions
before you have tried every possible solution strategy you know of yourself.
- May 4 2011: I will have extra office hours tomorrow, Thursday May 5, 1-3pm.
- April 20 2011: In section tomorrow the TA will discuss the midterm and, if time allows, the
review sections III.1 and III.2 in the book, about line integrals, Green's theorem, and related
topics which you have hopefully seen already in your multivariable calculus class. Read these
sections and make sure your memory is refreshed, because I will not spend much time on them in
class. Questions are always welcome of course.
- April 20 2011: There was a mistake in the solutions that I posted earlier this week (in the last
sentence of the answer to 2a). I've corrected the mistake and taken the opportunity to also give an
alternative way of solving 2a and 2b. Corrected solutions:
version 1 and
version 2.
- April 18 2011: Here are the solutions to this morning's midterm:
version 1 and
version 2. The grades will follow within a day or
two.
- April 15 2011: The first midterm next Monday will be held in Boelter 5249, so not
in the normal class room. Come ten minutes early so we can start at 9 am exactly. Bring your UCLA id
card.
- April 13 2011: As requested by the class I have moved my Monday office hours half an hour back,
i.e. the new times are 2:00-3:30 pm. Here is the updated
Course Organization Handout.
- April 8 2011: I have added a plan for the remaining lectures before the midterm to the
"Calendar and classes" section below. The material for the midterm will be everything we have done
up to and including the last lecture before the midterm.
- April 7 2011: I have added some links to pictures of the stereographic projection and Riemann
surface(s) in the "Calendar and classes" section below.
- March 31 2011: Another update to the
Course Organization Handout! Presumably the last one.
This time the TA's office hours have been added: Tuesdays 12:00-1:00 pm and Thursday 10:00-11:00 am.
- March 31 2011: As requested and discussed in class yesterday the date of the first midterm is
changed to Monday April 18. I have updated the
Course Organization Handout.
- March 29 2011: Thanks to the keen eye of one of you a typo in the homework exercises was
detected. Question 1(i) should read conjugate(zw) = conjugate(z).conjugate(w). In the old version it
had the same expression on the left and the right of the equal sign. I have updated the file below
in the homework section. Thank you for letting me know!
- March 28 2011: In class today it was decided that my office hours will be on Monday 1:30-3:00 pm
and Wednesday 10:15-11:45 am. I have updated the
Course Organization Handout. Please also note that
I added a few lines explaning what information to put on the first page of your homework
assignments.
- March 22 2011: This space will be used for announcements. Check back regularly to see if there
is news. The newest items will appear at the top of the list. Enjoy the course!
Homework
Homework will be collected before the start of the Friday lecture. Late homework
will not be accepted.
- HW 1, due Friday April 01: Homework week 1
- HW 2, due Friday April 08: Section I.3: exercises 1, 6;   I.4: ex. 1 (b)(d)(f),
2 (b)(d)(f), 3;   I.5: ex. 1 (c)(f), 2 (c)(f), 3;   I.6: ex. 1 (d), 2 (b)(d), 4.
- HW 3, due Friday April 15: I.7: 1(a), 3(c), 7, 9;    I.8: 1, 7, 8;
   II.1: 2, 15(a)(d)(e)(f);    II.2: 1(f), 3, 5.
- HW 4, due Friday April 22: II.3: 1, 6, 8;    II.4: 5, 6, 7;    II.5:
1(b)(d)(f), 2, 5, 6, 7.
- HW 5, due Friday April 29: II.6: 1, 2, 7;    II.7: 1(b)(d)(e)(f), 3, 11, 12;
   III.1: 2, 5, 7;    III.2: 1, 3, 4.
- HW 6, due Friday May 06: III.3: 3,5;    IV.1: 1(b), 2(a), 3(b), 8, 9;
   IV.2: 1(d), 4, 5;    IV.3: 1, 2, 4.
- HW 7, due Friday May 13: IV.4: 1(b)(f)(h), 2, 4;    IV.5: 3;    IV.6: 3
;    V.1: 2, 3, 7;    V.2: 3, 7, 8, 9.
- HW 8, due Friday May 20: V.3: 1(d)(e)(g)(h), 6, 7;    V.4: 1(c)(d)(e), 5, 9, 11
;    V.5: 1(c)(d);    V.6: 3.
- HW 9, due Friday May 27: Homework week 9
- HW 10, due Friday June 03: Homework week 10
Calendar and classes
- Monday Mar. 28: Introductory remarks, review of complex numbers in Section I.1 and started with
polar representation in Section I.2.
- Wednesday Mar. 30: Finished Section I.2 and started discussing stereographic projections, up to
the statement, but not yet the proof, of the theorem in the book in Section I.3. Here you can see
an illustration of the
stereographic projection.
- Friday Apr. 01: Proved the theorem in I.3 and continued with Section I.4 on square and
square root functions.
- Monday Apr. 04: We discussed the Riemann surface of the complex root in I.4 and continued with
Sections I.5 and I.6 on the complex exponential and its inverse the complex logarithm. I didn't
completely finish the latter. The picture which I wanted to show you (and still will next time) is
this picture of the
Riemann surface
associated to the complex root.
- Wednesday Apr. 06: Finished Section I.6 and continued with Section I.7 and I.8 on power
functions and phase factors. Here is a picture of the
Riemann surface associated
to the complex logarithm.
- Friday Apr. 08: Finished Section I.7 and discussed I.8 (trigonometric and hyperbolic
functions).
- Monday Apr. 11: I gave a review about analysis, as presented in Section II.1. I went over that
section quickly, since it is not specific to complex analysis. A lot, if not all, of it you have
probably seen before in the context of real numbers. Come and ask questions if anything is unclear.
- Wednesday Apr. 13: We discussed Section II.2 introducing the concept of a (complex) analytic
function and started with II.3 about the Cauchy-Riemann equations. We showed that an analytic
function satisfies the Cauchy-Riemann equations with continuous partial derivatives. Next time we
will continue with showing that the reverse also holds: if a complex function has continuous
partial derivatives with respect to the real and imaginary parts of its variable and satisfies the
Cauchy-Riemann equations, then it is analytic.
- Friday Apr. 15: Finished II.3 and discussed II.5 about harmonic conjugates. We saw how to find a
harmonic conjugate v if we are given a harmonic function u, sucht that u+iv is analytic. Note that I
skipped II.4 for the moment.
- Monday Apr. 18: Midterm 1: in class, during class time. The material for the first
midterm will include everything we have done (class, section, book, homework, etc.) up to and
including the last lecture before the first midterm. If I can stick to my plan that means up to and
including Section II.5, except II.4.
- Wednesday Apr. 20: We discussed II.4 on inverse mappings and the Jacobian.
- Friday Apr. 22: We discussed Section II.6 on conformal mappings.
- Monday Apr. 25: II.7 on Möbius transformations. I tried to show you this beautiful
video on Möbius transformations, but I
couldn't get internet access. If time allows, I'll show it next time, but have a look yourself.
After today's class a lot of what it shows should look familiar.
- Wednesday Apr. 27: : I reviewed Sections III.1 and III.2 which are about line
integrals, Green's theorem, independence of paths for line integrals, etc. Most if not all of this
is review from analysis, so I didn't spend a whole lot of time on it. After I made a start with
III.3 which is a very short section on harmonic conjugates in the context of the
closed differentials of Section II.2.
- Friday Apr. 29: We finished III.3 and then continued with Section IV.1 on complex line integrals
and made a start with IV.2 and the analogue of the fundamental theorem of calculus for analytic
functions.
- Monday May 02: We finished IV.2, continued with IV.3 on the very important Cauchy's theorem and
started with IV.4 on the famous Cauchy integral formula. I only got to state the formula, I left you
and your curiosity hanging until Wednesday for the proof. Of course nothing (I hope) prevents you
from reading the book.
- Wednesday May 04: We finished IV.4 and discussed IV.5 on Liouville's Theorem (another one of
those famous complex analysis theorems).
- Friday May. 06: I stated Morera's Theorem from IV.6 and some corollaries (I didn't spend time on
the proof; if you're interested ---and of course you are--- you can read it in the book) and then
continued with V.1 and V.2. Those sections are again of the review type, this time about infinite
series, and sequences and series of functions. I think this has (partly) been covered in Math 33B.
- Monday May 09: Finished V.2, about series of functions. Then we continued with V.3 on
power series.
- Wednesday May 11: I gave some examples for the theorems we discussed in V.3, then we continued
with V.4 on the power series expansion of analytic functions.
- Friday May 13: We covered V.5 about power series expansions at infinity and V.6 about
manipulations of power series. At the end I made a brief start with V.7 on the (order of)
zeros of analytic functions.
- Monday May 16: Midterm 2 in Boelter 5249, during class time. The material for the second
midterm will include everything we have done (class, section, book, homework, etc.) up to and
including the last lecture before the second midterm.
- Wednesday May 18: Almost finished V.7
- Friday May 20: Finished V.7 with some examples and started with VI.1 on the Laurent
decomposition and series expansion.
- Monday May 23: We finished VI.1 with another example and started with VI.2 with a classification
of isolated singularities and a closer look at the first of these classes: removable singularities.
- Wednesday May 25: We finished VI.2 by discussing poles and essential singularities in more
detail and continued with VI.3 on isolated singularities at infnity.
- Friday May 27: We discussed Section VI.4 on partial fractions decompositions and then continued
with VII.1 on the important Residue Theorem.
- Monday May 30: No lecture: Memorial Day
- Wednesday Jun. 01: We saw some examples of how to use the Residue theorem in section
VII.2 on integrals featuring rational functions. I will be skipping section VII.3 on integrals of
trigonometric functions, but if you're looking for more examples of how to use the Residue Theorem,
this is a good section to look at. At the end of this lecture I distributed the course evaluation
forms.
- Friday Jun. 03: We discussed Section VIII.1 on the Argument Principle. At the end I briefly went
over another example of how to use the Residue Theorem. Unfortunately I didn't have time to go over
everything in detail, but you can find the details in Section VII.4 on integrals with branch points.
- Tuesday Jun. 07: Final exam in Boelter 2444, 3-6 pm. The material for the final exam
will include everything we have done (class, section, book, homework, etc.) up to and including the
last lecture before the final exam.
Last updated: June 9, 201