Math 132: Complex Analysis for Applications, Spring 2011

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

• 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.