| Chapter | Problems to do | Submit | Target or due date | Comments |
|---|---|---|---|---|
| Arithmetic | 1-5 | 5 | ||
| 1 | All except 1.13 | 1.12 | Tuesday, January 29 | In problem 1.12, I did not find the phrase "use the results in 1.8" to be useful (or even meaningful, really). Perhaps I'm missing something. |
| 2 | All | 2.13 or 2.14 | Tuesday, February 5 | Your choice on which problem to submit. |
| 3 | 3.2, 3.3, 3.6, 3.7, 3.9 | None | Thursday, February 7 | See daily note for Tuesday, February 5. |
| 3 | 3.11, 3.12, 3.13 | None | Monday, February 11 | |
| 3 | 3.10, 3.14, 3.16 | None | Monday, February 18 | Look for typos in 3.10 |
| 5 | All | 5.12 | Friday, February 22 | For 5.5, you'll need to analyze each partial derivative at (0,0) using the limit of a difference quotient. |
| 6 | 6.2, 6.3, 6.4 | None | Monday, February 25 | |
| 6 | 6.7, 6.9, 6.10 | None | Tuesday, February 26 | |
| 7 | 7.1, 7.2*, 7.3, 7.4 | None | Thursday, February 28 | I'll start denoted certain problems with a * to indicate optional problems (that are typically more challenging). |
| 7 | 7.5, 7.6, 7.8-7.13 | None | Monday, March 3 | |
| 7 | 7.14, 7.15, 7.16 | 7.16 | Thursday, March 6 | |
| 4 | 4.1 | None | Friday, March 7 | |
| 8 | 8.1-8.4, 8.15 | None | Friday, March 7 | |
| 9 | 9.1, 9.2, 9.3 | None | Tuesday, March 11 | |
| 10 | 10.1, 10.2 | None | Tuesday, March 25 | |
| 10 | 10.4, 10.5, 10.7* | 10.3 | Friday, March 28 | |
| 11 | 11.1, 11.2 | None | Monday, March 31 | |
| 11 | 11.3 11.4 | None | Tuesday, April 1 | |
| 13 | 13.1, 13.2, 13.4 | None | Thursday, April 3 | |
| 13 | 13.6, 13.8, 13.9, 13.10, 13.11 | None | Monday, April 7 | |
| 14 | 14.1, 14.4, 14.5, 14.6, 14.7*, 14.8* | None | Friday, April 11 | |
| 15 | 15.1, 15.2 | None | Friday, April 18 | |
| 15 | 15.3, 15.4, 15.5, 15.7 | None | Monday, April 21 | In (ii)-(iv) of 15.4, D(0;1) should be H(D(0;1)) |
| 17 | 17.1, 17.3, 17.4 | None | Thursday, April 24 | |
| 17 | 17.5, 17.6, 17.8, 17.9, 17.12, 17.15* | None | Friday, April 25 | For 17.8 and 17.9, start with the odd subparts; do the even ones if you want/need more experience. |
| 18 | 18.2, 18.3, 18.4 | None | Tuesday, April 29 | For 18.2 and 18.3, start with the odd subparts; do the even ones if you want/need more experience. |
| 18 | 18.6, 18.8 | None | Thursday, May 1 | |
| 20 | 20.1, 20.2, 20.3 | None | Monday, May 5 | |
| * indicates an optional problem |
Here's the final exam. It is due by Friday, May 16 at 10 am. If you can bring it to my office anytime before then. If I am not it, just slide it under my door.
I have assigned some problems from Chapter 20.
Exam #4 will in in-class on Tuesday, May 6 from 8:00-9:20 am.
The final exam will be take-home. I'll distribute it on Tuesday, May 6. It will be due at the scheduled time for our final exam, 10 am on Friday, May 16.
I've assigned a few more problems from Chapter 18
We will not cover the material on meromorphic functions in Sections 17.18 to 17.20.
I've assigned a few problems from Chapter 17. I'll assign more after we develop a few more tools for efficiently anaylzing singularities.
We will not cover Sections 15.12 to 15.15. Some of you will want to read Section 15.12 on analytic continuation since this technique is used in some parts of analytic number theory.
We will also not cover Chapter 16 unless we have a bit of time left at the end of the semester.
In class today, we started talking about the ideas in Chatper 17. I have not yet assigned problems from Chapter 17.
I'm leaving it to you to finish off the proof that we started in class of the Identity Theorem and to read the Uniqueness Theorem (which follows by applying the Uniqueness Theorem to a difference of two functions). These theorems provide another way in which holomorphic complex-valued functions differ from differentiable real-valued functions. To illustrate this, you should try to build a differentiable real-valued function of two variables that is zero only on a line segment in the plane. Contrast this with the first example in Section 15.10.
I mentioned in class that there are typos in the statement and proof of Theorem 15.3. The statement of the theorem has at least one typo. In the proof, references to the four equivalent statements in the theorem are messed up. You should correct these on your own. If you have questions, ask by e-mail or in class.
I've assigned a few problems from Chapter 15.
The in-class component of Exam #3 will be on Tuesday from 8:00 to 9:20 am. The take-home component will be distributed at the same time and due at the start of class on Monday, April 21.
Exam #3 will have an in-class part on Tuesday, April 15 and a take-home part due Monday, April 21. The exam will cover things we've looked at since the previous exam. In the text, this corresponds to Chapters 10, 11, 13, and 14.
I'm leaving it to you to read the result in 14.7 and an application of this result in 14.8. We will not cover the ideas in the remainder of Chapter 14.
I've assigned problems from Chapter 14. In class tomorrow, I'll give you some background that might help with Exercise 14.5.
The in-class portion of Exam #3 will be next Tuesday from 8:00 to 9:20 am. I'll then distribute a take-home portion that will be due the following Monday.
I'll assign problems from Chapter 14 tomorrow (Wednesday) but you can wait until after Thursday's class to look at these.
I have not assigned new problems today
Exam #3 will be next week, probably on Tuesday. It might include a take-home component.
The proof of Liouville's Theorem that we looked at in class uses Cauchy's integral formula for the first derivative (CIF-1). In the text, Priestley gives a proof based on Cauchy's integral formula for the function itself (CIF-0).
I am leaving it up to you to read the complex analysis proof of the Fundamental Theorem of Algebra. This proof uses Liouville's Theorem.
In class, we also worked on this handout outlining two proofs of CIF-2.
I've assigned a few more problems from Chapter 13. The first two involve using Liouville's Theorem. The others involve using CIF-n to evaluate integrals.
I've skipped over Liouville's Theorem (13.3 in the text) and the complex analysis proof of the Fundamental Theorem of Calculus (13.4 in the text) for now. We'll come back to these once we've finished looking at Cauchy's integral formula for nth derivative.
We will not cover the subsection "Logarithms again" in Chapter 11. We will also not cover the "advanced track" in Chapter 12 other than what we discussed in class yesterday.
In class, we looked at a handout summarizing various versions of Cauchy's Theorem. We will routinely use Version 1C that is a bit stronger than what Priestley proves (essentially Version 1B). That is, we will use simple closed paths where Priestley restricts to closed contours.
We will take similar liberties with the deformation theorem. The version we looked at in class is a bit stronger than (1) and (2) of the result Priestley states and proves in 11.9. In similar fashion, we will use a version of (3) with general paths in place of circline paths. As a result, a problem such as 11.4 is much easier. On other other hand, it wouldn't hurt to restrict yourself to Priestley's version of (3) and to then follow the hint in 11.4.
In class, I distributed a handout on "the other" FTC for single and multivariable real analysis (i.e, from first and third semester calculus).
In Section 11.6, Priestley generalizes Cauchy's Theorem from the version we proved for triangle paths to a version for closed contours. Remember that, for Priestley, contour means a path that is the join of line segments and circular arcs. I'll briefly describe the proof of this generalization in class on Monday without going into full detail.
I've assigned a few problems from Chapter 11 and will assign more after class on Monday.
Some technical details on curves and paths are given in Chapter 4 of our text. Note that what Priestley calls a path is called a contour in many other references. Priestley reserves the word contour for the special case of a path that is the join of line segments and circular arcs.
The applet we looked at in class is The Mandelbrot/Julia Set Applet at Boston University. There are lots of others like it available on the web. You can also find freeware with similar capabilities and more.
Some details are available in our text in Problem 3.16 and the Appendix.
The text I had in class is A First Course in Chaotic Dynamical Systems by Robert L. Devaney.
Enjoy your break!
In class, Alex asked about doing an example of choosing a branch for something involving a cube root. I've written a handout with some details and an example.
At the very end of class, I defined branch point at infinity and gave a quick example. You don't need to use this concept but it can make analysis of branch cuts more efficient. If a multifunction has a branch point at infinity, then at least one branch cut in the form of a ray will be needed to define a branch. If a multifunction does not have a branch point, then a ray is not needed.
In class yesterday, we looked at examples of choosing a branch cut for each individual factors of a multifunction and then asked what overall branch cut resulted. In many cases, it is enough to pick the overall branch cut(s) without being specific about how to pick a branch cut for each factor.
Exam #2 will be next Thursday from 8:00 to 9:20 am. It will cover material we have looked at since the previous exam. In the text, this is Chapters 5, 6, 7, first part of 4, first part of 8, 9.
We've covered most of the ideas in Chapter 9, although in places I have not developed all of the technical details and notation in text.
For Problem 9.3, I've only assigned the parts that do not involve a branch point at infinity. We'll talk about branch points at infinity tomorrow if we have time.
Exam #2 will be next Thursday from 8:00 to 9:20 am. It will cover material we have looked at since the previous exam. In the text, this is Chapters 5, 6, 7, first part of 4, first part of 8, 9. I'll be clear tomorrow about expectations on material from Chapter 9.
This second attempt at developing this material went a bit better than the first attempt, up until the final moments of class. I'll sort out our confusion over the weekend.
Exam #2 will be next Thursday from 8:00 to 9:20 am.
Chapter 4 deals with paths in the complex plane. We covered just enough of this material in class today to get a feel for tangent vectors. This ideas should be familiar from multivariable calculus.
The main goal for today's class was to develop a general geometric interpretation for what a holomorphic function does as a mapping at a point. The main lesson is that a function f maps each tangent vector at z0 to a tangent vector at f(z0) that is scaled by a factor of |f′(z0)| and rotated by the angle arg(f′(z0)). Since all tangent vectors are scaled by the same factor and rotated through the same angle, the angle between any pair of tangent vectors is preserved under mapping by a holomorphic function. The general name for a mapping that preserves angles (but not necessarily distances) is conformal. So, we can translate our result into the statement that a holomorphic function is a conformal map.
Only one problem, 8.15, in Chapter 8 deals directly with the geometric interpretation of f′ we've developed. I've assigned a few problems that deal with "global" geometric interpretation of complex functions as mappings. The remainder of the problems deal with aspects of Chapter 8 that we will be skipping for now.
In class, we've talked about most of the ideas from Chapter 7 except for the definition of complex powers. Complex powers are introduced in 7.15 with more details in 7.17. We'll discuss these ideas on Monday.
I'll try to come up with an argument for the first inequality in Problem 7.3 (ii).
I'll assign more problems from Chapter 7 after class tomorrow.
I've assigned a few problems from Chapter 6. I'll assign more after class on Monday.
I've only assigned one problem from Chapter 5 so far. More to come after we get further into this material in class tomorrow.
I've assigned a few more problems from Chapter 3 that deal with some of the ideas we discussed today.
We will be skipping over Chapter 4 for now. We'll come back to this material when we need it.
Since the exam seemed a bit long, I'm assigning Problem 10 to be done on a take-home basis. Here's a copy of the exam for reference. This will be due at the beginning of class tomorrow (Friday). You are to work on this problem without using any references (including our text) or other assistance.
Exam #1 will be Thursday, February 14 from 8:00 to 9:20 am. Note that we will start earlier than our usual class session. Everything we have discussed in class or that is covered in Chapters 1 through 3 of the text is fair game.
Exam #1 will be Thursday, February 14 from 8:00 to 9:20 am. Note that we will start earlier than our usual class session. Everything we have discussed in class or that is covered in Chapters 1 through 3 of the text is fair game.
In class, we talked about continuity of functions before looking at limits of functions. In the big picture, continuity is a special case of limits.
I've assigned a few more problems from Chapter 3
I've assigned a few problems from Chapter 3. I'll assign more as we discuss more ideas covered in Chapter 3 over the next few days.
For Exercise 3.3, you don't need to write out a complete proof but you should, at the least, write down a few details of how a proof would go. For a direct proof (using the definition of open), fix z0 in the given set and then write down a choice for the radius r of a disk centered at z0 that will satisfy the defining condition. At the level we will operate, you don't need to write out the full details of showing that your choice of r does in deed satisfy the defining condition.
For Exercise 3.7, you should make sure you understand how the sets S and T are related geometrically. Doing so should make the statement to proved very obvious. You don't need to write out a detailed proof.
I do want everyone to get practice in developing ε-N proofs for limits of sequences (and, soon, ε-δ proofs for limits of functions). Some of you already have sufficient experience; others don't. You should either write out detailed proofs for Exercise 3.9 or be very confident that you can do so if required on, say, an exam.
The home of the video Mobius Transformations Revealed is here.
The software 3D XplorMath is available here. You might also want to check out the Virtual Math Museum put together by the same folks who make 3D XplorMath.
I'll assign more problems from Chapter 2 after we talk about Mobius transformations in class tomorrow.
You may (or should) have noticed that the Exercises section for each chapter starts with Exercises from the text. These are generally about filling in details of things stated but not proven in the text. One of the Exercises from the text for Chapter 2 asks you to prove that stereographic projection maps any circle on the Riemann sphere to circline (shorthand for circle or line) in the extended complex plane. Now, this exercise is in square brackets. In the Preface to the First Edition, our author writes "Not all students have the same mathematical background. To allow for this, I have adopted the convention that text enclosed in square brackets should be heeded by anyone to whom it makes sense but can safely be ignored by others." So, you can choose to work on this problem or not. Here's two comments that might be helpful:
In class, we've talked about two forms for equations of circles, the "center-radius form" and the "inverse point form". Sections 2.4 and 2.5 discuss how to describe a circular arc with an equation somewhat analogous to the "inverse point form". The basis for this comes from the geometric fact I stated at the beginning of class. I'm leaving it to you to read Sections 2.4 and 2.5 and to try the problems related to these ideas.
I've assigned a few problems from Chapter 2. These deal with understanding and manipulating various forms of equations for lines, circles, and arcs. I'll assign more problems from Chapter 2 after we have discussed the other topics in the chapter.
I have not yet assigned problems from Chapter 2. If you are bored, take a look at 2.1, 2.2, and 2.3. I'll post an assignment after class tomorrow.
I've put the due date for the first assignment as Thursday. Normally, I will give you a bit more time between assigning a problem to submit and having it due. In this case, the problem is pretty straightforward. Come talk with me today or tomorrow if you have questions.
The Mathematical Atlas describes the many fields and subfields of mathematics. The site has numerous links to other interesting and useful sites about mathematics.
If you are interested in the history of mathematics, a good place to start is the History of Mathematics page at the University of St. Andrews, Scotland.
Check out the Astronomy Picture of the Day.