## Complex Variables

### "...so many things are possible just as long as you don't know they're impossible."

Complex Variables

Here is the Mathematica file for your final exam!

2/6: The exercises assigned today are due on 2/11 except as noted. Written exercises should be written neatly or typed (not in your notes) and turned in at the beginning of class on the indicated date. Of course, there is nothing to turn in for reading assignments, but there could be a quiz or writing assignment over that material.

You are welcomed and encouraged to stop by my office for help on the homework. However, you may not ask questions over homework on the day that the assignmednt is due.

Do NOT assume that any assignment beyond today's is correct! I often change them the day of class.

Today's Mathematica File.

* DO THIS BEFORE OUR NEXT CLASS: Read the Vital Information section of the notes, pages 7-11.
* DO THIS BEFORE OUR NEXT CLASS: Read Section 1.0 in the notes.
* DO THIS BEFORE OUR NEXT CLASS: Read Section 1.1 in the notes.
* DO THIS BEFORE OUR NEXT CLASS: Do exercises 1-2 from Section 1.1.
* DO THIS BEFORE OUR NEXT CLASS: Check the blog for new entries. Even if I fail to put this in your assignment, it would be a good idea to check the blog at least every other day. As I read your homework or think about how class went, I will post insights and hints in the blog.

Due 2/11, written very neatly.
* Check the blog for new entries.
* From Section 1.1 in the notes, do exercises 3-8.

2/8: The exercises assigned today are due on 2/13. Today's Mathematica File.

* Read Ssection 1.2 in the notes.
* From Section 1.2, do exercises 1, 2, 3, 4acd, 5ac, 6ac, 7ac.
* Check the blog every day for new entries.

Week 2

2/11: Due 2/15. Today's Mathematica File.

* Read Section 1.3 in the notes.
* From Section 1.3, do exercises 1, 2b, 3, 4, and 5.
* Don't forget to check the blog!
* For Friday's class, you may want to bring your laptop with Mathematica to class. If you don't have a laptop or don't want to install Mathematica, you can use the computers in the classroom.

2/13: Due 2/18.

* Read Section 1.4 in the notes.
* From Section 1.4, do exercises 1, 2.
* Check the blog!

2/15: Due 2/20. Today's Mathematica file. You may want to download a new copy of this file. I updated it on February 15 at 12:00 noon (after class).

* Read Section 5.1 in the notes.
* Work through the cobwebs.nb file provided above, and jot down any questions that arise. There is nothing to turn in for this assignment.
* Do exercises 1, 3abdf, 4abdf, 5ade, 6ade, 9 from Section 5.1.

Week 3

2/18: Due 2/22.

* Read Section 1.5 in the notes.
* From Section 1.5, do exercises 1, 2, 3, 6, 8, 10.
* Check the blog every day for new entries.

2/20: Due 2/25. Last day to drop a class without a W appearing on your transcript. Here is a Mathematica file for sections 1.6-1.7.

* Read Section 1.6 in the notes.
* Do exercises 1-3 from Section 1.6.
* Check the blog every day for new entries.

2/22: Due 2/27. I uploaded yet another new version of the cobwebs.nb file. You should download a new copy.

* Read Section 1.7 in the notes.
* Do exercises 1-4 from Section 1.7.
* Check the blog! I'm going to stop putting "Check the blog" in every assignment, although you certainly should keep doing that!

Week 4

2/25: Due 3/11. Here is a Mathematica file specifically for complex linear functions. Here is today's code. We'll see if this works.

* Read Section 5.2 in the notes.
* Do exercises 1, 2abdef (use Mathematica, and sketch in your homework), 4, 6, 8, 10.
* DUE ON WEDNESDAY, MARCH 13: Find out as much as you can for problem 11. E-mail me a Mathematica file with evidence supporting your claims, and submit a written paper summarizing your findings. Feel free to work together on this and submit one file/paper for your group. You can use the cobwebs file, and you could also investigate the circle map for n=2, since that's equivalent to the function in the assignment (see exercises 5 and 6 from Section 5.1).

2/27:

* Study for the exam on Friday. There will be a pencil-and-paper portion of the exam, and a Mathematica portion of the exam.

3/1: Exam 1 over the material from 2/6-2/25.

Week 5 - Spring Break

Spring Break March 3/4-3/8!

Week 6

3/11: Due 3/15.

* Read Section 2.1 in the notes.
* Re-read Section 5.2 in the notes before Friday's class.
* Do exercises 1, 2, 6, 7 (you may use exercise 5 even though it isn't assigned) from Section 2.1.

3/13: Due 3/18.

* Read Section 2.2 in the notes.
* Do exercises 1, 4 from Section 2.2. If you want to use Mathematica to help with some of the integrals (once you have parameterized the curve), I think that's reasonable.

If any of you are interested in mathematics tutoring for next year, you can apply at this link.

3/15: Due 3/20.

* No reading for today's class. We will review Sections 5.1 and 5.2 through an activity. No homework to turn in on Wednesday!

Week 7

3/18: Due 3/22.

* Read Section 2.3 in the notes.
* Do exercises 1-4 from Section 2.3.
* Hopefully you are still checking the blog on a regular basis. Just a reminder!

3/20: Due 3/25.

* Read Section 2.4 in the notes.
* From Section 2.4 in the notes, do exercises 1, 2, 3ac, 4. For 2a, you may use Example 2.3.5.
* For Friday: resubmit the homework from Section 2.2 if you wish. Please include your original submission with your resubmission.

3/22: Due 3/27. Today's fractal, one example and another. More information on the fractal.

* Read Section 5.3 in the notes.
* From Section 5.3 in the notes, do exercises 2-7 in the notes. (You've already done Exercise 1.)

Week 8

3/25: Due 3/29.

* Read Section 2.5 in the notes.
* Do exercises 1-8 except for 6b from Section 2.5 in the notes. You can do 6b for extra credit, but Fisher's answer in the back of the book is not correct, at least in the editions I have.

3/27: Due 4/1.

* Read Section 2.6 in the notes.
* Do exercises 1-2 from Section 2.6 in the notes.

3/29: Due 4/3. You may want to download the cobwebs file again.

* Read Section 5.4 in the notes.
* From Section 5.4 in the notes, do exercises 1-4.
* Remember to read the blog often, preferably every day.

Week 9

4/1: Due 4/5.

* Read Section 2.7 in the notes.
* Do exercise 1 from Section 2.7 in the notes.
* Are you reading the blog regularly? You should be!

4/3: Due 4/8. April 12 is the last day to withdraw from a course.

* Read Section 3.1 in the notes.
* Do exercises 1, 2, 3a, 4, 6 from Section 3.1.

4/5: Due 4/15.

* Read Section 5.5 in the notes.
* Do exercise 1 from Section 5.5 in the notes. Have some fun with this! See the Help Page for a link.
* Remember to read the blog often, preferably every day.

Week 10

4/8: Review for the exam on Wednesday. I will provide a copy of this sheet to each of you.

4/10: Exam 2 over the material covered from 3/11-4/3.

4/12: Class cancelled due to MAA conference. Please help us make this conference a success by introducing talks, monitoring the registration table or book sale, or being a guide to lost mathematicians.

Week 11

4/15: Due 4/24.

* Read Section 3.2 in the notes.
* Do exercises 1-4, 5b, 6a, 7.
* Remember to read the blog often, preferably every day.

4/17: Due 4/26.

* Read Section 3.3 in the notes.
* Do exercise 1 from Section 3.3.
* Remember to read the blog often, preferably every day.

4/19: Easter Break!

Week 12

4/22: Easter Break!

4/24: Due 4/29.

* Do exercise 1 from Section 3.4.

4/26: Due 5/1. Today pictures: logistic and quadratic

* Do exercises 1, 3, 4, 6 at the end of Section 5.6 in the notes. There are a few typos in Exercise 1, which should read, "Prove that $$f_6:\mathbb R \rightarrow \mathbb R$$ by $$f_6(x):= 6x(1-x)$$ is topologically conjugate to $$Q_6:\mathbb R \rightarrow \mathbb R$$ by $$Q_{-6}(x):= x^2-6$$. Hint: $$h$$ will be the linear function mapping $$[-3,3]$$ onto $$[0,1]$$ with $$h(3)=0$$."

Week 13

4/29: Due 5/3.

* Do exercises 1-9 from Section 3.5 in the notes. The answers are provided for some of the exercises, so you need to show your work. Compute residues by hand, and check them with Mathematica.

5/1: Due 5/6.

* Do exercises 1, 2a, 2b, 4a, 5 from Section 3.6 in the notes.

5/3: Due 5/8.

* Do exercises 2, 3, 4 from Section 5.7, and the exercises below. There is a typo in Exercise 4. Replace the inverval $$[0,1]$$ with $$\Lambda_5$$ or $$\Lambda_6$$, where $$\Lambda_n$$ is the set of points that never leave the interval $$[0,1]$$ under iteration by $$f_n(x):= nx(1-x)$$.
* How many points of least period 151 does $$f(x):= 5x(1-x)$$ have?
* How many points of least period 64 does $$f(x):=5x(1-x)$$ have?
* Using exercise 4 from Section 5.7, how many points of least period 151 does $$f(x):= 6x(1-x)$$ have?
* Using exercise 4 from Section 5.7, how many points of least period 64 does $$f(x):= 6x(1-x)$$ have?

Week 14

5/6: Due 5/10. Today's Mathematica File

* Do exercises 1, 2, 3, 5abc, 6a, 7a.
* Course evaluations are now live. Please complete a course evaluation before May 19.

5/8: Due 5/13. A video for today. Another good video. Today's Mathematica File. Today's pictures: picture1, picture2, picture3.

* No exercise from this section.

5/10: This homework will not be collected, but the material will be on the third exam. Solutions will be posted immediately. For the third exam, you will be given a copy of this sheet. You will also be given a copy of this table of conformal mappings. Today's Mathematica File.

* Do exercises 1-7 from Section 4.3.

Week 15

5/13: This homework will not be collected, but the material will be on the third exam. Solutions will be posted immediately.

* No exercises from this section.

5/15: This homework will not be collected, but the material will be on the third exam. Solutions will be posted immediately.

* Read Section 4.1. NOT ON THE FINAL!
* Do exercise 1 from Section 4.1 in the notes.