Syllabus for Complex Variables

Professor: Mark McClure

Course purpose

Although this class is a major elective, the subject is really required knowledge for mathematicians. The complex numbers complete the reals in a very real sense and Calculus is in many ways most natural in this context. In this class we will:

• Improve our philosophical understanding of numbers,
• Explore the complex numbers algebraically and geometrically,
• Expand our understanding of calculus, and
• See some very cool contemporary mathematics.

Materials

• Text: We will use A First Course in Complex Analysis by Beck et al. This is an open text made freely available through a specialized license. You may download a PDF for free or purchase a hard copy online for about $10.
  • Our PDF
  • The textbook's main site
  • Purchase online
• Technology: We won't be using calculators and they will not be permitted on quizzes or exams. We will make some use of Mathematica, which UNCA students can download for free.
• Calculators: We won't be using calculators and they will not be permitted on quizzes or exams.
• LaTeX: I'll expect you to type up a few HW assignments using LaTeX. I recommend:
  • MacTeX for Macs
  • MikTeX for Windows
  • TeXLive for Linux
  • Overleaf online
• Mathematica: Mathematica won't be a major part of the course and no assignments will require that you use it. I will use Mathematica to demonstrate some concepts in front of the whole class, however, and you will likely find it quite handy to assist you in some computations. UNCA students can download it for free at the Wolfram user portal. If you don't have account, simply create one using your UNCA email address.
• Discourse: I'll be setting up a super fun online discussion forum for us. More on that later.

Evaluation

The standard 90-80-70-60 scale will guarantee you an A, B, C, or D. However, it is quite likely that the final scale will be shifted down from this. You will be apprised of your standing as the term progresses.

• Exams: There will be two exams during the semester worth about 100 points apiece. They will both be two day affairs. Likely dates for the exams are:
  • Wednesday-Friday, September 28-30
  • Wednesday-Friday, November 9-11
• Quizzes: There will be two quizzes - each, two weeks ahead of an exam:
  • Friday, September 16
  • Friday, October 28
• Final exam: There will be a comprehensive, final exam worth around 180 points at 11:30 AM on Friday, December 9.
• Discourse: Discourse has three built in trust levels and you earn 15 points per trust level. There will also be several mandatory questions that you get points just for doing so that Discourse will be worth 60-80 points or so. Note that this is a course requirement; those points are real. There will be a discussion on the site itself indicating exactly how it works.
• Homework: There will be two types of homework:
  • Typed up, turn in assignments.
  • Daily textbook assignments, which will not be collected but offer important practice. These will often be discussed on Discourse.
• Late work: In general, I don't accept late work.
• Cheating: I don't deal with cheating. If I suspect cheating strongly enough, I simply refer you to the provost and fail you for the class.

Advice

• Learning Mathematics: I believe that mathematics is a wondrous but challenging field. I assume that most people in this class have interest in mathematics and appreciate its unique challenges. There will be times of frustration ahead. Buckle down and work hard.
• Help: You are not in this endeavor alone. You have four major sources of help:
  • Me: I like to talk to people about mathematics. That's why I chose this profession. My full schedule with office hours is shown below. You will almost always find me in my office during my office hours but please feel free to approach me any time you have questions.
  • Your classmates: Most people learn mathematics best by talking through it with others. You will find that you can both learn from and help your fellow classmates. You should get to know one another very well.
  • Discourse: A kinda combination of the previous two that never sleeps!
  • The text: It is very important that you read the text and reread the text and raise questions about the text with me and others until you understand it. We are aiming for a deeper level of understanding than in a lower level course and we are trying to learn to communicate that understanding. You should emulate the text in your mathematical writing.