What is Complex Variables?

This course and the course text are between the level of the prerequisite Multivariate Calculus (320) and the capstone courses Algebra and Real Analysis (461-2 and 481-2).  In many ways this course is a continuation of  the calculus, but we are now doing the work using complex (rather than real) variables.  This involves a reexamination of the transcendental functions (for example, what should  ln(i), sin(i) and iⁱ mean?), then an application of differentiation and integration to functions of complex variables.  Look at the catalog description of the course:

Algebraic operations and geometry of complex numbers, definitions of limit, continuity, and analytic functions, differentiation, mappings of simple functions, line integrals, Cauchy integral formula, Laurent series, evaluation of real integrals using the residue theorem.

You have seen these ideas (though not all of these words) before: 'Analytic' is a generalization of 'differentiable,' 'Laurent series' are just a generalization of (complex) Taylor series...  This is both good news and bad news.  The good news is that since you have seen these ideas, what we learn this semester will not be totally new and should fit well in you recall from previous courses.  The bad news is that I will expect you to remember calculus--so you might need to keep your calculus book handy.

The title of our course is complex variables (as opposed to complex analysis) which indicates the department's intention that the course emphasize application rather than proof.  But it is a senior level math course, so you will get a small dose of proof (especially at the beginning of the course)!  We will, however, not do anywhere near as much proof as in Algebra or Real Analysis; and will will be much less formal about it.

  How do we succeed?

The suggestions are the same for most any mathematics class:

1. Come to class and do the homework
2. Always seek to understand, not just memorize.
3. Study with a friend.  Make a new friend if necessary!  More than half the class works in the Math Lab—so perhaps you can work there.
4. Use other complex variables books as references.  Get familiar with the library.
5. Stay ahead of the class in the text (we will move through it sequentially, skipping some of the optional sections but we will not "hop around" much).
6. Come by my office.  Do not hesitate because I look busy.  I rarely just sit around, but you are my top priority.
7. Do not fall behind.
8. Do the homework (maybe extra even). 

I want you to succeed and will gladly help you, but you must start by working on the homework and reading the text.  Dr. McCullough, Dr. L. Kolitsch, Dr. M. Harvey, Mr. Eskew and many of the other faculty member might also be able to answer some questions (if they have time), but they have not taught this course.

  How will we determine the grade?

Homework will be assigned daily but sporadically collected.  At the beginning of the following class period I may ask students to present parts of the homework as "board problems."  I expect you to use your homework as notes.  We will have about four, fifty minute tests and a comprehensive final.  The homework should give you an idea what the test question will look like.  The course grade will probably be determined as follows:

Letter grades will then be assigned at my discretion.

  What textbook will we use?

The text for this course is: Complex Variables for Mathematics and Engineering by John H. Matthews and Russell W. Howell, 4th edition.  Publisher-Jones and Bartlett.  ISBN: 0-7637-1425-9.  We will cover much of chapters one through nine.  See also the department syllabus.