Mathematics 430 and 630

Complex Variables (3)

Dr. Chris Caldwell

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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 *i ^{i}* 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 (especially calculus). The bad news is that I will expect you to remember som calculus--so you might need to keep your old 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 this is a senior level math course, so you will get
a dose of proof (especially at the beginning of the course)! We will not,
however, do *anywhere near* as much proof as in Abstract Algebra or Real Analysis;
and will will be much less formal about it.

The suggestions are the same for most any mathematics class:

- Come to class and do the homework
- Always seek to understand, not just memorize.
- Study with a friend. Make a new friend if necessary! Usually many of the class works in the Math Lab-so perhaps you can work there.
- Use other complex variables books as references. Get familiar with the library.
- 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).
- Come by my office. Do not hesitate because I look busy. I rarely just sit around, but you are my top priority.
- Do not fall behind.
- Always do the homework (and maybe even extra problems).

I want you to succeed and will gladly
help you, but you must start by working on the homework and reading the
text (even if you cn not do it, at leasts start). Many of the other
faculty member *might* also
be able to answer some questions (if they have time), but few have taught this course recenty.

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 excellent 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 (but at any time in the semeser I will gladly tell you were you are "grade-wise" and usually do so after every test).

The text for this course is: *Complex Variables and Applications* by Brown and Churchill, 4th edition. Publisher McGraw-Hill. ISBN: 978-0073051949. We will cover much of chapters
one through nine. See also the department
syllabus.