The goals of this course are to introduce students to working with the field of complex numbers, and with functions of a complex variable. We investigate the calculus of such functions and if time permits we look at applications to other areas of mathematics and/or other fields of study.
Complex numbers are constructed by adding the imaginary unit i to the field of real numbers, where i has the property that i2 = -1. The term "imaginary" is an unfortunate choice of words, causing far more problems than are necessary - more about this in class. In any case, the complex numbers are defined to be the smallest field containing all the real numbers and also containing i. It can be shown that every complex number can be expressed in the form z = a + bi, where a & b are real numbers.
This field has many interesting properties. Geometrically, it can be represented as a 2-dimensional plane, called the complex plane. Algebraicially, the addition of i allows for every quadractic equation to have roots (no surprise here if you recall the quadratic forumla.) Amazingly, it also allows for every polynomial equation of arbitrary degree to have roots (this is a surprise - it is a very important theorem called the fundamental theorem of algebra.)
Analytically, there are interesting properties as well. Suffice it to say that when considering complex numbers as variables of functions, the calculus of these functions is in some ways very much the same as ordinary calculus, but in some other ways much more profound. It is harder for a function over the complex numbers to be differentiable than for a function over the reals to be differentiable, and therefore those that are differentiable behave in special ways that are not always apparent in the real case. This leads to many hidden treasures that are not true in the real case, too many to outline here.
Historically, the complex numbers are very important also, because of their connection to power series and to Fourier series which in turn are important to the foundations of calculus. Finally, it turns out that the complex numbers are also very practical. There are many applications in physics and engineering, for example, to the study of fluid flow. Also, even when both the statement and solution to certain problems (such as differential equations or integral problems) are within the realm of real numbers, one can sometimes solve these problems more efficiently by bringing in complex numbers.
Ideally, we would like to cover Chapters 1-6 of the text, with additional topics from chapters 7-9 if time permits.