This course consists of two, essentially independent, five-week sections. the first is an introduction to complex numbers and complex-valued functions, and the second is presents some of the basic techniques in differential equations. The course is intended to give you an overview of these topics, and will not be able to treat them in great depth. After the course, however, you should be able to recognize the basics, and will know where to look for further information when you need it.
The first half of the course will develop the ideas of complex numbers, complex-valued functions of a single complex variable, and the basic calculus of these functions. It is interesting to note that one of the earliest applications of this material was developed by Steinmetz while he was here at Union College. Because complex numbers include what are called "imaginary numbers", many mistakenly feel that they have no bearing on the "real" world. This is not the case, and the mathematics of complex numbers is a powerful tool for engineers and physical scientists in general.
They also form a beautiful and important field in pure mathematics, with surprizingly rich connections to a variety of unexpected areas. For example, many of you will have heard of fractals, but you probably don't know that one of the most famous examples, the Mandelbrot set, is based on complex numbers. In this course, you should gain some sense of the elegance and mystery associated with this important field of mathematics.
The second half of the course deals with partial differential equations (PDEs), which are one of the main tools used for analyzing physical phenomena, from the flow of heat, to the dynamics of sound waves, to the motions of particles in magnetic fields. We will develop the basic tools for solving simple PDEs and apply these ideas to the classical PDEs: the wave equation, Laplace's equation, the heat equation, and the diffusion equation.
Many of the basic ideas originated with the work of Joseph Fourier in his study of the flow of heat. Now called Fourier series, his method of representing any initial distribution of heat as a sum of sine functions is of great importance in many fields. We shall develop some of the fundamental methods of Fourier series in learning to solve PDEs.
To give a picture of some recent results, we survey the theoretical work of Michel Lapidus and the experimental work of Bernard Sapoval on drums with fractal boundaries. It is a surprise that complicated boundary geometry can give rise to qualitatively new solutions of the wave equation. Far from something completed a century ago, PDEs is a fast-moving field, now branching off into very exciting directions much closer to applications accounting for the messiness of nature.