## Contact

Welcome to the Home Page for this course on Advanced Engineering Math in the Department of Integrative Engineering. On this site you will find a variety of useful resources, including lecture notes, problem sets, applications, demos that illustrate many of the concepts and applications covered in the course, and links to important resources outside this web page.

Topics:

• Complex Variables
• Analytic Functions
• Complex Differentiation and Integration
• Taylor and Laurant Series
• Fourier Analysis
• Fourier Series
• Fourier Transforms
• Laplace Transforms
• Use and Applications of Transforms
• Probability Theory
• Probability Axioms
• Probability Mass and Density Functions
• Conditional Probability
• Independence
• Expected Value
• Functions of a Random Variable
• Sampling and Occupancy Problems

This course provides an introduction to two very important fields of advanced mathematics: Complex Variables and Fourier Analysis. A daily schedule for the course may be found here. The basic concepts of these two areas of mathematics will be developed and illustrated with numerous examples and applications. This course also develops the fundamental principles of Probablility Theory. The importance of probability theory and the notion of randomness and noise are often not appreciated when one begins a study of probability. Therefore, one of the goals in this course will be to motivate and illustrate why probability theory is important and present numerous examples of when randomness and uncertainty arises in applications.

There is no doubt that each of these topics is fundamental and important in nearly all branches of science and engineering. In addition, it is generally believed that each topic considered by the student to be conceptually difficult and challenging since in each case the student is required to think and work with an entirely new set of concepts and principles. With hard work and a willingness to learn and seek help when difficulties are encountered, the student should be able to master the concepts in this course.

The basic goals of the course are:

1. To understand concepts of complex analysis, including functions of a complex variable, differentiation, the Cauchy-Reiman relations, branch points and branch cuts, mappings, Taylor and Laurant series, singularities, and integration. Â
2. To understand the theory and applications of Fourier analysis, including Fourier expansions, Fourier series and Fourier transforms, generalized functions and the impulse, the Laplace transform, and the Fourier analysis of signals.
3. To understand use and applications of probability theory including the concepts of a random variable, probability mass and density functions, statistical independence, expected value, functions of a random variable, and Monte Carlo simulations.

It is expected that the student has a solid foundation in calculus and understands the concepts of differentialtion, integration, the convergence of a series of numbers. One lecture will be given to review this material. In addition an understanding of calculus, additional prerequisites include:

1. A willingness to work hard and to think independently.
2. A commitment to put a time, outside of class, reading and working problems.
3. A desire to learn and to not be afraid to ask questions.