# 7: Fourier Series

• • Contributed by Marcia Levitus

Chapter Objectives

• Learn how to express periodic functions, identify them as even, odd or neither, and calculate their period.
• Compute the Fourier series of periodic functions.
• Understand the concept of orthogonal expansions and orthonormal functions.

• 7.1: Introduction to Fourier Series
If we want to produce a series which will converge rapidly, so that we can truncate if after only a few terms, it is a good idea to choose basis functions that have as much as possible in common with the function to be represented. If we want to represent a periodic function, it is useful to use a basis set containing functions that are periodic themselves like sines and cosines.
• 7.2: Fourier Series
A Fourier series is a linear combination of sine and cosine functions, and it is designed to represent periodic functions.
• 7.3: Orthogonal Expansions
The idea of expressing functions as a linear combination of the functions of a given basis set is more general than what we just saw. The sines and cosines are not the only functions we can use, although they are a particular good choice for periodic functions. There is a fundamental theorem in function theory that states that we can construct any function using a complete set of orthonormal functions.
• 7.4: Problems