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7: Fourier Series

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  • 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

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