The Trigonometric Fourier Series

In the previous chapter we obtained a set of formulas that we suspect will allow us to describe any “reasonable” periodic function as a (possibly infinite) linear combination of sines and cosines. Let us now see about actually *computing* with these formulas.

First, though, a little terminology and notation so that we can conveniently refer to this important set of formulas.

9.1 Defining the Trigonometric Fourier Series

Terminology and Notation

Let *f* be a periodic function with period *p* where *p* is some positive number. The *(trigonometric) Fourier series* for *f* is the infinite series

${A}_{0}+{\displaystyle \sum _{k=1}^{\infty}\left[{a}_{k}\mathrm{cos}\left(2\pi {\omega}_{k}t\right)+{b}_{k}\mathrm{sin}\left(2\pi {\omega}_{k}t\right)\right]}$ |
(9.1a) |

where, for *k* = 1, 2, 3, …,

${\omega}_{k}=\frac{k}{p},$ |
(9.1b) |

${A}_{0}=\frac{1}{p}{\displaystyle {\int}_{0}^{p}f\left(t\right)\text{\hspace{0.17em}}dt},$ |
(9.1c) |

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