9

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

A0+k=1[akcos(2πωkt)+bksin(2πωkt)]

(9.1a)

where, for k = 1, 2, 3, …,

ωk=kp,

(9.1b)

A0=1p0pf(t)dt,

(9.1c)

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