Principles of Fourier Analysis, 2nd Edition

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

$A_0+{\displaystyle \sum _{k=1}^{\infty }\left[a_k\cos \left(2\pi {\omega }_{k}t\right)+b_k\sin \left(2\pi {\omega }_{k}t\right)\right]}$

(9.1a)

where, for k = 1, 2, 3, …,

${\omega }_k=\frac{k}{p},$

(9.1b)

$A_0=\frac{1}{p}{\displaystyle {\int }_0^{p}f\left(t\right)dt},$

(9.1c)