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454 CHAPTER 13 Fourier Series

When each a n = 0, these equations enable us to write the phase angle form of the Fourier
series of f (x) on [−p/2, p/2]:

∞
1
a 0 + c n cos(nω 0 x + δ n ),
2
n=1
where

ω 0 = 2π/p,c n = a + b , and δ n =−arctan(b n /a n ).
2
2
n
n
This phase angle form is also called the harmonic form of the Fourier series for f (x)
on [−p/2, p/2]. The term cos(nω 0 x + δ n ) is called the nth harmonic of f , c n is the nth
harmonic amplitude, and δ n is the nth phase angle of f .

If f has fundamental period p, then in the expressions for the coefficients a n and b n , we can
compute the integrals over any interval [α,α + p], since any interval of length p carries all of the
information about a p-periodic function.
This means that the Fourier coefficients of p-periodic f can be obtained as

2 α+p
a n = f (x)cos(nω 0 x)dx
p α
and
2 α+p
b n = f (x)sin(nω 0 x)dx
p α
for any number α.

EXAMPLE 13.16
Let

2
f (x) = x for 0 ≤ x < 3
and suppose f has fundamental period p = 3. A graph of f is shown in Figure 13.17.
Since f is 3-periodic, and we are given an algebraic expression for f (x) only on [0,3),we
will use this interval to compute the Fourier coefficients of f . That is, use p = 3 and α = 0inthe
preceding discussion. We also have ω o = 2π/p = 2π/3.
The Fourier coefficients are
2 3 2
a 0 = x dx = 6,
3 0

2 3 2nπx 9
2
a n = x cos dx = ,
2
3 0 3 n π 2

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October 14, 2010 14:57 THM/NEIL Page-454 27410_13_ch13_p425-464