The Fourier coefficients

12.2 THE FOURIER COEFFICIENTS
we can write any function as the sum of a sine series and a cosine series.
All the terms of a Fourier series are mutually orthogonal, i.e. the integrals, over
one period, of the product of any two terms have the following properties:
x0 +L
2πrx
2πpx
sin
cos
dx = 0 for all r and p,
(12.1)
L
L
x0

x0 +L
for r = p = 0,
L
2πrx
2πpx
cos
(12.2)
cos
dx = 12 L for r = p > 0,

L
L
x0
0
for r = p,

x0 +L
for r = p = 0,
0
2πrx
2πpx
sin
(12.3)
sin
dx = 12 L for r = p > 0,

L
L
x0
0
for r = p,
where r and p are integers greater than or equal to zero; these formulae are easily
derived. A full discussion of why it is possible to expand a function as a sum of
mutually orthogonal functions is given in chapter 17.
The Fourier series expansion of the function f(x) is conventionally written
∞ 2πrx
2πrx
a0 +
ar cos
f(x) =
+ br sin
,
(12.4)
2
L
L
r=1
where a0 , ar , br are constants called the Fourier coefficients. These coefficients are
analogous to those in a power series expansion and the determination of their
numerical values is the essential step in writing a function as a Fourier series.
This chapter continues with a discussion of how to find the Fourier coefficients
for particular functions. We then discuss simplifications to the general Fourier
series that may save considerable effort in calculations. This is followed by the
alternative representation of a function as a complex Fourier series, and we
conclude with a discussion of Parseval’s theorem.
12.2 The Fourier coefficients
We have indicated that a series that satisfies the Dirichlet conditions may be
written in the form (12.4). We now consider how to find the Fourier coefficients
for any particular function. For a periodic function f(x) of period L we will find
that the Fourier coefficients are given by
2πrx
2 x0 +L
f(x) cos
ar =
dx,
(12.5)
L x0
L
x0 +L
2πrx
2
f(x) sin
dx,
(12.6)
br =
L x0
L
where x0 is arbitrary but is often taken as 0 or −L/2. The apparently arbitrary
factor 12 which appears in the a0 term in (12.4) is included so that (12.5) may
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FOURIER SERIES
apply for r = 0 as well as r > 0. The relations (12.5) and (12.6) may be derived
as follows.
Suppose the Fourier series expansion of f(x) can be written as in (12.4),
f(x) =
∞ 2πrx
2πrx
a0 +
ar cos
+ br sin
.
2
L
L
r=1
Then, multiplying by cos(2πpx/L), integrating over one full period in x and
changing the order of the summation and integration, we get
x0 +L
f(x) cos
x0
2πpx
L
dx =
2πpx
a0 x0 +L
cos
dx
2 x0
L
∞
x0 +L
2πrx
2πpx
ar
cos
cos
dx
+
L
L
x0
r=1
x0 +L
∞
2πrx
2πpx
br
sin
cos
dx.
+
L
L
x0
r=1
(12.7)
We can now find the Fourier coefficients by considering (12.7) as p takes different
values. Using the orthogonality conditions (12.1)–(12.3) of the previous section,
we find that when p = 0 (12.7) becomes
x0 +L
f(x)dx =
x0
a0
L.
2
When p = 0 the only non-vanishing term on the RHS of (12.7) occurs when
r = p, and so
x0 +L
2πrx
ar
f(x) cos
dx = L.
L
2
x0
The other Fourier coefficients br may be found by repeating the above process
but multiplying by sin(2πpx/L) instead of cos(2πpx/L) (see exercise 12.2).
Express the square-wave function illustrated in figure 12.2 as a Fourier series.
Physically this might represent the input to an electrical circuit that switches between a
high and a low state with time period T . The square wave may be represented by
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−1 for − 12 T ≤ t < 0,
f(t) =
+1 for 0 ≤ t < 12 T .
In deriving the Fourier coefficients, we note firstly that the function is an odd function
and so the series will contain only sine terms (this simplification is discussed further in the
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