62 Chapter 1 Fourier Series and Integrals
We can now summarize our results. In order for the proposed equality
f(x)=a 0 +
∑∞
n= 1
(
ancos(nx)+bnsin(nx)
)
(2)
to hold, thea’s andb’s must be chosen according to the formulas
a 0 = 21 π
∫π
−π
f(x)dx, (3)
an=
1
π
∫π
−π
f(x)cos(nx)dx, (4)
bn=^1
π
∫π
−π
f(x)sin(nx)dx. (5)
When the coefficients are chosen this way, the right-hand side of Eq. (1) is
called theFourier seriesoff.Thea’s andb’s are calledFourier coefficients.We
have not yet answered question (b) about equality, so we write
f(x)∼a 0 +
∑∞
n= 1
(
ancos(nx)+bnsin(nx)
)
to indicate that the Fourier seriescorrespondstof(x). See the CD for an ani-
mated example.
Example.
Suppose thatf(x)is periodic with period 2πand is given by the formula
f(x)=xin the interval−π<x<π(see Fig. 2). According to our formulas,
a 0 =^1
2 π
∫π
−π
f(x)dx=^1
2 π
∫π
−π
xdx= 0 ,
an=π^1
∫π
−π
f(x)cos(nx)dx=π^1
∫π
−π
xcos(nx)dx
=
1
π
[cos(nx)
n^2 +
xsin(nx)
n
]∣∣
∣∣
π
−π
= 0 ,
bn=π^1
∫π
−π
f(x)sin(nx)dx=π^1
∫π
−π
xsin(nx)dx
=
1
π
[sin(nx)
n^2 −
xcos(nx)
n
]∣∣
∣∣
π
−π
=π^1 (−^2 π)ncosnπ=^2 n(− 1 )n+^1.