62 Chapter 1 Fourier Series and Integrals
We can now summarize our results. In order for the proposed equalityf(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−πf(x)dx, (3)an=1
π∫π−πf(x)cos(nx)dx, (4)bn=^1
π∫π−πf(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−πf(x)dx=^1
2 π∫π−πxdx= 0 ,an=π^1∫π−πf(x)cos(nx)dx=π^1∫π−πxcos(nx)dx=
1
π[cos(nx)
n^2 +xsin(nx)
n]∣∣
∣∣
π−π= 0 ,
bn=π^1∫π−πf(x)sin(nx)dx=π^1∫π−πxsin(nx)dx=
1
π[sin(nx)
n^2 −xcos(nx)
n]∣∣
∣∣
π−π
=π^1 (−^2 π)ncosnπ=^2 n(− 1 )n+^1.