60 Chapter 1 Fourier Series and Integrals
Figure 1 A periodic function of periodp.functions have the common period 2π, although each has other periods as
well.
Iffis periodic with period 2π, then we attempt to representfin the form
of an infinite series
f(x)=a 0 +∑∞
n= 1(
ancos(nx)+bnsin(nx))
. (1)
Each term of the series has period 2π, so if the sum of the series exists, it will
be a function of period 2π. There are two questions to be answered: (a) What
values musta 0 ,an,bnhave? (b) If the appropriate values are assigned to the
coefficients, does the series actually represent the given functionf(x)?
On the face of it, the first question is tremendously difficult, for Eq. (1) rep-
resents an equation in an infinite number of unknowns. But a reasonable an-
swer can be found easily by using theorthogonality^1 relations shown in Table 1.
We may summarize those relations by saying: The definite integral (over the
interval−πtoπ) of the product of any two different functions from the series
in Eq. (1) is zero.
The fundamental idea is that if the equality proposed in Eq. (1) is to be a real
equality, then both sides must give the same result after the same operation.
The orthogonality relations then suggest operations that simplify the right-
hand side of Eq. (1). Namely, we multiply both sides of the proposed equation
by one of the functions that appears there and integrate from−πtoπ.(We
must assume that the integration of the series can be carried out term by term.
This is sometimes difficult to justify, but we do it nonetheless.)
Multiplying both sides of Eq. (1) by the constant 1 (=cos( 0 x))andinte-
grating from−πtoπ,wefind
∫π−πf(x)dx=∫π−πa 0 dx+∑∞
n= 1∫π−π(
ancos(nx)+bnsin(nx))
dx.(^1) The wordorthogonalityshould not be thought of in the geometric sense.