1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

86 Chapter 1 Fourier Series and Integrals
∫b

a

f(x)g(x)dx=

∫b

a

a 0 g(x)dx

+

∑∞

n= 1

∫b

a

(

ancos(nx)+bnsin(nx)

)

g(x)dx. (3)

In Theorems 3 and 4, the functionf(x)is only required to be sectionally
continuous. It is not necessary that the Fourier series off(x)converge at all.
Nevertheless, the theorems guarantee that the series on the right converges and
equals the integral on the left in Eqs. (2) and (4).
One important application of Theorem4 was the derivation of the formulas
for the Fourier coefficients in Section1. An application of Theorems 3 and 4
is given in what follows.

Example.
The periodic functiong(x)whose formula in the interval 0<x< 2 πis

g(x)=x, 0 <x< 2 π

has the Fourier series

g(x)∼π− 2

∑∞

n= 1

sin(nx)

n

.

By applying Theorems 1 and 2, we find that the functionf(x)defined byf(x)=
[π−g(x)]/2 has the series

f(x)∼

∑∞

n= 1

sin(nx)

n

.

This manipulation would be simple algebra if the correspondence∼were an
equality.
The functionf(x)satisfies the hypotheses of Theorem 3. Thus we may inte-
grate the preceding series from 0 tobto obtain

∫b

0

f(x)dx=

∑∞

n= 1

1 −cos(nb)

n^2

.

Theorem 3 guarantees that this equality holds for anyb.Intheintervalfrom0
to 2πwe have the formulaf(x)=(π−x)/2. Hence

∫b

0

f(x)dx=πb

2

−b

2

4

=

∑∞

n= 1

1 −cos(nb)

n^2

, 0 ≤b≤ 2 π.