1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

66 Chapter 1 Fourier Series and Integrals

of period 2a, by using the following definitions:

̄f(x)=f(x), −a<x<a,

̄f(x)=f(x+ 2 a), − 3 a<x<−a,

̄f(x)=f(x− 2 a), a<x< 3 a

and so on, up and down thex-axis. Notice that the argument offon the right-
hand side always falls in the interval−a<x<a,wherefwas originally given.
Graphically, this kind of extension amounts to making a template of the graph
offon−a<x<aand then copying from the template in abutting intervals
of length 2a.
For the extended function with period 2a, the formulas for the Fourier co-
efficients become

a 0 =

1

2 a

∫a

−a

̄f(x)dx,

an=

1

a

∫a

−a

̄f(x)cos

(nπx

a

)

dx,

bn=^1

a

∫a

−a

̄f(x)sin

(nπx

a

)

dx.

(2)

If we are concerned withf(x)only in the interval−a<x<awhere it was
originally given, the process of periodic extension is strictly formal, because
the formulas for the coefficients involvefonly on the original interval. Thus,
we may write

f(x)∼a 0 +

∑∞

n= 1

ancos

(nπx

a

)

+bnsin

(nπx

a

)

, −a<x<a.

The inequality forxdraws attention to the fact thatfwas defined only on the
interval−atoa.

Example.
Supposef(x)=xin the interval− 1 <x<1. The graph of its periodic exten-
sion (with period 2) is seen in Fig. 3, and the Fourier coefficients are

a 0 = 0 , an= 0 ,

bn=

∫ 1

− 1

xsin(nπx)dx=−2cos(nπ)

nπ

=^2

π

(− 1 )n+^1

n

.

The sine and cosine functions that appear in a Fourier series have some
special symmetry properties that are useful in evaluating the coefficients. The
graph of the cosine function is symmetric about the vertical axis, and that of
the sine is antisymmetric. We formalize these properties with a definition.