1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

70 Chapter 1 Fourier Series and Integrals

If the series on the right converge, theyactually represent periodic functions
with period 2a. The cosine series would represent theeven periodicextension
off— the periodic extension offe; and the sine series would represent theodd
periodicextension off.
When the problem at hand is to represent the functionf(x)in the interval
0 <x<a, where it was originally given, we may use either the Fourier sine
series or the cosine series because bothfeandfocoincide withfin the interval.
Thus we may summarize by saying: Iff(x)is given for 0<x<a,then

f(x)∼a 0 +

∑∞

n= 1

ancos

(nπx

a

)

, 0 <x<a,

a 0 =^1

a

∫a

0

f(x)dx, an=^2

a

∫a

0

f(x)cos

(

nπx

a

)

dx

and

f(x)∼

∑∞

n= 1

bnsin

(nπx

a

)

, 0 <x<a,

bn=^2 a

∫a

0

f(x)sin

(

nπx

a

)

dx.

These two representations are calledhalf-range expansions, and the series are
called the Fourier cosine and Fourier sine series off,respectively.Weshall
need these, more than any other kind of Fourier series, in the applications we
make later in this book.

Example.
Let us suppose that the functionfhas the formula

f(x)=x, 0 <x< 1.

Then the odd periodic extension offis as shown in Fig. 5, and the Fourier sine
coefficients offare

bn= 2

∫ 1

0

xsin(nπx)dx=−n^2 πcos(nπ).

The even periodic extension offis shown in Fig. 6. The Fourier cosine co-
efficients are

a 0 =

∫ 1

0

xdx=

1

2 ,

an= 2

∫ 1

0

xcos(nπx)dx=−n (^22) π 2

(

1 −cos(nπ)

)

.