1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

106 Chapter 1 Fourier Series and Integrals

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1.9 Fourier Integral

In Sections 1 and 2 of this chapter we developed the representation of a pe-
riodic function in terms of sines and cosines with the same period. Then, by
means of periodic extension, we obtained series representations for functions
defined only on a finite interval. Now we must deal with nonperiodic functions
defined forxbetween−∞and∞. Can such functions also be represented in
terms of sines and cosines? We make some transformations that suggest an
answer.
Supposef(x)is defined for−∞<x<∞and is sectionally smooth in every
finite interval. Then for any positivea,f(x)can be represented in the interval
−a<x<aby its Fourier series:

f(x)=a 0 +

∑∞

n= 1

ancos

(

nπx

a

)

+bnsin

(

nπx

a

)

, −a<x<a,

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. (1)

Example 1.
Let

f(x)=

{

e−x, 0 <x,

0 , x<0.

For anya>0, we have the Fourier series forf(x)on the interval−a<x<a:

f(x)=a 0 +

∑∞

1

ancos

(nπx

a

)

+bnsin

(nπx

a

)

, −a<x<a,

a 0 =^1 −e

−a

2 a , an=

1 −e−acos(nπ)

a( 1 +(nπ/a)^2 ),

bn=(^1 −e

−acos(nπ))nπ

a^2 ( 1 +(nπ/a)^2 ). (2)

(The series converges to 1/2atx=0andtoe−a/2atx=a.)