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116 Chapter 1 Fourier Series and Integrals

EXERCISES

1.Use the complex form

an−ibn=

1

π

∫π

−π

f(x)e−inxdx, n= 0 ,

to find the Fourier series of the function

f(x)=eαx, −π<x<π.

2.Find the complex Fourier series for the “square wave” with period 2π:

f(x)=

{ 1 , 0 <x<π,

− 1 , −π<x<0.

3.Find the complex Fourier integral representation of the following func-

tions:

a.f(x)=

{

e−x, x>0,

0 , x<0;

b.f(x)=

{

sin(x), 0 <x<π,

0 , elsewhere.

4.Find the complex Fourier integral for

a.f(x)=

{

xe−x, 0 <x,

0 , x<0;

b.f(x)=e−α|x|sin(x).

5.Relate the functions and series that follow by using complex form and Tay-

lor series.

a. 1 +

∑∞

n= 1

rncos(nx)= 1 −^12 −rcosrcos(x()x+)r 2 , 0 ≤r<1;

b.

∑∞

n= 1

sin(nx)

n!

=ecos(x)sin

(

sin(x)

)

.

6.Show by integrating that

∫π

−π

einxe−imxdx=

{

0 , n=m,

2 π, n=m,

and develop the formula for the complex Fourier coefficients using this idea

of orthogonality.