1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

126 Chapter 1 Fourier Series and Integrals

e.odd periodic extension;

f.the one corresponding tof(x)=x,−a<x<0.

8.Perform the same task as in Exercise 7, butf(x)=0, 0<x<a.

9.Find the Fourier series of the function given by

f(x)=

{ 0 , −a<x<0,

2 x, 0 <x<a.

Sketch the graph off(x)and its periodic extension. To what values does

the series converge atx=−a,x=−a/2,x=0,x=a,andx= 2 a?

10.Sketch the odd periodic extension and find the Fourier sine series of the

function given by

f(x)=

{

1 , 0 <x<π 2 ,

1

2 ,

π

2 <x<π.

To what values does the series converge atx=0,x=π/2,x=π,x=

3 π/2, andx= 2 π?

11.Sketch the even periodic extension of the function given in Exercise 10.

Find its Fourier cosine series. To what values does the series converge at

x=0,x=π/2,x=π,x= 3 π/2, andx= 2 π?

12.Find the Fourier cosine series of the function

g(x)=

{ 1 −x, 0 <x<1,

0 , 1 <x<2.

Sketch the graph of the sum of the cosine series.

13.Find the Fourier sine series of the function defined byf(x)= 1 − 2 x,

0 <x<1. Sketch the graph of the odd periodic extension off(x),and

determine the sum of the sine series at points where the graph has a

jump.

14.Following the same requirements as in Exercise 13, use the cosine series

and the even periodic extension.

15.Find the Fourier series of the function given by

f(x)=







0 , −π<x<−π 2 ,

sin( 2 x), −π 2 <x<π 2 ,

0 , π 2 <x<π.

Sketch the graph of the function.