1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

78 Chapter 1 Fourier Series and Integrals

a.f(x)=|x|+x, − 1 <x<1;

b.f(x)=xcos(x), −π 2 <x<π 2 ;

c. f(x)=xcos(x), − 1 <x<1;

d.f(x)=

{ 0 , 1 <x<3,

1 , − 1 <x<1,

x, − 3 <x<−1.

3.TowhatvaluedoestheFourierseriesoff converge iff is acontinuous,

sectionally smooth, periodic function? Give an example.

4.State convergence theorems for the Fourier sine and cosine series that arise

from half-range expansions.

5.A function is given on the interval 0<x<2 by the formula

f(x)=

{x, 0 <x<1,

1 −x, 1 <x<2.

a.Sketch the odd periodic extension ̄f 0 (x)for− 4 <x<4.

b.Explain why ̄f 0 (x)is sectionally smooth.

c. Determine the value that the sine series offconverges to at these points:

x=1,x=2,x= 9 .6,x=− 3 .8.

6.For the same function given in Exercise 5, answer the same questions for

f ̄e(x), the even periodic extension offand its cosine series.

7.The series

∑∞

n= 1

(− 1 )n

n^2 cos(nx)

converges to a functionf(x)whose formula on the interval−π<x<πis

f(x)=A+Bx+Cx^2.

DetermineA,B,andC.

8.The series

∑∞

n= 1

1

n^3

sin(nx)

converges to a continuous periodic function. On the interval 0<x< 2 π,

this function coincides with a polynomialp(x)of degree 3. Find the polyno-

mial. Hint: Determine pointsxon the interval 0<x< 2 πwherep(x)=0.

Use this information to get a form forp(x).