1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

84 Chapter 1 Fourier Series and Integrals

d.f(x)=sin(x)+|sin(x)|, −π<x<π;

e.f(x)=x+|x|, −π<x<π;

f. f(x)=x(x^2 − 1 ), − 1 <x<1;

g.f(x)= 1 + 2 x− 2 x^3 , − 1 <x<1.

2.The Fourier series of the function

f(x)=sin(x)

x

, −π<x<π,

converges at every point. To what value does the series converge atx=0?

atx=π? The convergence is uniform. Why?

3.Determine whether the sine and cosine series of the following functions

converge uniformly. Sketch.

a.f(x)=sinh(x), 0 <x<π;

b.f(x)=sin(x), 0 <x<π;

c. f(x)=sin(πx), 0 <x<^12 ;

d.f(x)= 1 /( 1 +x), 0 <x<1;

e.f(x)= 1 /( 1 +x^2 ), 0 <x<2.

4.Ifanandbntend to zero asntends to infinity, show that the series

a 0 +

∑∞

n= 1

e−αn

(

ancos(nx)+bnsin(nx)

)

converges uniformly(α > 0 ).

5.For each of the following coefficients, use Theorem 1 to decide whether

convergence of the associate Fourier series is uniform.

a.an=sin

(^2) (nπ/ 2 )
n^2 π^2
, bn=0;
b.an= 0 , bn=^1 −cos(nπ)
nπ

;

c. a 1 = 0 , an=^2 (^1 +cos(nπ))

n^2 − 1

(n≥ 2 ), bn=0;

d.an= 0 , bn=

1

cosh(nπ/ 2 ).