1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

64 Chapter 1 Fourier Series and Integrals

b. f(x)=

{ 0 , − 1 <x≤0,

x, 0 <x<1, f(x+^2 )=f(x);

c. f(x)=

{ 0 , −π<x≤0,

1 , 0 <x≤ 2 π, f(x+^3 π)=f(x);

d. f(x)=

{ 0 , −π<x≤0,

sinx, 0 <x≤π, f(x+^2 π)=f(x).

3.Show that the constant functionf(x)=1 is periodic with every possible

periodp>0.

4.Carry out the details of deriving the equation foram.

5.Supposef(x)has periodp. Show that for anyc, the following equation

holds. Hint: Think of the integral as the net signed area.

∫c+p

c

f(x)dx=

∫p

0

f(x)dx.

6.Supposef(x),g(x)are periodic with a common periodp. Show thataf(x)+

bg(x)andf(x)·g(x)also are periodic with periodp(a,bare constants).

7.Find the Fourier series of each of the following periodic functions. Integra-

tion is not necessary: Use trigonometric identities.

a. f(x)=cos^2 (x);

b. f(x)=sin(x−π/ 6 );

c. f(x)=sin(x)cos( 2 x).

8.Ve r i f y t h a t s i n(πx/a)and cos(πx/a)are periodic with period 2a.

1.2 Arbitrary Period and Half-Range Expansions

In Section 1 we found a way to represent a periodic function of period
2 πwith a Fourier series. It is not necessary to restrict ourselves to this pe-
riod. In fact, we may broaden the idea of Fourier series to include func-
tions of any period by a simple rescaling of the variables. Let us suppose
that a functionfis periodic with period 2a.(Weuse2ain place ofpfor
later convenience.) Then we may relatef toaseriesofthefunctions1,
sin(πx/a),cos(πx/a),sin( 2 πx/a),cos( 2 πx/a),...,allhavingperiod2a,in
the form

f(x)∼a 0 +

∑∞

n= 1

ancos

(

nπx

a

)

+bnsin

(

nπx

a

)

.

The coefficients of this Fourier series may be determined either by scaling from
the formulas of Section 1 or through the concept of orthogonality. In either