1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

Chapter 1 447

b.

4

π

[

sin

(πx

2

)

+

1

3 sin

( 3 πx

2

)

+

1

5 sin

( 5 πx

2

)

+···

]

;

c. 121 −π^12

[

cos( 2 πx)−^14 cos( 4 πx)+^19 cos( 6 πx)−···

]

.

3. ̄f(x)=f(x− 2 na),2na<x< 2 (n+ 1 )a,

f(x)∼a 0 +

∑∞

1

ancos(nπx/a)+bnsin(nπx/a),

a 0 =^1

2 a

∫ 2 a

0

f(x)dx,an=^1

a

∫ 2 a

0

f(x)cos(nπx/a)dx,

bn=

1

a

∫ 2 a

0

f(x)sin(nπx/a)dx.

5. Odd: (a), (d), (e); even: (b), (c); neither: (f ).


6. a.

2

π

(

sin(πx)−

1

2 sin(^2 πx)+···

)

;

b. This function is its own Fourier series;

c.

4

π^2

(

sin(πx)−

1

9 sin(^3 πx)+

1

25 sin(^5 πx)−···

)

.

9. Iff(−x)=−f(x)andf(x)=f(a−x)for 0<x<a, sine coefficients
   with even indices are zero. Example: square wave.


10. a.f(x)= 1 =

2

π

∑∞

1

1 −cos(nπ)

n sin

(nπx

a

)

;

b.f(x)=

a

2 −

2 a

π^2

∑∞

1

1 −cos(nπ)

n^2 cos

(nπx

a

)

=^2 πa

∑∞

1

−cos(nπ)

n sin

(

nπx

a

)

;

c.f(x)=

∑∞

1

(− 1 )n+^1 sin( 1 )^2 nπ

(nπ)^2 − 1

sin(nπx), 0 <x< 1

=

∑∞

1

(

(− 1 )ncos( 1 )− 1

) 2

(nπ)^2 − 1 cos(nπx),^0 <x<^1 ;

d.f(x)=^2

π

[

1 −

∑∞

1

1 +cos(nπ)

n^2 − 1

cos(nx)

]

=sin(x).

13. Even, yes. Odd, yes only iff( 0 )=f(a)=0.