1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

Chapter 1 453

13.f(x)=

∑∞

n= 1

bnsin(nπx),bn= 2

(

1 +cos(nπ)

)

/nπ.

15.f(x)=

∑∞

n= 1

bnsin(nx),

b 2 =^1

2

,otherbn=4sin(nπ/^2 )

π( 4 −n^2 )

.

17.

∑N

1

cos(nx)=Re

∑N

1

einx=Re

eix−eiNx

1 −eix =Re

eix/^2 −ei(^2 N−^1 )x/^2

e−ix/^2 −eix/^2.

The denominator is now− 2 isin(x/ 2 ).

19.f(x)=

∑∞

n= 1

bnsin(nx),bn=^2 asin(na+π)

n^2 a^2 −π^2

.

21.f(x)=

∫∞

0

(

sin(λa)

λπ cos(λx)+

1 −cos(λa)

λπ sin(λx)

)

dλ.

23.f(x)=

∫∞

0

2sin(λπ )

π( 1 −λ^2 )sin(λx)dλ (x>0).

29. Use

∫∞

0

sin(λt)

λ

dλ=π

2

.

31. These answers are not unique.

a.

∑∞

n= 1

bnsin(nπx),bn= 2 /nπ;

b.a 0 +

∑∞

n= 1

ancos(nπx),a 0 =^1

2

,an= 2

(

1 −cos(nπ)

)

/n^2 π^2 ;

c.

∫∞

0

B(λ)sin(λx)dλ,B(λ)= 2

(

λ−sin(λ)

)/(

πλ^2

)

;

d.

∫∞

0

A(λ)cos(λx)dλ,A(λ)= 2

(

1 −cos(λ)

)/(

πλ^2

)

.

The integrals of parts c. and d. converge to 0 forx>1.

33. Uses=6inEq.(7)ofSection8.

ˆa 0 = 0. 78424 , aˆ 4 =− 0. 00924 ,

ˆa 1 = 0. 22846 , aˆ 5 = 0. 00744 ,

ˆa 2 =− 0. 02153 , aˆ 6 =− 0. 00347 ,

ˆa 3 = 0. 01410.