1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

Miscellaneous Exercises 127

16.Show that the function given by the formulaf(x)=(π−x)/2, 0<x<
2 π, has the Fourier series

f(x)=

∑∞

1

sin(nx)

n ,^0 <x<^2 π.

Sketchf(x)and its periodic extension.

17.Use complex methods and a finite geometric series to show that

∑N

n= 1

cos(nx)=

sin

(

(N+^12 )x

)

−sin(^12 x)

2sin(^12 x)

.

Then use trigonometric identities to identify

∑N

n= 1

cos(nx)=

sin(^12 Nx)cos

( 1

2 (N+^1 )x

)

sin(^12 x).

18.Identify the partial sums of the Fourier series in Exercise 16 as

SN(x)=

∑N

n= 1

sin(nx)

n.

The series of Exercise 17 isS′N(x). Use this information to locate the max-

ima and minima ofSN(x)in the interval 0≤x≤π.Findthevalueof

SN(x)at the first point in the interval 0<x<πwhereS′N(x)=0 for

N=5. Compare to(π−x)/2atthatpoint.

19.Find the Fourier sine series of the function given by

f(x)=

{

sin

(πx

a

)

, 0 <x<a,

0 , a<x<π,

assuming that 0<a<π.

20.Find the Fourier cosine series of the function given in Exercise 19.

21.Find the Fourier integral representation of the function given by

f(x)=

{ 1 , 0 <x<a,

0 , x<0orx>a.