1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

100 Chapter 1 Fourier Series and Integrals

EXERCISES

1.Verify Lemma 2. Multiply through by 2 sin(^12 y). Use the identity

sin

( 1

2 y

)

cos(ny)=

1

2

(

sin

((

n+

1

2

)

y

)

−sin

((

n−

1

2

)

y

))

.

Note that most of the series then disappears. (To see this, write out the

result forN=3.)

2.Verify Lemma 1 by integrating the sum term by term.

3.Letf(x)=f(x+ 2 π)andf(x)=|x|for−π<x<π.Notethatfis con-

tinuous and has a corner atx=0. Sketch the functionφ(y)as defined in

Eq. (16) ifx=0. Findφ( 0 +)andφ( 0 −).

4.Letfbe the odd periodic extension of the function whose formula isπ−x

for 0<x<π.Inthiscase,fhas a jump discontinuity atx=0. Taking

x=0, sketch the functions

φR(y)=f(x+y)−f(x+)

2sin(^12 y)

cos

(

1

2 y

)

(y> 0 ),

φL(y)=

f(x+y)−f(x−)

2sin(^12 y)

cos

( 1

2 y

)

(y< 0 ).

(These functions appear if the integrands in Eq. (22) are developed as in

Part 3 of the proof.)

5.Consider the functionfthat is periodic with period 2πand has the formula

f(x)=|x|^3 /^4 for−π<x<π.

a.Show thatfis continuous atx=0 but is not sectionally smooth.

b.Show that the functionφ(y)(from Eq. (16), withx=0) is sectionally

continuous,−π<x<π, except for a bad discontinuity aty=0.

c. Show that the Fourier coefficients ofφ(y)tend to 0 asnincreases, de-

spite the bad discontinuity.

1.8 Numerical Determination of Fourier Coefficients

There are many functions whose Fourier coefficients cannot be determined
analytically because the integrals involved are not known in terms of easily
evaluated functions. Also, it may happen that a function is not known explic-
itly but that its value can be found at some points. In either case, if a Fourier