1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

112 Chapter 1 Fourier Series and Integrals

a.f(x)=

{

sin(x), −π<x<π,

0 , |x|>π;

b.f(x)=

{

sin(x), 0 <x<π,

0 , otherwise;

c. f(x)=

{

|sin(x)|, −π<x<π,

0 , otherwise.

6.Show that ifkandKare positive, then the following are true:

a.

∫∞

0

e−kxsin(x)dx= 1 +^1 k 2 ;

b.

∫∞

0

1 −e−Kx

x sin(x)dx=tan

− (^1) (K);
c.

∫∞

0

sin(x)

x

dx=π

2

.

(Part (a) by direct integration, (b) by integration of (a) with respect tok

over the interval 0 toK,(c)bylimitof(b)asK→∞.)

7.Starting from Exercise 6c, show that

∫∞

0

sin(λz)

λ

dλ=

{π/ 2 , 0 <z,

0 , z=0,

−π/ 2 , z<0.

Is this the Fourier integral of some function?

8.Change the variable of integration in the formulas forAandB, and justify

each step of the following string of equalities. (Do not worry about chang-

ing order of integration.)

f(x)=^1

π

∫∞

0

∫∞

−∞

f(t)

(

cos(λt)cos(λx)+sin(λt)sin(λx)

)

dt dλ

=π^1

∫∞

−∞

f(t)

∫∞

0

cos

(

λ(t−x)

)

dλdt

=π^1

∫∞

−∞

f(t)

[

ωlim→∞sin(ω(t−tx−x))

]

dt

=ωlim→∞

1

π

∫∞

−∞

f(t)

sin(ω(t−x))

t−x dt.

The last integral is calledFourier’s single integral.Sketchthefunction

sin(ωv)/vas a function ofvfor several values ofω. What happens near