1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

128 Chapter 1 Fourier Series and Integrals

22.Find the Fourier sine and cosine integral representations of the function

given by

f(x)=

{a−x

a

, 0 <x<a,

0 , a<x.

23.Find the Fourier sine integral representation of the function

f(x)=

{

sin(x), 0 <x<π,

0 ,π<x.

24.Find the Fourier integral representation of the function

f(x)=

{ 1 /, α <x<α+,

0 , elsewhere.

25.Use integration by parts to establish the equality

∫∞

0

e−λcos(λx)dλ=

1

1 +x^2.

26.TheequationinExercise25isvalidforallx. Explain why its validity

implies that

2

π

∫∞

0

cos(λx)

1 +x^2 dx=e

−λ,λ> 0.

27.Integrate both sides of the equality in Exercise 25 from 0 totto derive

the equality

∫∞

0

e−λsin(λt)

λ dλ=tan

− (^1) (t).
28.Does the equality in Exercise 27 imply that
2
π

∫∞

0

tan−^1 (t)sin(λt)dt=e

−λ

λ

?

29.FromExercise27derivetheequality

∫∞

0

1 −e−λ

λ sin(λx)dλ=

π

2 −tan

− (^1) (x), x> 0.
30.Without using integration, obtain the Fourier series (period 2π)ofeach
of the following functions: