1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

392 Chapter 6 Laplace Transform

16.Suppose that the functionf(t)is periodic with period 2a. Show that the

Laplace transform offis given by the formula

F(s)=

G(s)

1 −e−^2 as,

where

G(s)=

∫ 2 a

0

f(t)e−stdt.

(Hint: See Section 6.1, Exercise 6.)

17.Apply the extended Heaviside method to the inversion of a transform

with the form

F(s)= G(s)

1 −e−^2 as

,

whereG(s)does not become infinite for any value ofs.

18.Show that for a periodic functionf(t)the quantities

cn=^1

2 a

G

(

inπ

a

)

(G(s)is defined in Exercise 16) are the complex Fourier coefficients.

19.How is it possible to determine that a Laplace transformF(s)corre-

sponds to a periodicf(t)?

20.Is this function the transform of a periodic function?

F(s)=^1

s^2 +a^2

.

21.Use the method of Exercise 16 to find the transform of the periodic ex-

tension of

f(t)=

{ 1 , 0 <t<π,

− 1 ,π<t< 2 π.

22.Same as Exercise 21, but use the function of Exercise 15.

23.Use the method of Exercise 16 to find the transform of

f(t)=

∣∣

sin(t)

∣∣

.

24.Find the transform of the solution of the problem