y direct integration, find the Laplace transform F $($s$)$ and the region of convergence of F $($s$)$ for

the following signals where a and b are positive real numbers:

$($a$)(2$ points$)$ $($t$)$

$($b$)(2$ points$)$ u$($t$)$

$($c$)(3$ points$)$ e$$atu$($t$)$

$($d$)(4$ points$)$ cos$($bt$)$ u$($t$)$

$($e$)(4$ points$)$ sin$($bt$)$ u$($t$)$

$2.(5$ points$)$ Compare the Fourier and Laplace transforms of the signals

f$($t$)=$ $($t$)$

g$($t$)=$ u$($t$).$

Explain

F $($$)=$ F $($s$)|$s$=$ $,$

while

G$($$)6=$ G$($s$)|$s$=$ $.$

Problem $46$: $(20$ points$)$

Let F $($s$)=$ L$\{$f$($t$)\}$ denote the unilateral Laplace transform of f$($t$).$ Prove the following properties of the Laplace

transform, where to $0$ is a real constant and so is a complex constant.

$1.(4$ points$)$ Right shift in time:

L$\{$f$($t $$ to$)$u$($t $$ to$)\}=$ F $($s$)$e$$sto $,$ to $>0$

$2.(4$ points$)$ Multiplication by t:

L$\{$tf$($t$)\}=$ d

ds F $($s$)$

$3.(4$ points$)$ Frequency shift:

L

$\{$

eso tf$($t$)$

$\}$

$=$ F $($s $$ so$)$

$4.(8$ points$)$ Time differentiation property:

L

$\{$ df

dt

$\}$

$=$ sF $($s$)$ f$(0)$

and

L

$\{$ d$2$f

dt$2$

$\}$

$=$ s$2$F $($s$)$ sf$(0)$f$(0)$

Problem $47$: $(20$ points$)$

Using the elementary transform pairs derived in Problem $43$ and the properties derived in Problem $44,$ find the

Laplace transform of the following signals where to$,$ a$,$ and b are positive real parameters.

$1.(4$ points$)$ u$($t $$ to$)$

$2.(4$ points$)$ tu$($t$)$

$3.(4$ points$)$ te$$atu$($t$)$

$4.(4$ points$)$ e$$at cos$($bt$)$ u$($t$)$

$5.(4$ points$)$ e$$at sin$($bt$)$ u$($t$)$

Note that this approach, particular in the case of the signals considered in parts $4$ and $5,$ is much easier than finding

the Laplace transform by direct integration.