Problem $45$: $(20$ points$)1.(15$ points$)$ By 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$=$ $.$