Question

Laplace proof problem

Answer 1

http://assets.openstudy.com/updates/attachments/528e9920e4b04247257181b6-usukidoll-1385078944897-5.3.png

Answer 2

oh yeah we were doing #9

Answer 3

\[f_a(x)=\begin{cases}f(x-a)&x\geq a\\0&0\leq x<a\end{cases}\\\qquad=f(x-a)h(x-a)\]

Answer 4

\[\mathcal L\{f_a(x) \}=\int \limits_0^\infty f_a(x) e^{-sx}\mathrm dx\\
\qquad=\int \limits_0^\infty f(x-a)h(x-a) e^{-sx}\mathrm dx\\
\qquad =0+\int \limits_a^\infty f(x-a) e^{-sx}\mathrm dx\\
\qquad =\int \limits_a^\infty f(x-a) e^{-sx}\mathrm dx\\
\qquad\qquad\text{let } u=x-a\\
\:\\
%\qquad =f(x-a) e^{-sx}\Big|_a^\infty\\
%\qquad=\hat f(s+a)\]