# Step Functions

• Apr 13th 2010, 01:25 AM
TheBerkeleyBoss
Step Functions
Hi all, I'm having a bit of trouble solving this problem. The instructions say to find the Laplace transform of the given function.

The given function:

$f(t)=\left\{
\begin{array}{l l}
0,& t<2\\
(t-2)^3, & t \geq 2
\end{array}\right.$

Phew! Trying to create that in LaTeX was probably just as difficult :)

• Apr 13th 2010, 01:54 AM
HallsofIvy
Quote:

Originally Posted by TheBerkeleyBoss
Hi all, I'm having a bit of trouble solving this problem. The instructions say to find the Laplace transform of the given function.

The given function:

$f(t)=\left\{
\begin{array}{l l}
0,& t<2\\
(t-2)^3, & t \geq 2
\end{array}\right.$

Phew! Trying to create that in LaTeX was probably just as difficult :)

Well, by the definition of "Laplace Transform" this is $\int_2^\infty e^{-st}(t- 2)^3 dt$. That should be doable with integration by parts- three times.
• Apr 14th 2010, 12:48 AM
bandedkrait
You can solve this by redefining the function in terms of the unit step function $u(t)$

$u2(t)=0$ $t<2$
$u2(t)=1$ $t>=2$

So the given function may be written as

$f(t)=u2(t)(t-2)^3$

Now, an important property regarding laplace transforms of unit step functions is

$L(ua(t)g(t-a)) = e^{-as}L(g(t))$

Therefore here, $g(t)=t^3$ and $a=2$

So the laplace transform can be evaluated using the above property