# Step Functions

• Mar 25th 2009, 10:18 AM
drewkx152
Step Functions
Hey amigos :)

Step Functions.. I have quite a few sample problems, however they are all so nice as to have either a full period as the subscript on u, or something like pi/2 so i can convert from sine to cosine.

I'm running into a problem with this one:

Laplace Transform of {u(subscript(pi/4))*e^(3t)*cost}

I have quite a few equations to work with but found them to no avail. My best attempt was using {u(subscript c)(t)*g(t)} = e^(-cs) * Laplace Transform {g(t+c)}

But I find that I am then working with e^(3(t+pi/4))*cos(t-pi/4).. which i simply do not have anything close to those on my tables, and we're not supposed to integrate for these, we're supposed to use the laws..
• Mar 26th 2009, 06:54 AM
CaptainBlack
Quote:

Originally Posted by drewkx152
Hey amigos :)

Step Functions.. I have quite a few sample problems, however they are all so nice as to have either a full period as the subscript on u, or something like pi/2 so i can convert from sine to cosine.

I'm running into a problem with this one:

Laplace Transform of {u(subscript(pi/4))*e^(3t)*cost}

I have quite a few equations to work with but found them to no avail. My best attempt was using {u(subscript c)(t)*g(t)} = e^(-cs) * Laplace Transform {g(t+c)}

But I find that I am then working with e^(3(t+pi/4))*cos(t-pi/4).. which i simply do not have anything close to those on my tables, and we're not supposed to integrate for these, we're supposed to use the laws..

$
\mathcal{L} [u_{\pi/4}(t)e^{3t}] = \int_{\pi/4}^{\infty}e^{3t}e^{-st} \ dt=\int_{\pi/4}^{\infty}e^{(3-s)t}\ dt$
$
= \left[\frac{e^{(3-s)t}}{3-s}\right]_{\pi/4}^{\infty}
=\frac{e^{(3-s)\pi/4}}{s-3}
$

(you don't need to do the integral as by suitable change of variable this converts into a function of $s$ times the LT of a known function.

CB