# Thread: Step Functions

1. ## 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..

2. 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