# Math Help - Laplace Transformation

1. ## Laplace Transformation

Hi,

I want to find the Laplace inverse of the following function:

$1/(s^2 +2zws + w^2)$ , where z and w are constants.

I know that the result should have an exponintial part and a sin or cos, but I don't know how to factor the function to use the table.

Thanks,

2. This is what I get after completing the square:
$
\frac {1}{(s+ \frac{zw}{2})^2 - (\frac{zw}{2})^2 + w^2}$

I still don't know what to do next

3. Originally Posted by Musab
This is what I get after completing the square:
$
\frac {1}{(s+ \frac{zw}{2})^2 - (\frac{zw}{2})^2 + w^2}$

I still don't know what to do next
Um, that's not completing the square...

if you complete the square, you get

$\displaystyle \frac 1{(s + zw)^2 + w^2(1 - z^2)}$

now what?

4. Thx

I think you got the signs wrong, it should be $\displaystyle \frac 1{(s + zw)^2 + w^2(-z^2 + 1)}$

Now I should do the following:

$\frac{1}{(w^2(-z^2+1))^{0.5}} . \frac{(w^2(-z^2+1))^{0.5}}{(s + zw)^2 + w^2(-z^2 + 1)}$

and use the laplace table
is that right ?

5. Originally Posted by Musab
Thx

I think you got the signs wrong, it should be $\displaystyle \frac 1{(s + zw)^2 + w^2(-z^2 + 1)}$
yes

Now I should do the following:

$\frac{1}{(w^2(-z^2+1))^{0.5}} . \frac{(w^2(-z^2+1))^{0.5}}{(s + zw)^2 + w^2(-z^2 + 1)}$

and use the laplace table
is that right ?
and yes