# Inverse Laplace Transform

• ns5032
In summary, the conversation is discussing the inverse Laplace transform of a complex function with an exponential term. The speaker is seeking help on finding the inverse transform and mentions using partial fractions and the Dirac function. The conversation also mentions using the Laplace transform of integrals or taking the sum of all the residues of the function.

#### ns5032

My goodness.. I have not come across an inverse Laplace transform like this. My teacher let's us just use a chart to figure them out, but this is definitely not on there. How do I find the inverse Laplace transform of:

{ (1/2)+[(5e^-6s)/(4s^2)] } / (s+5)

I already used partial fractions to split up the denominator, so there is one more inverse laplace that I need to do on top of this one, but I figure if I get this one, then I can get the other one as well. Any help??!

you will need to use the dirac function for the exponential, the rest is pretty standard.

The e^-6s part means that this part of your signal is delayed in time by 6 seconds. so you can do the inverse transform without it and then when you get your time signal for this part, delay it by 6 seconds. So replace t with t-6 in your answer for the time equation and you should be good.

Do the inverse transform without it? Like... just take that whole term out and treat it like it is zero?

You'll need to use this:

$$L(f(t-a)u(t-a)) = e^{-as}F(s)$$

I don't think dirac delta function comes into play here.

Or i also think that applying the laplace transforms of integrals would work here.

An easy way to take the inverse Laplace Transform (if you have some knowledge of Complex Calculus) is to take the sum of all the residues of the function e^(zt) f(z), where you are taking the inverse Laplace transform of f(z) and z is the complex variable.