How Do You Apply and Invert the Unit Step Laplace Transform?

[Deathfish]

Someone give me a crash course on Unit Step Laplace Transform and Inverse Laplace Transform. Need help on the formula:

U(t-a)f(t-a) to (e^-as)F(s)

from what i can deduce, the way to use it is

(Laplace)

- Find a from U(t-a)
- group (t-a) together
- replace (t-a) with ____ (?) What is a suitable symbol?
- work out Laplace
- multiply Laplace with e^-as

(Inverse Laplace)

- Factor out e^-as
- find out what is a from e^-as
- inverse Laplace
- replace t with (t-a)
- multiply with U(t-a)

Just need to know how to use it, not how it works in detail. Tell me if I got it right and any where to improve on. Also need to know a suitable symbol to replace (t-a) as above. Thanks!

 

[lippyka]

Yes, you have the basic idea correct. The symbol you use to replace (t-a) is usually ξ. The equation you are looking at is a special case of the Laplace Transform of a shifted function, which is given by: F(s) = e^-as * L{f(ξ+a)} Where L{f(ξ)} is the Laplace Transform of f(ξ). To invert this equation, you simply note that a = -Ln(e^-as), and then apply the inverse Laplace transform to the resulting expression. For example, if we have the equation: F(s) = e^-2s * L{f(ξ+2)} Then the inverse Laplace transform is given by: f(t) = U(t-2) * L^{-1}{F(s)} Where L^{-1}{F(s)} is the inverse Laplace transform of F(s).