# Inverse laplace of 1s

1. Feb 5, 2013

### indianaronald

This is very tricky for me. How to find the inverse laplace of 1s. I haven't been taught the integral method of inverse. Only the formula based , splitting terms kind of thing. I used matlab and found it was dirac delta. But how do I get to it without using the integral for inverse?

2. Feb 6, 2013

### antibrane

When I look at this in Mathematica I get a derivative of the delta function, in other words:
$$\mathcal{L}^{-1}\left\{s\right\} = \frac{d}{dt}\delta(t)$$

3. Feb 6, 2013

### indianaronald

Hey yeah, figured that out. It's actually by this property:

inverse( df/dt)= s F(s) where laplace(f) = F(s)

laplace ( dirac delta )=1 ( known property )

laplace( d(diracdelta)/dt ) = s*(1)

hence, inverse( s) = d(diracdelta)/dt

4. Feb 6, 2013

### jfgobin

Indianaronald,

Almost, be careful that:

$\mathcal{L}\left\{ \frac{\mathrm{d}f}{\mathrm{dt}} \left(t\right)\right\}=s\mathcal{L}\left\{f\left(t\right)\right\}$

Not the inverse as you mentioned it.

5. Feb 6, 2013

### indianaronald

yeah yeah. That's what I meant. Typo.