Translation theorem

When discussing Laplace transforms, the first translation theorem (also first shifting theorem) is:

$$\mathcal{L}\{e^{at}f(t)\}=F(s-a)$$

for any $a\in \mathbb{R}$ for $s>a+c$. This can be proved by the definition of the Laplace transform.

The second translation theorem is:

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

where $\mathcal{U}$ is the unit step function.

Questions

We may sometimes run into a problem where there is a clear unit shift at one point but not throughout the entire function (in $s$). In cases like this, it’s good to define some $\tilde{s}$ as a change of variables and rewrite. We can usually simplify from that.