# Laplace Transformations Translation Theorems

We will now cover some more complicated Lapalce Transformations:

## First Translation Theorem

The First Translation Theorem states:

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

The inverse of the theorem is simply:

$$\mathcal {L}^{-1} \{ F(s-a) \} = e^{at} f(t)$$

Evaluate $$\mathcal {L} \{ e^{2t} \sin (\pi t) \}$$.

Evaluate $$\mathcal {L}^{-1} \left \{ \cfrac{1}{(s+5)^3} \right\}$$.

## Second Translation Theorem

The Second Translation Theorem states:

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

The inverse of the theorem is simply:

$$\mathcal {L}^{-1} \{ e^{-as}F(s) \} = \mathcal {L} \{ f(t-a) \mathcal {U} (t-a) \}$$

Where $$\mathcal {U} (t-a)$$ is the heaviside function or unit step function which is an "on/off" function:

$$\mathcal{U} (t-a) = \begin{cases} 0, & \text{if } 0 \le t \lt a \\ 1, & \text{if } t \ge a \end{cases}$$

Evaluate $$\mathcal {L} \{ f(t) \}$$ if $$f(t) = \begin{cases} \sinh {(t)}, & \text{if } 0 \le t \lt \frac{\pi}{8} \\ \sinh {(t)} + \cos {(t - \frac{\pi}{8})}, & \text{if } t \ge \frac{\pi}{8} \end{cases}$$.

Evaluate $$\mathcal {L}^{-1} \{ e^{-5s} \cfrac{s}{s^2 + 10} \}$$.

## Derivative Theorem

The Derivative Theorem state:

$$\mathcal {L} \{ t^n f(t) \} = (-1)^n \cfrac{d^n}{ds^n} F(s)$$

You are less likely to use the inverse as the derivative of $$F(s)$$ may be simplified making it challenging to identify that this theorm can be used.

Evaluate $$\mathcal{L} \{ t^2e^{3t} \}$$.