# Laplace Transformations Translation Theorems Part 2

We will now cover some more complicated Lapalce Transformations:

## Convolution Theorem

The Convolution Theorem states:

$$\mathcal {L} \{ f \ast g \} = F(s) G(s)$$

Where $$f \ast g$$ is the convolution operator:

$$f \ast g = \displaystyle \int_0^t f(\tau) g(t-\tau) d\tau$$

The inverse of the theorem is simply:

$$\mathcal {L}^{-1} \{ F(s) G(s) \} = f \ast g$$

Evaluate $$\mathcal {L} \{ e^{-2t} \ast t^4 \}$$.

Evaluate $$\mathcal {L}^{-1} \{ \frac{s}{s^4-9} \}$$.

## Integro-Differential Equations

A special case is when $$g(t) = 0$$ which gives:

$$f \ast 1 = \int_0^t {f(\tau) d\tau}$$

The Lapalce Transformation is:

$$\mathcal{L} \{ \int_0^t {f(\tau) d\tau}\} = \cfrac{F(s)}{s}$$

The inverse Laplace is:

$$\mathcal {L}^{-1} \{ \cfrac{F(s)}{s} \} = \int_0^t {f(\tau) d\tau}$$

This allows us to solve integro-differential equations.

Solve $$f(t) = 2t -4 \int_0^t \sin (\tau) f(t-\tau) d\tau$$.

## Periodic Functions

Consider a piecewise continuous functions with period $$T$$ such as:

$$f(t) = \begin{cases} 0, & 0 \le t \lt \frac{T}{2} \\ 1, & \frac{T}{2} \lt t \le T \end{cases}$$

The Lapalce Transformation is:

$$\mathcal{L} \{f\} = \cfrac{1}{1-e^{-sT}} \displaystyle \int_0^T {e^{-st} f(t)} dt$$

Evaluate $$\mathcal{L} \{f\}$$ for the function above for a period of 4.

## Dirac Function

The Dirac function is a special unit impluse where the "on" time is squeezed to only a moment in time:

$$\delta (t-t_0) = \lim\limits_{a \to \infty} \delta_a (t-t_0)$$

Where the unit impluse is defined as:

$$\delta_a (t-t_0) = \begin{cases} 0, & 0 \le t \lt t_0 - a \\ \frac{1}{2a}, & t_0 - a \le t \lt t_0 + a \\ 0, & t_0 + a \ge t \end{cases}$$

That means the Dirac Function is infinitiy when $$t = t_0$$. It is found in applications with temporary external forces or voltage, for example. The Lapalce Transformation is:

$$\mathcal{L} \{\delta (t-t_0) \} = e^{-st_0}$$

Solve $$y'' + 2y' +2y = \delta(t-\pi)$$ subject to $$y(0) = 1, y'(0)=0$$.