# Laplace Transform of Function of t minus a/Examples/Example 2/Proof 1

## Example of Use of Laplace Transform of Function of t minus a

Let $\laptrans f$ denote the Laplace transform of the real function $f$.

Let $f: \R \to \R$ be the function defined as:

$\forall t \in \R: \map f t = \begin {cases} \map \cos {t - \dfrac {2 \pi} 3} & : t \ge \dfrac {2 \pi} 3 \\ 0 & : t < \dfrac {2 \pi} 3 \end {cases}$

Then:

$\laptrans {\map f t} = s \exp \dfrac {-2 \pi s} 3 \dfrac 1 {s^2 + 1}$

## Proof

 $\displaystyle \laptrans {\map f t}$ $=$ $\displaystyle \exp \dfrac {-2 \pi s} 3 \laptrans {\cos t}$ Laplace Transform of Function of t minus a $\displaystyle$ $=$ $\displaystyle \exp \dfrac {-2 \pi s} 3 \dfrac s {s^2 + 1}$ Laplace Transform of Cosine

and the result follows.

$\blacksquare$