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

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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.

$\blacksquare$


Sources