Laplace Transform of Function of t minus a/Examples
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Examples of Use of Laplace Transform of Function of t minus a
Let $\laptrans f$ denote the Laplace transform of the real function $f$.
Example $1$
- $\laptrans {\paren {t - 2}^3} = \dfrac {6 e^{-2 s} } {s^4}$
where $t > 2$.
Example $2$
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:
- $\laptrans {\map f t} = s \exp \dfrac {-2 \pi s} 3 \dfrac 1 {s^2 + 1}$