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

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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 1

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

and the result follows.

$\blacksquare$


Proof 2

\(\ds \laptrans {\map f t}\) \(=\) \(\ds \int_0^\infty e^{-s t} \map f t \rd t\) Definition of Laplace Transform
\(\ds \) \(=\) \(\ds \int_0^{\frac {-2 \pi s} 3} e^{-s t} \map f t \rd t + \int_{\frac {-2 \pi s} 3}^\infty e^{-s t} \map f t \rd t\)
\(\ds \) \(=\) \(\ds \int_0^{\frac {-2 \pi s} 3} e^{-s t} \times 0 \rd t + \int_{\frac {-2 \pi s} 3}^\infty e^{-s t} \map \cos {t - \dfrac {2 \pi} 3} \rd t\) Definition of $f$
\(\ds \) \(=\) \(\ds \int_0^\infty e^{-s \paren {u + 2 \pi / 3} } \map \cos u \rd u\) Integration by Substitution: $t = u + \dfrac {2 \pi} 3$
\(\ds \) \(=\) \(\ds \exp \dfrac {-2 \pi s} 3 \int_0^\infty e^{-s u} \map \cos u \rd u\)
\(\ds \) \(=\) \(\ds \exp \dfrac {-2 \pi s} 3 \laptrans {\cos u}\) Definition of Laplace Transform
\(\ds \) \(=\) \(\ds \exp \dfrac {-2 \pi s} 3 \dfrac s {s^2 + 1}\) Laplace Transform of Cosine

and the result follows.

$\blacksquare$


Sources