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
- 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Solved Problems: Translation and Change of Scale Properties: $10$