Laplace Transform of Constant Mapping
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Theorem
Let $a \in \R$ be a real constant.
Let $f_a: \R \to \R$ or $\C$ be the constant mapping, defined as:
- $\forall t \in \R: \map {f_a} t = a$
Let $\laptrans {f_a}$ be the Laplace transform of $f_a$.
Then:
- $\laptrans {\map {f_a} t} = \dfrac a s$
for $\map \Re s > a$.
Proof 1
\(\ds \laptrans {\map {f_a} t}\) | \(=\) | \(\ds \laptrans a\) | Definition of Constant Mapping | |||||||||||
\(\ds \) | \(=\) | \(\ds a \, \laptrans 1\) | Linear Combination of Laplace Transforms | |||||||||||
\(\ds \) | \(=\) | \(\ds a \frac 1 s\) | Laplace Transform of 1 | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac a s\) |
$\blacksquare$
Proof 2
\(\ds \laptrans {\map {f_a} t}\) | \(=\) | \(\ds \laptrans a\) | Definition of Constant Mapping | |||||||||||
\(\ds \) | \(=\) | \(\ds a \, \laptrans 1\) | Linear Combination of Laplace Transforms | |||||||||||
\(\ds \) | \(=\) | \(\ds a \int_0^{\to +\infty} e^{-s t} \rd t\) | Definition of Laplace Transform | |||||||||||
\(\ds \) | \(=\) | \(\ds \intlimits {- \frac a s e^{-st} } {t \mathop = 0} {t \mathop \to +\infty}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 0 - \paren {- \frac a s}\) | Complex Exponential Tends to Zero, Exponential of Zero | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac a s\) |
$\blacksquare$
Examples
Example $1$
Let $\map f t$ be the real function defined as:
- $\forall t \in \R: \map f t = \begin {cases} 0 & : t < 0 \\ 5 & : 0 \le t < 3 \\ 0 & : t \ge 3 \end {cases}$
Then the Laplace transform of $f$ is given by:
- $\laptrans {\map f t} = \dfrac {5 \paren {1 - e^{-3 s} } } s$