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:

$\laptrans {\map f t} = \dfrac {5 \paren {1 - e^{-3 s} } } s$