# Laplace Transform of Constant Mapping

## 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:

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