Laplace Transform of Constant Mapping/Proof 1

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

\(\ds \laptrans {\map {f_a} t}\)

\(=\)

\(\ds \laptrans a\)

Definition of Constant Mapping

\(\ds \)

\(=\)

\(\ds a \, \laptrans 1\)

Linear Combination of Laplace Transforms

\(\ds \)

\(=\)

\(\ds a \frac 1 s\)

Laplace Transform of 1

\(\ds \)

\(=\)

\(\ds \frac a s\)

$\blacksquare$