Laplace Transform of 1

Theorem

Let $f: \R \to \R$ be the function defined as:

$\forall t \in \R: \map f t = 1$

Then the Laplace transform of $\map f t$ is given by:

$\laptrans {\map f t} = \dfrac 1 s$

for $\map \Re s > 0$.

Proof 1

 $\ds \laptrans {\map f t}$ $=$ $\ds \laptrans 1$ Definition of $\map f t$ $\ds$ $=$ $\ds \int_0^{\to +\infty} e^{-s t} \rd t$ Definition of Laplace Transform $\ds$ $=$ $\ds \lim_{L \mathop \to \infty} \int_0^L e^{-s t} \rd t$ Definition of Improper Integral $\ds$ $=$ $\ds \lim_{L \mathop \to \infty} \intlimits {-\frac 1 s e^{-s t} } 0 L$ Primitive of Exponential Function $\ds$ $=$ $\ds \lim_{L \mathop \to \infty} \paren {-\frac 1 s e^{-s L} - \paren {-\frac 1 s} }$ Exponential of Zero $\ds$ $=$ $\ds \lim_{L \mathop \to \infty} \paren {\frac {1 - e^{-s L} } s}$ simplification $\ds$ $=$ $\ds \frac 1 s$ Complex Exponential Tends to Zero

$\blacksquare$

Proof 2

$(1): \quad \laptrans {\map {f'} t} = s \laptrans {\map f t} - \map f 0$

under suitable conditions.

Then:

 $\ds \map f t$ $=$ $\ds 1$ $\ds \leadsto \ \$ $\ds \map {f'} t$ $=$ $\ds 0$ $\ds \map f 0$ $=$ $\ds 1$ $\ds \leadsto \ \$ $\ds \laptrans 0$ $=$ $\ds 0$ $\ds$ $=$ $\ds s \laptrans 1 - 1$ from $(1)$, substituting for $\map f t$ and $\map f 0$ $\ds \leadsto \ \$ $\ds s \laptrans 1$ $=$ $\ds 1$ $\ds \leadsto \ \$ $\ds \laptrans 1$ $=$ $\ds \dfrac 1 s$

$\blacksquare$