Laplace Transform of 1
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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
From Laplace Transform of Derivative:
- $(1): \quad \laptrans {\map {f'} t} = s \laptrans {\map f t} - \map f 0$
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 $\map f 0$ | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds s \laptrans 1\) | \(=\) | \(\ds 1\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \laptrans 1\) | \(=\) | \(\ds \dfrac 1 s\) |
$\blacksquare$
Sources
- 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Laplace Transforms of some Elementary Functions: $1$.
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 32$: Table of Special Laplace Transforms: $32.25$