Integral of Laplace Transform
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Theorem
Let $f: \R \to \R$ or $\R \to \C$ be a continuous function on any interval of the form $0 \le t \le A$.
Let $\laptrans f = F$ denote the Laplace transform of $f$.
Then:
- $\ds \laptrans {\dfrac {\map f t} t} = \int_s^{\to \infty} \map F u \rd u$
wherever $\ds \lim_{t \mathop \to 0} \dfrac {\map f t} t$ and $\laptrans f$ exist.
Proof
Let $\map g t := \dfrac {\map f t} t$.
Then:
\(\ds \map f t\) | \(=\) | \(\ds t \map g t\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \laptrans {\map f t}\) | \(=\) | \(\ds \laptrans {t \map g t}\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \laptrans {\map f t}\) | \(=\) | \(\ds -\dfrac \d {\d s} \laptrans {\map g t}\) | Derivative of Laplace Transform | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map F s\) | \(=\) | \(\ds -\dfrac {\d G} {\d s}\) | $\map F s := \laptrans {\map f t}$, and so on | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map G s\) | \(=\) | \(\ds -\int_{-\infty}^s \map f u \rd u\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_s^\infty \map f u \rd u\) |
The result follows.
$\blacksquare$
Examples
Example $1$
Let $\sin$ denote the real sine function.
Let $\laptrans f$ denote the Laplace transform of a real function $f$.
Then:
- $\laptrans {\dfrac {\sin t} t} = \arctan \dfrac 1 s$
Also see
Sources
- 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Some Important Properties of Laplace Transforms: $8$. Division by $t$: Theorem $1 \text{-} 13$
- 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Solved Problems: Division by $t$: $21$
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 32$: Table of General Properties of Laplace Transforms: $32.16$