Integral of Laplace Transform

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.

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 \ \ $

$\ds \laptrans {\map f t}$

$=$

$\ds -\dfrac \d {\d s} \laptrans {\map g t}$

$\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.

$\blacksquare$

Let $\sin$ denote the real sine function.

Let $\laptrans f$ denote the Laplace transform of a real function $f$.

Then:

$\laptrans {\dfrac {\sin t} t} = \arctan \dfrac 1 s$

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 $t$: $21$
1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Appendix $\text A$: Table of General Properties of Laplace Transforms: $14.$
1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 32$: Table of General Properties of Laplace Transforms: $32.16$
2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 33$: Laplace Transforms: Table of General Properties of Laplace Transforms: $33.16$