Second Derivative of Laplace Transform

Theorem

Let $f: \R \to \R$ or $\R \to \C$ be a continuous function, twice differentiable on any closed interval $\closedint 0 a$.

Let $\laptrans f = F$ denote the Laplace transform of $f$.

Then, everywhere that $\dfrac {\d^2} {\d s^2} \laptrans f$ exists:

$\dfrac {\d^2} {\d s^2} \laptrans {\map f t} = \laptrans {t^2 \, \map f t}$

$\ds \dfrac {\d^2} {\d s^2} \laptrans {\map f t}$

$=$

$\ds \map {\frac \d {\d s} } {\dfrac \d {\d s} \laptrans {\map f t} }$

Definition of Second Derivative

$\ds $

$=$

$\ds \map {\frac \d {\d s} } {-\laptrans {t \, \map f t} }$

$\ds $

$=$

$\ds -\frac \d {\d s} \laptrans {t \, \map f t}$

$\ds $

$=$

$\ds -\paren {-\laptrans {t \paren {t \, \map f t} } }$

$\ds $

$=$

$\ds \laptrans {t^2 \, \map f t}$

$\blacksquare$

1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Appendix $\text A$: Table of General Properties of Laplace Transforms: $9.$
1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 32$: Table of General Properties of Laplace Transforms: $32.11$
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: $33.11$