# Second Derivative of Laplace Transform

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## Contents

## 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:

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

## Proof

\(\displaystyle \dfrac {\d^2} {\d s^2} \laptrans {\map f t}\) | \(=\) | \(\displaystyle \map {\frac \d {\d s} } {\dfrac \d {\d s} \laptrans {\map f t} }\) | Definition of Second Derivative | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \map {\frac \d {\d s} } {-\laptrans {t \, \map f t} }\) | Derivative of Laplace Transform | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle -\frac \d {\d s} \laptrans {t \, \map f t}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle -\paren {-\laptrans {t \paren {t \, \map f t} } }\) | Derivative of Laplace Transform | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \laptrans {t^2 \, \map f t}\) |

$\blacksquare$

## Also see

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 32$: Table of General Properties of Laplace Transforms: $32.11$