Second Derivative of Laplace Transform
Jump to navigation
Jump to search
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
\(\ds \dfrac {\d^2} {\d s^2} \laptrans {\map f t}\) | \(=\) | \(\ds \map {\frac \d {\d s} } {\dfrac \d {\d s} \laptrans {\map f t} }\) | Definition of Second Derivative | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\frac \d {\d s} } {-\laptrans {t \, \map f t} }\) | Derivative of Laplace Transform | |||||||||||
\(\ds \) | \(=\) | \(\ds -\frac \d {\d s} \laptrans {t \, \map f t}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -\paren {-\laptrans {t \paren {t \, \map f t} } }\) | Derivative of Laplace Transform | |||||||||||
\(\ds \) | \(=\) | \(\ds \laptrans {t^2 \, \map f t}\) |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 32$: Table of General Properties of Laplace Transforms: $32.11$