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 $\dfrac {\d^2} {\d^2 s} \laptrans f$ exists:


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

Also see

 * Derivative of Laplace Transform
 * Higher Order Derivatives of Laplace Transform