Laplace Transform of Second Derivative

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 $f$ be twice differentiable.

Let $f'$ be continuous and $f''$ piecewise continuous with one-sided limits on said intervals.

Let $f$ and $f'$ be of exponential order.

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

Then $\laptrans {f''}$ exists for $\map \Re s > a$, and:


 * $\laptrans {\map {f''} t} = s^2 \laptrans {\map f t} - s \, \map f 0 - \map {f'} 0$

Also see

 * Laplace Transform of Higher Order Derivatives