On Laplace derivative

Abstract

If f: ℝ → ℝ is integrable in a right neighbourhood of x ∈ ℝ and if there are real numbers α0, α1, ..., αn−1 such that the limit lim $$ \mathop {\lim }\limits_{s \to \infty } s^{n + 1} \int_0^\delta {e^{ - st} } \left[ {f(x + t) - \sum\limits_{i = 0}^{n - 1} {\frac{{t^i }} {{i!}}\alpha _i } } \right]dt $$ exists, then this limit is called the right-hand Laplace derivative of f at x of order n and is denoted by LDn+f(x). There is a corresponding definition for the left-hand derivative and if they are equal the common value is the Laplace derivative LDnf(x).