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Keywords:
Peano derivative; generalized Peano derivative; Laplace derivative; Laplace transform; Tauberian theorem
Peano derivative; generalized Peano derivative; Laplace derivative; Laplace transform; Tauberian theorem
Summary:
A function $f:\Bbb R \rightarrow \Bbb R$ is said to have the $n$-th Laplace derivative on the right at $x$ if $f$ is continuous in a right neighborhood of $x$ and there exist real numbers $\alpha_0, \ldots, \alpha_{n-1}$ such that $s^{n+1}\int_0^\delta e^{-st}[f(x+t)-\sum_{i=0}^{n-1}\alpha_i t^i/i!]\,dt$ converges as $s\rightarrow +\infty$ for some $\delta>0$. There is a corresponding definition on the left. The function is said to have the $n$-th Laplace derivative at $x$ when these two are equal, the common value is denoted by $f_{\langle n\rangle }(x)$. In this work we establish the basic properties of this new derivative and show that, by an example, it is more general than the generalized Peano derivative; hence the Laplace derivative generalizes the Peano and ordinary derivatives.
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