On Laplace derivative |
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Authors: | S N Mukhopadhyay S Ray |
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Institution: | 1. University Teachers’ co-op. Housing, Krishnapur Road, Burdwan, 713104, West Bengal, India 2. Department of Mathematics, Siksha Bhavan, Visva-Bharati, Santiniketan, 731235, West Bengal, India
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Abstract: | If f: ? → ? is integrable in a right neighbourhood of x ∈ ? and if there are real numbers α 0, α 1, ..., α n?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 LD n + f(x). There is a corresponding definition for the left-hand derivative and if they are equal the common value is the Laplace derivative LD n f(x). In this paper, it is shown that the basic properties of the Peano derivatives are also possessed by this derivative (cf. 5]). |
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