Abstract
In this paper, we introduce a new sort of fractional derivative. For this, we consider the Cauchy's integral formula for derivatives and modify it by using Laplace transform. So, we obtain the fractional derivative formula F(α)(s) = L{(–1)(α)L–1{F(s)}}. Also, we find a relation between Weyl's fractional derivative and the formula above. Finally, we give some examples for fractional derivative of some elementary functions.
Highlights
Cauchy's integral formula for derivatives is given by the following relation F n s = n! 2 i CF w dw w s n 1, s int C, nIt calculates the derivative of order n of an analytic function when n is a nonnegative integer.it seems to calculate the derivative of fractional order when we write > 0 instead of n : F s =2 i C w s 1, s int C . (1)
Raina and Koul [1] proved in 1979 that the Laplace transform of the function x f x is equal to th derivative, in the sense of Weyl, of the Laplace transform of f. This means that the fractional derivative of a function F with the inverse Laplace transform can be calculated by the following formula sW F s = L x L 1 F
We move the contour integral in the formula (1) to an infinite vertical line, and we prove the relation (2)
Summary
We introduce a new sort of fractional derivative. We consider the Cauchy's integral formula for derivatives and modify it by using Laplace transform. We obtain the fractional derivative formula F(α)(s) = L{(– 1)(α)L–1{F(s)}}. We find a relation between Weyl's fractional derivative and the formula above. We give some examples for fractional derivative of some elementary functions.
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