Laplace’s transform of fractional order via the Mittag-Leffler function and modified Riemann-Liouville derivative. (English) Zbl 1181.44001

The Laplace transform of fractional order \(\alpha\) is determined as \[ \lim_{M\to\infty} \alpha\int^M_0 E_\alpha(-s^\alpha x^\alpha) f(x)(M- x)^{\alpha-1} dx \] with the Mittag-Leffler function \[ E_\alpha(u)= \sum^\infty_{k=0} u^k/\Gamma(\alpha k+ 1). \] Replacing derivatives by fractional derivatives, the usual properties of the Laplace transform and also its inversion formula are transferred to this generalized transform.


44A10 Laplace transform
44A20 Integral transforms of special functions
26A33 Fractional derivatives and integrals
33E12 Mittag-Leffler functions and generalizations
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