# Mittag Leffler function Laplace transforms with Mathematica

Mathematica seems not to to know the basic Laplace and inverse Laplace relation

$$\mathcal L(E_\alpha[−λt^α],t)(s)=\frac{s^{α-1}}{λ+s^α}$$

surrounding the Mittag Leffler function (MittagLefflerE). The evaluations

LaplaceTransform[MittagLefflerE[alpha, -lambda t^alpha], t, s]

Integrate[ Exp[-s t] MittagLefflerE[alpha, -lambda t^alpha], {t, 0, Infinity}]

InverseLaplaceTransform[s^(alpha - 1)/(lambda + s^alpha), s, t]

1/(2*Pi*I)* Integrate[ Exp[s t] s^(alpha - 1)/(lambda + s^alpha), {s, -Infinity, Infinity}]


all fail. However, when setting $\alpha=1$ the correct Laplace transform $1/(\lambda +s)$ of the expontential function is recovered.

Has anybody had more success with this kind computations?

• It seems to me that the current symbolic support for general Mittag-Leffler functions is quite limited. – J. M. will be back soon May 12 '17 at 12:31

The failure of LaplaceTransform and other functions to evaluate for MittagLefflerE in your examples is a limitation that we hope to address in a future release.