The Fractional Derivative of the Dirac Delta Function and Additional Results on the Inverse Laplace Transform of Irrational Functions

Motivated from studies on anomalous relaxation and diffusion, we show that the memory function $M\left(t\right)$ of complex materials, that their creep compliance follows a power law, $J\left(t\right)\sim t^q$ with $q\in {\mathbb{R}}^{+}$, is proportional to the fractional derivative of the Dirac delta function, $\frac{{\mathrm{d}}^q\delta (t-0)}{\mathrm{d}{t}^q}$ with $q\in {\mathbb{R}}^{+}$. This leads to the finding that the inverse Laplace transform of $s^q$ for any $q\in {\mathbb{R}}^{+}$ is the fractional derivative of the Dirac delta function, $\frac{{\mathrm{d}}^q\delta (t-0)}{\mathrm{d}{t}^q}$. This result, in association with the convolution theorem, makes possible the calculation of the inverse Laplace transform of $\frac{s^q}{s^{\alpha }\mp \lambda }$ where $\alpha <q\in {\mathbb{R}}^{+}$, which is the fractional derivative of order q of the Rabotnov function $\epsilon$${}_{\alpha -1}(\pm \lambda ,t)=t^{\alpha -1}{E}_{\alpha ,\alpha }(\pm \lambda t^{\alpha })$. The fractional derivative of order $q\in {\mathbb{R}}^{+}$ of the Rabotnov function, $\epsilon$${}_{\alpha -1}(\pm \lambda ,t)$ produces singularities that are extracted with a finite number of fractional derivatives of the Dirac delta function depending on the strength of q in association with the recurrence formula of the two-parameter Mittag–Leffler function.