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Motivated from studies on anomalous diffusion, we show that the memoryfunction $M(t)$ of complex materials, that their creep compliance follows apower law, $J(t)\sim t^q$ with $q\in \mathbb{R}^+$, is the fractionalderivative of the Dirac delta function,$\frac{\mathrm{d}^q\delta(t-0)}{\mathrm{d}t^q}$ with $q\in \mathbb{R}^+$. Thisleads 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 associationwith the convolution theorem, makes possible the calculation of the inverseLaplace transform of $\frac{s^q}{s^{\alpha}\mp\lambda}$ where$\alpha<q\in\mathbb{R}^+$ which is the fractional derivative of order $q$ ofthe Rabotnov function $\varepsilon_{\alpha-1}(\pm\lambda,t)=t^{\alpha-1}E_{\alpha, \alpha}(\pm\lambda t^{\alpha})$. The fractionalderivative of order $q\in \mathbb{R}^+$ of the Rabotnov function,$\varepsilon_{\alpha-1}(\pm\lambda, t)$ produces singularities which areextracted with a finite number of fractional derivatives of the Dirac deltafunction depending on the strength of $q$ in association with the recurrenceformula of the two-parameter Mittag-Leffler function.

The fractional derivative of the Dirac delta function and new results on the inverse Laplace transform of irrational functions