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arXiv · 2103.04284

On the asymptotic of Wright functions of the second kind

Richard Paris·Armando Consiglio·Francesco Mainardi
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Abstract

The asymptotic expansions of the Wright functions of the second kind, introduced by Mainardi [see Appendix F of his book {\it Fractional Calculus and Waves in Linear Viscoelasticity}, (2010)], $$ F_\sigma(x)=\sum_{n=0}^\infty \frac{(-x)^n}{n! \g(-n\sigma)}~,\quad M_\sigma(x)=\sum_{n=0}^\infty \frac{(-x)^n}{n! \g(-n\sigma+1-\sigma)}\quad(0<\sigma<1)$$ for $x\to\pm\infty$ are presented. The situation corresponding to the limit $\sigma\to1^-$ is considered, where $M_\sigma(x)$ approaches the Dirac delta function $\delta(x-1)$. Numerical results are given to demonstrate the accuracy of the expansions derived in the paper, together with graphical illustrations that reveal the transition to a Dirac delta function as $\sigma\to 1^-$.

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Richard Paris, Armando Consiglio, Francesco Mainardi. 2021-03-07. On the asymptotic of Wright functions of the second kind. https://doi.org/10.1515/fca-2021-0003

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