On the asymptotic of Wright functions of the second kind

TL;DR

This paper derives asymptotic expansions for Wright functions of the second kind, analyzes their behavior as the parameter approaches one, and demonstrates the transition to a Dirac delta function through numerical and graphical methods.

Contribution

It provides new asymptotic formulas for Wright functions of the second kind and explores their limiting behavior as the parameter approaches one, including numerical validation.

Findings

Asymptotic expansions for $F_\sigma(x)$ and $M_\sigma(x)$ are derived for large $|x|$.

The limit $\sigma o 1^-$ causes $M_\sigma(x)$ to approach the Dirac delta function.

Numerical results confirm the accuracy of the asymptotic formulas and illustrate the transition to a delta function.

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 $\sigma\to 1^-$.