The asymptotic expansions of the Wright functions of the second kind, introduced by Mainardi [see Appendix F of his book Fractional Calculus and Waves in Linear Viscoelasticity (2010)], Fσ(x)=∑n=0∞(−x)nn!Γ(−nσ) , Mσ(x)=∑n=0∞(−x)nn!Γ(−nσ+1−σ) (0<σ<1)$$F_\sigma(x)=\sum\limits_{n=0}^\infty \frac{(-x)^n}{n! {\mathrm{\Gamma}}(-n\sigma)}~,\quad M_\sigma(x)=\sum\limits_{n=0}^\infty \frac{(-x)^n}{n! {\mathrm{\Gamma}}(-n\sigma+1-\sigma)}\quad(0 \lt \sigma \lt 1) $$ for x → ± ∞ are presented. The situation corresponding to the limit σ → 1 − is considered, where M σ ( x ) approaches the Dirac delta function δ ( 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 σ → 1 − .

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关于第二种赖特函数的渐近性

Mainardi引入了第二种Wright函数的渐近展开式（请参见他的书《线性粘弹性的分数微积分和波动》（2010）的附录F），Fσ（x）= ∑n =0∞（-x）nn！ Γ（-nσ），Mσ（x）= ∑n =0∞（-x）nn！Γ（-nσ+1-σ）（0 <σ<1）$$ F_ \ sigma（x）= \ sum \ limit_ {n = 0} ^ \ infty \ frac {（-x）^ n} {n！{\ mathrm {\ Gamma}}（-n \ sigma）}〜，\ quad M_ \ sigma（x）= \ sum \ limits_ {n = 0} ^ \ infty \ frac {（-x）^ n} {n ！给出了{\ mathrm {\ Gamma}}（-n \ sigma + 1- \ sigma）} \ quad（0 \ lt \ sigma \ lt 1）$$ x→±∞。考虑对应于极限σ→1-的情况，其中Mσ（x）接近狄拉克δ函数δ（x-1）。给出了数值结果以证明本文导出的展开的准确性，并给出了图解图，这些图表明了向Diracδ函数的转换为σ→1-。