On the asymptotics of wright functions of the second kind

Richard B. Paris; Armando Consiglio; Francesco Mainardi

doi:10.1515/fca-2021-0003

On the asymptotics of wright functions of the second kind

Richard B. Paris, Armando Consiglio, Francesco Mainardi^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

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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 Fractional Calculus and Waves in Linear Viscoelasticity (2010)],

Fσ(x)=◀∑▶,Mσ(x)=∞∑n=0(−x)n/n!Γ(−nσ+1−σ) (0<σ<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⁻.

Original language	English
Pages (from-to)	54-72
Number of pages	19
Journal	Fractional Calculus and Applied Analysis
Volume	24
Issue number	1
Early online date	29 Jan 2021
DOIs	https://doi.org/10.1515/fca-2021-0003
Publication status	Published - 23 Feb 2021

Keywords

Wright function
Auxiliary Wright function
Asymptotic expansions
Exponentially small expansions

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AU - Consiglio, Armando

AU - Mainardi, Francesco

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N2 - 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)=◀∑▶,Mσ(x)=∞∑n=0(−x)n/n!Γ(−nσ+1−σ) (0<σ<1) for x → ± ∞ are presented. The situation corresponding to the limit σ → 1− is considered, where Mσ(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−.

AB - 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)=◀∑▶,Mσ(x)=∞∑n=0(−x)n/n!Γ(−nσ+1−σ) (0<σ<1) for x → ± ∞ are presented. The situation corresponding to the limit σ → 1− is considered, where Mσ(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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M3 - Article

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JO - Fractional Calculus and Applied Analysis

JF - Fractional Calculus and Applied Analysis

SN - 1311-0454

IS - 1

ER -

On the asymptotics of wright functions of the second kind

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