Asymptotics of the Mittag-Leffler function Ea(z) on the negative real axis when a→1

Richard Paris

doi:10.1007/s13540-022-00031-5

Asymptotics of the Mittag-Leffler function E_a(z) on the negative real axis when a→1

Richard Paris^*

^*Corresponding author for this work

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

Abstract

We consider the asymptotic expansion of the one-parameter Mittag-Leffler functionE_a(−x) for x → +∞ as the parameter a → 1. The dominant expansion when0 < a < 1 consists of an algebraic expansion of O(x⁻¹) (which vanishes whena = 1), together with an exponentially small contribution that approaches e^−x as a → 1. Here we concentrate on the form of this exponentially small expansion whena approaches the value 1. Numerical examples are presented to illustrate the accuracyof the expansion so obtained.

Original language	English
Pages (from-to)	735-746
Number of pages	12
Journal	Fractional Calculus and Applied Analysis
Volume	25
Issue number	2
Early online date	20 Apr 2022
DOIs	https://doi.org/10.1007/s13540-022-00031-5
Publication status	Published - 20 Apr 2022

Keywords

Mittag-Leffler function
Asymptotic expansion
Exponentially small expansion
Stokes lines

Access to Document

10.1007/s13540-022-00031-5

Cite this

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abstract = "We consider the asymptotic expansion of the one-parameter Mittag-Leffler functionEa(−x) for x → +∞ as the parameter a → 1. The dominant expansion when0 < a < 1 consists of an algebraic expansion of O(x−1) (which vanishes whena = 1), together with an exponentially small contribution that approaches e−x as a → 1. Here we concentrate on the form of this exponentially small expansion whena approaches the value 1. Numerical examples are presented to illustrate the accuracyof the expansion so obtained.",

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N2 - We consider the asymptotic expansion of the one-parameter Mittag-Leffler functionEa(−x) for x → +∞ as the parameter a → 1. The dominant expansion when0 < a < 1 consists of an algebraic expansion of O(x−1) (which vanishes whena = 1), together with an exponentially small contribution that approaches e−x as a → 1. Here we concentrate on the form of this exponentially small expansion whena approaches the value 1. Numerical examples are presented to illustrate the accuracyof the expansion so obtained.

AB - We consider the asymptotic expansion of the one-parameter Mittag-Leffler functionEa(−x) for x → +∞ as the parameter a → 1. The dominant expansion when0 < a < 1 consists of an algebraic expansion of O(x−1) (which vanishes whena = 1), together with an exponentially small contribution that approaches e−x as a → 1. Here we concentrate on the form of this exponentially small expansion whena approaches the value 1. Numerical examples are presented to illustrate the accuracyof the expansion so obtained.

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DO - 10.1007/s13540-022-00031-5

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JO - Fractional Calculus and Applied Analysis

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