Abstract
We consider the asymptotic expansion of the generalised exponential integral involving the Mittag-Leffler function introduced recently by Mainardi and Masina [Fract. Calc. Appl. Anal. 21 (2018) 1156–1169]. We extend the definition of this function using the two-parameter Mittag-Leffler function. The expansions of the similarly extended sine and cosine integrals are also discussed. Numerical examples are presented to illustrate the accuracy of each type of expansion obtained.
Highlights
IntroductionThe asymptotic expansion of this function will be obtained for large complex z with the parameters α, β held fixed
The complementary exponential integral Ein(z) is defined by Ein(z) = Z z 1 − e−t t ∞ dt =(−)n−1 zn (z ∈ C) nn! n =1 ∑and is an entire function
The asymptotic expansion of this function will be obtained for large complex z with the parameters α, β held fixed
Summary
The asymptotic expansion of this function will be obtained for large complex z with the parameters α, β held fixed. To determine the asymptotic expansion of Einα,β (z) for large complex z with the parameters α and β held fixed, we shall find it convenient to consider the related function defined by. The asymptotic expansion of Einα,β (z) defined in (4) can be constructed from that of F (χ) with the parameter γ = 1 It is sufficient, for real α, β, to consider 0 ≤ arg z ≤ π, since the expansion when arg z < 0 is given by the conjugate value. This can be seen to agree with (26) after a little rearrangement
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