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Regularized Integral Representations of the Reciprocal Gamma Function

1
Environment, Health and Safety, IMEC, 3001 Leuven, Belgium
2
Neuroscience Research Flanders, 3001 Leuven, Belgium
Fractal Fract 2019, 3(1), 1; https://doi.org/10.3390/fractalfract3010001
Received: 16 November 2018 / Revised: 29 December 2018 / Accepted: 8 January 2019 / Published: 12 January 2019
(This article belongs to the Special Issue The Craft of Fractional Modelling in Science and Engineering 2018)
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Abstract

This paper establishes a real integral representation of the reciprocal Gamma function in terms of a regularized hypersingular integral along the real line. A regularized complex representation along the Hankel path is derived. The equivalence with the Heine’s complex representation is demonstrated. For both real and complex integrals, the regularized representation can be expressed in terms of the two-parameter Mittag-Leffler function. Reference numerical implementations in the Computer Algebra System Maxima are provided. View Full-Text
Keywords: Gamma function; reciprocal Gamma function; Mittag-Leffler function; integral equation Gamma function; reciprocal Gamma function; Mittag-Leffler function; integral equation
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Prodanov, D. Regularized Integral Representations of the Reciprocal Gamma Function. Fractal Fract 2019, 3, 1.

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