Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 25 December 2018 doi:10.20944/preprints201812.0310.v1 Peer-reviewed version available at Fractal Fract 2019, 3, 1; doi:10.3390/fractalfract3010001 Article Regularized Integral Representations of the Reciprocal G Function Dimiter Prodanov Correspondence: Environment, Health and Safety, IMEC vzw, Kapeldreef 75, 3001 Leuven, Belgium;
[email protected];
[email protected] Version December 25, 2018 submitted to Preprints 1 Abstract: This paper establishes a real integral representation of the reciprocal G function in terms 2 of a regularized hypersingular integral. The equivalence with the usual complex representation is 3 demonstrated. A regularized complex representation along the usual Hankel path is derived. 4 Keywords: gamma function; reciprocal gamma function; integral equation 5 MSC: 33B15; 65D20, 30D10 6 1. Introduction 7 Applications of the Gamma function are ubiquitous in fractional calculus and the special function 8 theory. It has numerous interesting properties summarized in [1]. It is indispensable in the theory of 9 Laplace transforms. The history of the Gamma function is surveyed in [2]. In a previous note I have 10 investigated an approach to regularize derivatives at points, where the ordinary limit diverges [5]. 11 This paper exploits the same approach for the purposes of numerical computation of singular integrals, 12 such as the Euler G integrals for negative arguments. The paper also builds on an observations in [4]. 13 The present paper proves a real singular integral representation of the reciprocal G function. The 14 algorithm is implemented in the Computer Algebra System Maxima for reference and demonstration 15 purposes.