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.
Highlights
Applications of the Gamma function in fractional calculus and the special function theory are ubiquitous
The function is indispensable in the theory of Laplace transforms
A classical reference on the Gamma function is given by Artin [2]
Summary
Applications of the Gamma function in fractional calculus and the special function theory are ubiquitous. The reciprocal Gamma function is a normalization constant in all of the classical fractional derivative operators: the Riemann–Liouville, Caputo, and Grünwald–Letnikov. Methods for the fast computation of the reciprocal Gamma function for arbitrary arguments may be beneficial for numerical applications of fractional calculus. This paper exploits the same approach for the purposes of numerical computation of singular integrals, such as the Euler Γ integrals for negative arguments. The present paper proves a real singular integral representation of the reciprocal Γ function. The paper provides an integral representation of the Gamma function for negative numbers related to the Cauchy–Saalschütz integral [6]. The regularization procedure can be expressed in terms of the two-parameter Mittag-Leffler function
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