Regularized integral representations of the reciprocal Gamma function

This paper establishes a real integral representation of the reciprocal $\Gamma$ function in terms of a regularized hypersingular integral. The equivalence with the usual complex representation is demonstrated. A regularized complex representation along the usual Hankel path is derived.


Introduction
Applications of the Gamma function are ubiquitous in fractional calculus and the special function theory. It has numerous interesting properties summarized in [1]. The history of the Gamma function is surveyed in [2]. In a previous note I have investigated an approach to regularize derivatives at points, where the ordinary limit diverges [5]. 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 paper also builds on an observations in [4].
The present paper proves a real singular integral representation of the reciprocal Γ function. The algorithm is implemented in the Computer Algebra System Maxima for reference and demonstration purposes.
As a second contribution, the paper provides an integral representation of the Gamma function for negative numbers. Finally, the paper demonstrates an equivalent regularized complex representation based on the regularization of the Heine integral.

Preliminaries and notation
The reciprocal Gamma function is an entire function. Starting from the Euler's infinite product definition, the reciprocal Gamma function can be defined by the infinite product: Proceeding from the Euler's reflection formula for negative arguments the reciprocal Gamma function is simply It is plot is presented in Fig. 1. The Heines complex representation for the reciprocal Gamma function is well known and is given below: Here Ha − denotes the Hankel contour in the complex ζ-plane with a cut along the negative real semi-axis arg ζ = π and Ha + is its reflection. The contour is depicted in Fig. 2. The integrand has a simple pole at τ = 0. The Hölder exponent at 0 can be computed in the closed interval [0, ] as

Theoretical Results
Theorem 1 (Real Reciprocal Gamma representation). Let z > 0, z / ∈ Z and n = [z]. Then where the integrals are over the real axis.
Proof. First, we establish two preliminary results. Consider the following limit of the form 0/0 and apply n times l'Hôpital s'rule: Another application of l'Hôpital s'rule leads to Therefore, Secondly, consider the limit z > n Therefore, in order for both limits to vanish simultaneously n < z < n + 1. Therefore, n = [z]. Let {z} = z − [z]. Then by the above results. Therefore, by reduction by the Euler's reflection formula. We take the imaginary part of the integral since Γ({z}) is real and the middle expression is imaginary. Therefore, 1 Γ(z + 1) = 1 (z) n Γ({z}) = Im 1 π I n+1 (z + 1)

Corollary 1. By change of variables it holds that
The latter result can be useful for computations with large arguments of Γ. Finally, it is instructive to demonstrate the correspondence between the complexanalytical representation and the hyper-singular representation.
Theorem 2 (Regularized complex reciprocal Gamma representation). For z > 0, z / ∈ Z and n = [z] Proof. The proof technique follows [3]. We evaluate the line integral along the Hankel contour: The contour is depicted in Fig. 2. The integral can be split in three parts sin (δk) r k k! cos (δz) Therefore, The integral on the arc BCD is given by the Cauchy Residue Theorem. By the above observation the residue at τ = 0 is given by the limit Furthermore, after integration by parts since M z = 0. Therefore, the claim follows by reduction to n = 0.
3.1. Applications. As a concrete application of Th. 2 consider the Laplace transform pair The inverse Laplace transform can be calculated simply as as On a second place, the ratio of two Gamma functions can be represented as Proof.
Finally, for negative arguments: Proof. By the reflection formula Therefore, The latter result is equivalent to the classical CauchySaalschütz integral representation. Indeed, by integration by parts which is the CauchySaalschütz integral.

Numerical Implementation
A reference implementation in the Computer Algebra System Maxima is given in Listings 1 and 2. The numerical integration code uses the library Quadpack, distributed with Maxima. The reference implementation given in this section uses a routine for semi-infinite interval integration with tunable relative error of approximation (i.e. the epsrel parameter). A plot of the reciprocal Γ function computed from Listing 1 is presented in Fig. 3. Fig. 4 represents a plot of Γ(−z) computed