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Regularized Integral Representations of the Reciprocal Gamma Function

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Regularized Integral Representations of the Reciprocal Gamma Function fractal and fractional Article Regularized Integral Representations of the Reciprocal Gamma Function Dimiter Prodanov 1,2 1 Environment, Health and Safety, IMEC, 3001 Leuven, Belgium; [email protected] or [email protected] 2 Neuroscience Research Flanders, 3001 Leuven, Belgium Received: 16 November 2018; Accepted: 8 January 2019; Published: 12 January 2019 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. Keywords: Gamma function; reciprocal Gamma function; Mittag-Leffler function; integral equation MSC: 33B15; 65D20; 30D10 1. Introduction Applications of the Gamma function in fractional calculus and the special function theory are ubiquitous. For example, the function is indispensable in the theory of Laplace transforms. The history of the Gamma function is surveyed in [1], in which the author further states that “of the so-called ’higher mathematical functions’, the Gamma function is undoubtedly the most fundamental”. A classical reference on the Gamma function is given by Artin [2]. The Gamma function has numerous remarkable properties that are surveyed in [3]. For example, one of its classical applications is the formula for the volume of an n-dimensional ball. The reciprocal Gamma function is a normalization constant in all of the classical fractional derivative operators: the Riemann–Liouville, Caputo, and Grünwald–Letnikov. The reciprocal Gamma function is also prominent in the analytic number theory and its various connections to other transcendental functions (for example, the Riemann zeta function). Therefore, methods for the fast computation of the reciprocal Gamma function for arbitrary arguments may be beneficial for numerical applications of fractional calculus. In the special function theory, the Gamma function is used explicitly in the definitions of the Fox type and Fox–Wright type of special functions, which are related to actions of fractional differential operators [4]. In a previous work, I 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 G integrals for negative arguments. The present paper proves a real singular integral representation of the reciprocal G function. The algorithm was implemented in the computer algebra system Maxima for reference and demonstration purposes. As a second application, the paper provides an integral representation of the Gamma function for negative numbers related to the Cauchy–Saalschütz integral [6]. The algorithm was also implemented in Maxima. Finally, the paper demonstrates an equivalent regularized complex representation based Fractal Fract. 2019, 3, 1; doi:10.3390/fractalfract3010001 www.mdpi.com/journal/fractalfract Fractal Fract. 2019, 3, 1 2 of 11 on the regularization of the Heine integral. The regularization procedure can be expressed in terms of the two-parameter Mittag-Leffler function. 2. 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: 1 z (z + 1) ... (z + n) := lim G(z) n!¥ nz n! Proceeding from the Euler’s reflection formula for negative arguments, the reciprocal Gamma function is simply 1 sin pz = − G(z + 1) (1) G(−z) p The plot of the above function is presented in Figure1. Figure 1. 1/G(−z) computed from Equation (1). 2.1. Real Representations The Euler’s Gamma function integral representation is valid for real z > 0 or complex numbers such that Re z > 0 Z ¥ G(z) = e−ttz−1 dt 0 However, for negative z, the integral diverges. A less well-known integral representation for Re z < 0 is the Cauchy–Saalschütz integral [7] (Chapter 3), also along the real line: Z ¥ e−t − e (−t) (− ) = n G z z+ dt 0 t 1 where en() is given by Definition3. Fractal Fract. 2019, 3, 1 3 of 11 2.2. Complex Representations Hankel’s representation of the Gamma function is given as Z 1 − G(z) = ettz 1dt, z 2/ Z 2i sin (pz) Ha− Here, Ha− denotes the Hankel contour in the complex z-plane with a cut along the negative real semi-axis arg z = p and circulation in the positive direction. The contour is depicted in Figure2. y A B x E D C Arrows indicate the direction of propagation along the contour. Figure 2. The Hankel contour Ha−. The Heine’s complex representation of the reciprocal Gamma function is well known and is given below [6] (p. 161): 1 (−1)−z Z e−t 1 Z et = = z dt z dt G(z) 2pi Ha+ t 2pi Ha− t Ha+ is the reflection of Ha− across the origin. The integrand has a simple pole at t = 0. The Hölder exponent at 0 can be computed in the closed interval [0, e] as log eee−z e lim = −z + lim = −z e!0 log e e!0 log e Therefore, for k > [z], it holds that lim ekeee−z = 0 and the order of the residue is the integer part e!0 [z]. This observation is indicative of the statement of the main result of the paper. 2.3. Auxiliary Notation Definition 1. For a real number z, the notation [z] will mean the integral part of the number, while fzg := z − [z] will denote the non-integral remaining part. Definition 2. The falling factorial is defined as n−1 (z)n := ∏ (z − k) k=0 Fractal Fract. 2019, 3, 1 4 of 11 Definition 3. Let n xk en(x) := ∑ k=0 k! be the truncated Taylor polynomial under the convention e−1(x) = 0. 3. Theoretical Results Theorem 1 (Real Reciprocal Gamma Representation). Let z > 0, z 2/ Z and n = [z]. Then 1 sin pz Z ¥ e−x − e (−x) 1 Z 0 ex − e (x) = n−1 = n−1 z dx Im z dx G(z) p 0 x p −¥ x 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: ex − e (x) 1 ex − 1 = n = Lz lim z lim z−n x!0 x (z)n x!0 x Another application of l’Hôpital’s rule leads to 1 x n+1−z Lz = lim e x (z)n (z − n) x!0 Therefore, 8 0, z < n + 1 <> 1 = + Lz = G(n+1) , z n 1 > : ¥, z > n + 1 Secondly, consider the limit x x n k−z e − en(x) e x Mz = lim = Mz = lim − lim x!−¥ z x!−¥ z ∑ x!−¥ ( + ) x x k=0 G k 1 Therefore, 8 ¥, z < n <> 1 = Mz = G(n+1) , z n > : 0, z > n Therefore, in order for both limits to vanish simultaneously, n < z < n + 1. Therefore, n = [z]. Let fzg = z − [z]. In the following, we take xz as its principal value. Then, x− ( ) x− ( ) ¥ ( x− ( ))0 I (z + ) = R 0 e en x dx = e en x + 1 R 0 e en x dx n+1 1 −¥ xz+1 zxz z −¥ xz x 0 1 R 0 e −en−1(x) 1 = z −¥ xz dx = z In(z) by the above results. Therefore, by reduction, 1 1 1 R 0 ex In+ (z + 1) = I0(z − n) = I0(fzg) = dx 1 (z)n (z)n (z)n −¥ xfzg −x ( −f g) = 1 R ¥ e dx = e−ipfzg G 1 z (z)n 0 (−x)fzg (z)n Fractal Fract. 2019, 3, 1 5 of 11 Therefore, e−ipfzg G(fzg)I + (z + 1) = G(1 − fzg)G(fzg) n 1 (z)n = e−ipfzg p = p (cot pfzg − i) (z)n sin pfzg (z)n by Euler’s reflection formula. We take the imaginary part of the integral, since G(fzg) is real and the middle expression is imaginary. Therefore, 1 1 1 = = Im In+1(z + 1) G(z + 1) (z)nG(fzg) p Therefore, 1 1 Z 0 ex − e (x) = n−1 Im z dx G(z) p −¥ x Finally, by change of variables x 7! −x, 1 1 Z 0 e−x − e (−x) sin pz Z ¥ e−x − e (−x) = − n−1 = n−1 Im z dx z dx G(z) p ¥ (−x) p 0 x Corollary 1. By change of variables, it holds that k ! Z ¥ 1 n−1 k 1 sin pz 1 − −u z (−1) u z = u z 2 e − du ( ) ∑ G z pz 0 k=0 k! The latter result can be useful for computations with large arguments of G. Corollary 2 (Modified Euler Integral of the Second Kind). By change of variables, it holds that, for z > 0, ! 1 sin pz Z 1 1 n−1 (log u)k sin pz Z 1 1 − e (log u) /u = u − du = n−1 du ( ) z ∑ z G z p 0 u (log u) k=0 k! p 0 (log 1/u) Finally, it is instructive to demonstrate the correspondence between the complex-analytical representation and the hypersingular representation. Theorem 2 (Regularized Complex Reciprocal Gamma Representation). For Re z > 0, z 2/ Z and n = [Re z], 1 1 Z et − e (t) = n−1 z dt G(z) 2pi Ha− t where t 2 C . Proof. The proof technique follows [8]. We evaluate the line integral along the Hankel contour: 1 Z et − e (t) ( ) = n−1 In z z dt 2pi Ha− t with kernel t e − e − (t) Ker(t) = n 1 , Re z > 0 tz Fractal Fract.
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