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 Γ integrals for negative arguments. The present paper proves a real singular integral representation of the reciprocal Γ 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 Γ(z) n→∞ nz n!

Proceeding from the Euler’s reﬂection formula for negative arguments, the reciprocal Gamma function is simply 1 sin πz = − Γ(z + 1) (1) Γ(−z) π The plot of the above function is presented in Figure1.

Figure 1. 1/Γ(−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 ∞ Γ(z) = e−ττz−1 dτ 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−τ − e (−τ) (− ) = n Γ z z+ dτ 0 τ 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 − Γ(z) = eττz 1dτ, z ∈/ Z 2i sin (πz) Ha−

Here, Ha− denotes the Hankel contour in the complex ζ-plane with a cut along the negative real semi-axis arg ζ = π and circulation in the positive direction. The contour is depicted in Figure2.

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−τ 1 Z eτ = = z dτ z dτ Γ(z) 2πi Ha+ τ 2πi Ha− τ Ha+ is the reﬂection of Ha− across the origin. The integrand has a simple pole at τ = 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 {z} := 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 ∈/ Z and n = [z]. Then

1 sin πz Z ∞ e−x − e (−x) 1 Z 0 ex − e (x) = n−1 = n−1 z dx Im z dx Γ(z) π 0 x π −∞ 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,  0, z < n + 1  1 = + Lz = Γ(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 Γ k 1

Therefore,  ∞, z < n  1 = Mz = Γ(n+1) , z n   0, z > n Therefore, in order for both limits to vanish simultaneously, n < z < n + 1. Therefore, n = [z]. Let {z} = 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({z}) = dx 1 (z)n (z)n (z)n −∞ x{z} −x ( −{ }) = 1 R ∞ e dx = e−iπ{z} Γ 1 z (z)n 0 (−x){z} (z)n Fractal Fract. 2019, 3, 1 5 of 11

Therefore,

e−iπ{z} Γ({z})I + (z + 1) = Γ(1 − {z})Γ({z}) n 1 (z)n = e−iπ{z} π = π (cot π{z} − i) (z)n sin π{z} (z)n by 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 1 1 = = Im In+1(z + 1) Γ(z + 1) (z)nΓ({z}) π

Therefore, 1 1 Z 0 ex − e (x) = n−1 Im z dx Γ(z) π −∞ x Finally, by change of variables x 7→ −x,

1 1 Z 0 e−x − e (−x) sin πz Z ∞ e−x − e (−x) = − n−1 = n−1 Im z dx z dx Γ(z) π ∞ (−x) π 0 x

Corollary 1. By change of variables, it holds that

k ! Z ∞ 1 n−1 k 1 sin πz 1 − −u z (−1) u z = u z 2 e − du ( ) ∑ Γ z πz 0 k=0 k!

The latter result can be useful for computations with large arguments of Γ.

Corollary 2 (Modiﬁed Euler Integral of the Second Kind). By change of variables, it holds that, for z > 0, ! 1 sin πz Z 1 1 n−1 (log u)k sin πz Z 1 1 − e (log u) /u = u − du = n−1 du ( ) z ∑ z Γ z π 0 u (log u) k=0 k! π 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 ∈/ Z and n = [Re z], 1 1 Z eτ − e (τ) = n−1 z dτ Γ(z) 2πi Ha− τ where τ ∈ C .

Proof. The proof technique follows [8]. We evaluate the line integral along the Hankel contour:

1 Z eτ − e (τ) ( ) = n−1 In z z dτ 2πi Ha− τ with kernel τ e − e − (τ) Ker(τ) = n 1 , Re z > 0 τz Fractal Fract. 2019, 3, 1 6 of 11

The contour is depicted in Figure2. The integral can be split into three parts, Z Z Z Z Ker(τ)dτ = Ker(τ)dτ + Ker(τ)dτ + Ker(τ)dτ Ha AB BCD DE along the rays AB, DE, and the arc BCD, respectively. Along the ray AB, where τ = reiδ, the kernel becomes n−1 iδk−iδz k ! 1 iδ e r = e −iδzr − KerA z e ∑ r k=0 k! Along the ray DE, where ξ = re−iδ, the kernel becomes

n−1 −iδk+iδz k ! 1 −iδ e r = e +iδzr − KerB z e ∑ r k=0 k!

Therefore, changing the direction of DE, the rays can be added as

iπδ −iπδ Kδ = KerAe − KerBe −i − −i k k i k k = e δ − iδz+e iδr + n−1 e δ r iδz − 2iδ−iδz n−1 e δ r + eiδr+2iδ rz e ∑k=0 k! e e ∑k=0 k! e

= For δ π ! 2i sin (πz) n−1 (−r)k = −r − Kπ z e ∑ r k=0 k! Therefore, 1 Z ∞ sin(πz) Z ∞ e−r − e (−r) = n−1 lim Kπ dr z dr δ→π 2πi 0 π 0 r by Azrelá’s theorem. The integral on the (closure of the) arc BCD is given by the Cauchy Residue Theorem. By the above observation, the residue at τ = 0 is given by the limit

1−z τ Res[Ker](τ) = lim τ (e − en−1(τ)) = lim τLz = 0, z ≤ n + 1 τ→0 τ→0 since Lz = 0. Therefore, in the limit where the arc closes to a circle, I Ker(τ)dτ = 0 BCD

Furthermore, after integration by parts,

τ Z τ e − en−1(τ) 1 e − en−2(τ) 1 In(z) = − + dτ = I − (z) z−1 − z−1 n 1 τ Ha− 2πi(z − 1) Ha τ z − 1 since Mz = 0. Therefore, the claim follows by reduction to n = 0.

Remark 1. These results demonstrate that the regularized complex contour integral can be collapsed to an integral along the real line. The same principle can be used for the regularization of other nonlinear functions; for example, the Beta function or the Wright function [8].

It is interesting to note a connection to the Mittag-Lefﬂer function.

Definition 4 (Mittag-Leffler function). The two-parameter Mittag-Leffler function [9] under the present convention is denoted by ∞ zk E (z) := a,b ∑ ( + ) k=0 Γ ak b Fractal Fract. 2019, 3, 1 7 of 11

Notably, the regularizing kernel can be expressed in terms of the two-parameter Mittag-Lefﬂer function of integer arguments. The basis for the computation is the following auxiliary result:

Proposition 1. x n e − en−1(x) = x E1,n+1(x)

Proof. By direct computation,

∞ xk ∞ xj+n ex − e (x) = = = xnE (x) n−1 ∑ ( + ) ∑ ( + + ) 1,n+1 k=n Γ k 1 j=0 Γ j n 1

Therefore, it directly follows that

Corollary 3. For Re z > 0, z ∈/ Z and n = [Re z],

Z Z ∞ 1 1 n−z sin πz n−z = τ E1,n+1(τ) dτ = x E1,n+1(−x) dx Γ(z) 2πi Ha− π 0 where τ ∈ C and x ∈ R in the last integral.

In a similar way, for an integer argument n, we have

Corollary 4. For n ∈ N

I τ I 1 1 1 e − en−1(τ) 1 E1,n+1(τ) = = dτ = dτ, τ ∈ C n! Γ(n + 1) 2πi τn+1 2πi τ where the circulation of the contour is counterclockwise around the origin.

Proof. The computation follows from the Cauchy formula for derivatives:

k! I f (τ) f (k)(a) = dτ k+1 2πi Ca (τ − a)

t Therefore, for a = 0 and substitution f (t) = e − en−1(t),

f (n)(0) 1 1 I f (τ) 1 I eτ − e (τ) = = = n−1 n+1 dτ n+1 dτ n! n! 2πi C0 τ 2πi τ

iφ For n = 1, let τ = eeiφ. The integrand has the form Q = ieee . Therefore, in limit Z π iφ Z π lim ieee dφ = idφ = 2πi e→0 −π −π Finally, we observe that eτ − e (τ) E (τ) n−1 = 1,n+1 τn+1 τ Fractal Fract. 2019, 3, 1 8 of 11

4. Applications

4.1. Laplace Transform Pairs

Consider the Laplace transform Ls : f (t) ÷ F(s). As a concrete application of Theorem2, consider the pair

k Γ(k + 1) Ls : t ÷ , k > 0 sk+1 The inverse Laplace transform can be calculated simply as

ts ts −1 Γ(k+1) 1 R e −e[k](ts) sin π(k+1) R 0 e −e[k](ts) k L : ÷ − Γ(k + 1) ds = Γ(k + 1) ds = t t sk+1 2πi Ha sk+1 πi −∞ sk+1 by change of variables. The latter result can be used for numerical inversion of Laplace transforms.

4.2. Ratios of Gamma Functions Secondly, the ratio of two Gamma functions can be represented as

Proposition 2. Let A, B > 0. Then,

Γ(A) 1 Z 1 Z 1 1 − e (log u) /u = n−1 (− )A−1 B log v du dv Γ(B) π 0 0 (log u) where n = [B].

Proof. ( + ) − ( ) Γ A 1 = 1 R 1 1 en−1 log u /u du R 1 (− log v)Adv Γ(B) π 0 (log u)B 0 − ( ) = 1 R 1 R 1 1 en−1 log u /u (− log v)Adu dv π 0 0 (log u)B

4.3. The Cauchy–Saalschütz Integral Finally, for negative arguments:

Proposition 3. For z > 0, it holds that

1 Z ∞ e−x − e (−x) (− ) = − n−1 Γ z z dx z 0 x

Proof. By the reﬂection formula, π Γ(1 − z)Γ(z) = = −zΓ(−z)Γ(z) sin πz Therefore, π 1 1 Z ∞ e−x − e (−x) (− ) = − = − n−1 Γ z z dx z sin πz Γ(z) z 0 x Fractal Fract. 2019, 3, 1 9 of 11

Remark 2. The latter result is equivalent to the classical Cauchy–Saalschütz integral representation [6] (p. 157), [7] (Chapter 3, p. 4). Indeed, by integration by parts,

−x −x 1 R ∞ e −en−1(−x) 1 R ∞ d(e −en(−x)) Γ(−z) = − z 0 xz dx = z 0 xz −x ∞ −x −x e −en(−x) −z R ∞ e −en(−x) R ∞ e −en(−x) = z − z+1 dx = z+1 dx x 0 z 0 x 0 x which is the Cauchy–Saalschütz integral.

4.4. The Grünwald–Letnikov Derivative The Grünwald–Letnikov derivative is deﬁned [10] as

x−a [ h ] q 1 m Γ(q + 1) x − a D∓ f (x) := lim (±1) f (x ± mh), h = , q > 0 → q ∑ ( + ) ( − + ) h 0 h m=0 Γ m 1 Γ q m 1 N

For numerical applications, the limit is approximated as the ﬁnite difference for some large N ∈ N . Therefore, Theorem1 can be directly applied to compute the approximation for small m. For large m, on the other hand, the Stirling asymptotic formula can be used. Pursuing such application, however, goes beyond the scope of the present paper.

5. Numerical Implementation A reference implementation in the computer algebra system Maxima is given in Listings1 and2. 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 a tunable relative error of approximation (i.e., the epsrel parameter). A plot of the reciprocal Γ function computed from Listing1 is presented in Figure3.

Figure 3. 1/Γ(z) computed from Theorem1.

Figure4 represents a plot of Γ(−z) computed from Listing1. Fractal Fract. 2019, 3, 1 10 of 11

Figure 4. Γ(−z) computed from Proposition3.

Listing 1: The Maxima code corresponding to Theorem1.

1 Kg( z ) : =block([ u, ret, k, n, fr:1, ss:0], i f not numberp(z) then return(’Kg(z)) , i f integerp(z) then if z=0 then return(0) else return(1/(z − 1)!) e l s e( 6 f r : sin (%pi ∗z)/%pi , n : f i x ( z ) , ss : sum( (−u)^k/k!,k,0, n−1) , r e t : f r ∗ first(quad_qagi ( (exp(−u)− ss)/ u^(z) , u, 0, inf, ’epsrel=1d−8)) ), 11 f l o a t ( r e t ) );

Listing 2: The Maxima code corresponding to Proposition3.

Kgn( z ) : =block([ u, ret, k, n, fr:1, ss:0], 3 i f not numberp(z) then return(’Kgn(z)) , i f integerp(z) then return(0) e l s e( i f z<0 then z:−z , f r :−1/z , 8 n : f i x ( z ) , ss : sum( (−u)^k/k!, k, 0, n−1) , r e t : f r ∗ first(quad_qagi ( (exp(−u)− ss)/ u^(z) , u, 0, inf, ’epsrel=1d−8)) ), f l o a t ( r e t ) 13 );

Acknowledgments: The work has been supported in part by grants from Research Fund—Flanders (FWO), contract number VS.097.16N, and the COST Association Action CA16212 INDEPTH. Plots were prepared with the computer algebra system Maxima. Conﬂicts of Interest: The author declares no conﬂict of interest. Fractal Fract. 2019, 3, 1 11 of 11

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