• Γ(1) = 1,

• Γ(z + 1) = zΓ(z),

• Γ(n) = (n − 1)! for any positive integer n,

• The Euler’s reflection formula is Γ(1 − z)Γ(z) = sin ππz , z ̸∈ Z, (︁ 1 )︁ • Γ 2 = √π, ∫︂ 1 (︃ )︃z−1 • Γ(z) = log dt,

• Γ(z) = 2 t2z−1e−t2 dt, 190 Alasmry et al. | K. Multidiscip. Int. Sci. J. 3(3) (2026) 188-195

∫︂ ∞ α−1 −λt Γ(α) t e dt = α , for λ > 0, • 0 λ • Legendre’s duplication formula is (︃ )︃ 1 1−2z Γ(z)Γ z + = 2 √π Γ(2z). (5)

• The Stirling’s approximation Γ(z) ∼ √2πzz+1/2e−z as z → ∞ in |arg(z)| < π. This is precise in the sense that the ratio of the approximation to the true value approaches 1 in the limit as |z| goes to infinity.

For Re(a) > 0 and Re(b) > 0, ∫︂ 1 B(a, b) = ta−1(1 − t)b−1 dt. (6)

Its relationship with the gamma function is Γ(a)Γ(b) B(a, b) = . (7) Γ(a + b)

Define Γ′(z) (m) dm ψ(z) = , ψ (z) = m ψ(z). (8) Γ(z) dz

Differentiating the logarithm of the recurrence relation gives

ψ(z + 1) = ψ(z) + . (9)

For positive integers, ψ(n) = Hn−1 − γE, (10) ∑︁n −1 where H0 = 0, Hn = k=1 k , and γE is the Euler–Mascheroni constant.

The rising factorial is (a)n = a(a + 1) · · · (a + n − 1), (a)0 = 1. (11)

Where the quotient is defined, Γ(a + n) (a)n = . (12) Γ(a)

The confluent hypergeometric series is ∑︂ ∞ (a)n zn 1F1(a; b; z) = , (13) n=0 (b)n n!

with b / ∈ {0, −1, −2, . . .}.

This connection prepares the discussion of Coulomb functions [13, 14, 15].

For a > 0 and x ≥ 0, the lower incomplete gamma function is ∫︂ x γ(a, x) = ta−1e−t dt. (14)

For a > 0 and x ≥ 0, the upper incomplete gamma function is

Γ(a, x) = ta−1e−t dt. (15)

These restrictions provide a simple real-variable starting point. Extensions to complex arguments require branch conventions and analytic continuation [3, Sec. 8.2] and [10].

Now, we have

For x > 0, ∂γ(a, x) a−1 −x ∂Γ(a, x) a−1 −x = x e , = −x e . (16) ∂x ∂x

Integration by parts gives

γ(a + 1, x) = aγ(a, x) − xae−x, (17) Γ(a + 1, x) = aΓ(a, x) + xae−x. (18)

Special Values and Series. For integers n ≥ 1, n ∑︂ −1 xk Γ(n, x) = (n − 1)!e−x . (19)

For a > 0 and x ≥ 0, ∑︂ ∞ (−x)n γ(a, x) = xa . (20) n=0 n!(a + n)

The error function is related by (︃ )︃ γ , x = √π erf(√x). (21)

Complementarity and Regularization. Splitting Euler’s integral into x gives the following:

γ(a, x) + Γ(a, x) = Γ(a). (22)

Define γ(a, x) Γ(a, x) P(a, x) = , Q(a, x) = . (23) Γ(a) Γ(a)

Therefore, P(a, x) + Q(a, x) = 1. (24)

For a > 0 and x ≥ 0, both functions lie between zero and one.

The reciprocal 1/Γ(z) is entire. Its product representation is 1 ∏︂ ∞ (︂ z )︂ = zeγEz 1 + e−z/n, z ∈ C \ Z− 0 . (25) Γ(z) n=1 n 192 Alasmry et al. | K. Multidiscip.Int. Sci. J. 3(3) (2026) 188-195

Meaning of the Terminology. Here, “impossible integrals” means integrals not conveniently evaluated using elementary functions. It does not mean that the integrals are mathematically undefined or unevaluable. A General Gamma Integral. For µ > 0, p > 0, and a > 0, substitution of t = µxp gives ∫︂ ∞ (︃ )︃ a−1 −µxp 1 −a/p a x e dx = µ Γ . (26) 0 p p

Examples include ∫︂ ∞ (︃ )︃ −x4 1 1 e dx = Γ , (27) 0 4 4 2 −βx2 √π x e dx = 3/2 , β > 0. (28) 0 4β

For b > 0, the finite-range version is ∫︂ b (︃ )︃ a−1 −µxp 1 −a/p a p x e dx = µ γ , µb . (29) 0 p p

Beta-Type Integrals. For 0 < a < b, ∫︂ ∞ a−1 x Γ(a)Γ(b − a) b dx = . (30) 0 (1 + x) Γ(b)

Coulomb Wave Functions. (see [13, 14])

A standard representation of the regular Coulomb function is

Fℓ(η, ρ) = Cℓ(η)ρℓ+1 e−iρ 1F1(ℓ + 1 − iη; 2ℓ + 2; 2iρ), (31)

where 2ℓ e−πη/2∥Γ(ℓ + 1 + iη)∥ Cℓ(η) = . (32) Γ(2ℓ + 2)

Here ℓ is a nonnegative integer, η is real, and ρ > 0.

The Coulomb phase shift is σℓ(η) = ph Γ(ℓ + 1 + iη), (33) with phase chosen continuously from zero at η = 0 [3, Sec. 33.2].

Clarify that Cℓ is the conventional normalization factor for the Coulomb solution, not an assertion that a continuum scattering wave is square-integrable.

Canonical Energy Integrals. (see [15])

Suppose a model has density of states

g(E) = AEa−1, E > 0, a > 0. (34)

For inverse temperature β = (kBT)−1,

−βE Γ(a) Z(β) = g(E) e dE = A a . (35) If A is independent of β, then ∂ log Z a ⟨E⟩ = − = . (36) ∂β β

This is a model-dependent result: it applies to the stated power-law density of states, not to every physical system.

The Gamma Distribution. (see [14])

Using shape a > 0 and rate λ > 0, define

λa a−1 −λx fX(x) = x e , x > 0. (37) Γ(a)

The rate convention must be distinguished from the scale convention, where the scale is 1/λ.

Normalization follows directly from Euler’s integral.

Moments and Tail Probabilities. For real r > −a,

r −r Γ(a + r) E[X ] = λ . (38) Γ(a)

Hence, E[X] = , Var(X) = 2 . (39)

The cumulative and survival probabilities are

Pr(X ≤ x) = P(a, λx), Pr(X > x) = Q(a, λx). (40)

Related Distributions. (see [13] )

The exponential distribution corresponds to a = 1. For an integer shape, the gamma distribution is also called an Erlang distribution. A chi-squared distribution with ν degrees of freedom has shape ν/2 and rate 1/2.

This part explains why the gamma function appears naturally when integration and differentiation are extended to noninteger orders (see [6, 7, 16, 17]).

From Repeated Integration to Fractional Integration. For a positive integer n, repeated integration gives ∫︂ x n 1 n−1 I0+f(x) = (x − t) f(t) dt. (41) (n − 1)! 0

Replacing (n − 1)! by Γ(n) motivates the following extension.

Definition 1. For α > 0, the Riemann–Liouville fractional integral is ∫︂ x α 1 α−1 I0+f(x) = (x − t) f(t) dt, (42) Γ(α) 0

whenever the integral exists.

194 Alasmry et al. | K. Multidiscip. Int. Sci. J. 3(3) (2026) 188-195

For µ > −1, α > 0, and x > 0, α µ Γ(µ + 1) µ+α I0+x = x . (43) Γ(µ + α + 1)

Derive this formula by setting t = xu and applying Equation (7).

For example, 1/2 2√x I0+ 1 = . (44)

The Semigroup Property. Under suitable integrability assumptions,

I0+ α I0+ β f = I0+ α+βf, α, β > 0. (45)

The proof uses an interchange of integrals and the beta identity. The gamma normalization is essential to obtaining the correct composition law.

Riemann–Liouville and Caputo Derivatives. Let n − 1 < α < n, where n is a positive integer.

The Riemann–Liouville derivative is α dn n−α D0+f(x) = n I0+ f(x). (46) dx

The Caputo derivative is CD0+ α f(x) = I0+ n−αf (n)(x). (47)

Assume sufficient regularity for the stated operations; for example, the Caputo construction can be introduced for functions whose (n − 1)st derivative is absolutely continuous on the interval under consideration.

Why the Definitions Are Not Interchangeable. For 0 < α < 1,

α x−α C α D0+1 = , D0+1 = 0. (48) Γ(1 − α)

This example shows that fractional derivatives are not uniquely specified by their order alone. The operator definition and lower terminal must also be stated.

Mittag–Leffler Functions and Fractional Relaxation. (see [5, 6])

The Mittag–Leffler Function of two parameters define by

∑︂ ∞ zk Eα,β(z) = , α > 0. (49) Γ(αk + β)

Write Eα(z) = Eα,1(z). The identity E1(z) = ez connects this family to the ordinary exponential. Now, we consider For 0 < α < 1 and λ > 0, the Caputo problem

CD0+ α y(t) = −λy(t), y(0) = y0, (50)

has the standard solution y(t) = y0Eα(−λtα). (51)

This example links gamma-function coefficients to the solution of a fractional differential equation (see [16, 17]).