Gamma Function

📐 Definition

For complex numbers $z$ with $\Re(z) > 0$ , the Gamma function is defined by the improper integral

\Gamma(z) = \int_0^\infty t^{z-1} e^{-t}\, dt

This definition extends the factorial function to non-integer and complex arguments via analytic continuation.

⚙️ Key Properties

Functional Equation (Factorial Property)

The Gamma function satisfies the fundamental recurrence relation

\Gamma(z+1) = z\,\Gamma(z)

which implies $\Gamma(n+1) = n!$ for all $n \in \mathbb{N}$ .

Normalization

At $z = 1$ , the Gamma function is normalized as

\Gamma(1) = 1

This choice uniquely determines the function given the functional equation and analyticity.

Poles and Residues

The Gamma function has simple poles at the non-positive integers:

z = 0, -1, -2, \dots

with residues given by

\operatorname{Res}\bigl(\Gamma(z), z = -n\bigr) = \frac{(-1)^n}{n!}, \qquad n = 0, 1, 2, \dots

Reflection Formula

The Gamma function satisfies Euler’s reflection formula

\Gamma(z)\,\Gamma(1-z) = \frac{\pi}{\sin(\pi z)}

This identity relates values across the line $\Re z = \tfrac12$ and plays a central role in analytic continuation.

Multiplication Formula (Gauss)

For any positive integer $m$ ,

\Gamma(mz) = (2\pi)^{\frac{1-m}{2}} m^{mz-\frac12} \prod_{k=0}^{m-1} \Gamma\!\left(z + \frac{k}{m}\right)

This formula generalizes the duplication identity and underlies many special-value evaluations.

Asymptotic Behavior (Stirling’s Formula)

As $|z| \to \infty$ with $|\arg z| < \pi$ , the Gamma function satisfies

\Gamma(z) \sim \sqrt{2\pi}\, z^{z-\frac12} e^{-z}

More precisely,

\Gamma(z) \sim \sqrt{2\pi}\, z^{z-\frac12} e^{-z} \left( 1 + \frac{1}{12z} + \frac{1}{288z^2} - \cdots \right)

This asymptotic expansion is fundamental in approximation theory and mathematical physics.

Zeros

The Gamma function has no zeros in the complex plane.

This fact distinguishes $\Gamma(z)$ from many other special functions and is crucial in the study of its logarithmic derivative.

🎯 Special Values

At selected arguments, the Gamma function simplifies to closed-form expressions:

$z$	$\Gamma(z)$ (exact)	Approximate value
$1$	$1$	$1.0$
$2$	$1$	$1.0$
$\tfrac12$	$\sqrt{\pi}$	$1.7724538509$
$\tfrac32$	$\tfrac12\sqrt{\pi}$	$0.8862269255$
$n \in \mathbb{N}$	$(n-1)!$	—

Digamma Function Logarithmic derivative: ψ(z) = Γ'(z)/Γ(z)

Polygamma Functions Higher-order logarithmic derivatives: ψ^(n)(z)

Stirling Series Asymptotic expansion of Γ(z) for large arguments

Riemann Zeta Function Connected through functional equations

Usage in Oakfield

Historical Foundations

📜 Early Foundations (18th Century)

Leonhard Euler introduced the Gamma function as a continuous extension of the factorial, establishing its integral representation and key functional properties.
🔬 19th-Century Development

Adrien-Marie Legendre introduced the modern notation $\Gamma(z)$ and refined its normalization. Gauss and others developed its complex-analytic theory, including multiplication formulas and asymptotics.
🌍 Modern Perspective

Today, the Gamma function is ubiquitous across mathematics and physics, underpinning probability theory, special functions, number theory, and quantum field theory.

📚 References

NIST Digital Library of Mathematical Functions (DLMF), §5
Abramowitz & Stegun, Handbook of Mathematical Functions
Whittaker & Watson, A Course of Modern Analysis