Gamma Function

Why we want a continuous factorial

Factorial n! = n · (n−1) · … · 2 · 1 is defined only for non-negative integers. The values jump: 0! = 1, 1! = 1, 2! = 2, 3! = 6, 4! = 24, 5! = 120. What about 2.5!? Or (−1/3)!? Or i!?

Euler asked this question in 1729. He sought a function that (i) agrees with factorial at the integers and (ii) is smoothly defined in between. The answer is the gamma function, and the integral representation Γ(z) = ∫₀^∞ t^(z−1) e^(−t) dt is the cleanest closed form.

The notational shift Γ(n) = (n−1)! (rather than n!) is Legendre's, dating to 1809. It's an irritating offset by 1 that nobody loves but everyone uses.

The integral definition

Γ(z)  =  ∫₀^∞ t^(z−1) · e^(−t) dt,    Re(z) > 0.

This is Euler's integral of the second kind. The exponential decay e^(−t) kills any polynomial growth, so the integral converges for any z with Re(z) > 0. To extend to other complex z, we use the recurrence (next section) — or, more deeply, the Weierstrass product or Hankel contour.

The functional equation Γ(z+1) = z·Γ(z)

Integrate by parts on Γ(z+1) = ∫₀^∞ tᶻ e^(−t) dt with u = tᶻ, dv = e^(−t) dt:

Γ(z+1)  =  ∫₀^∞ tᶻ e^(−t) dt
        =  [−tᶻ e^(−t)]₀^∞  +  z · ∫₀^∞ t^(z−1) e^(−t) dt
        =  0  +  z · Γ(z)
        =  z · Γ(z).

The boundary term vanishes — at t = 0 the tᶻ factor wins for Re(z) > 0; at t = ∞ the e^(−t) factor wins. The recurrence is the same step that builds factorial: n! = n · (n−1)!.

Combined with Γ(1) = ∫₀^∞ e^(−t) dt = 1, the recurrence gives Γ(2) = 1·1 = 1, Γ(3) = 2·1 = 2, Γ(4) = 3·2 = 6, …, Γ(n) = (n−1)! for every positive integer.

The half-integer values and Γ(1/2) = √π

For Γ(1/2), substitute t = u² (so dt = 2u du, t^(−1/2) = 1/u):

Γ(1/2)  =  ∫₀^∞ t^(−1/2) e^(−t) dt
        =  ∫₀^∞ (1/u) · e^(−u²) · 2u du
        =  2 ∫₀^∞ e^(−u²) du
        =  √π.

The last step is the Gaussian integral. Applying the recurrence gives all half-integer values:

Γ(3/2)   =  (1/2) · Γ(1/2)  =  (1/2) √π
Γ(5/2)   =  (3/2) · Γ(3/2)  =  (3/4) √π
Γ(n+1/2) =  ((2n)! / (4ⁿ · n!)) · √π   (general formula)

The appearance of √π is why π shows up in formulas you'd expect to be √π-free — like the volume of a sphere, the normalizing constant of the normal distribution, and Stirling's approximation.

Where Γ blows up

The recurrence Γ(z) = Γ(z+1)/z reveals the poles. At z = 0: Γ(0) = Γ(1)/0 = 1/0 — diverges. At z = −1: Γ(−1) = Γ(0)/(−1) — diverges (since Γ(0) is already infinite, but as a pole). By induction, Γ has simple poles at every non-positive integer 0, −1, −2, −3, …, with residue (−1)ⁿ/n! at z = −n.

Between the poles, Γ takes positive values for z > 0 and oscillates sign in each interval (−n−1, −n) for negative z. The graph of Γ on the real line is the canonical picture — smooth positive curve through factorial points for z > 0, with sharp asymptotes at every non-positive integer.

Bohr-Mollerup — Γ is the unique log-convex extension

Many functions interpolate the factorial through (1, 0!), (2, 1!), (3, 2!), …. Why is Γ the "right" one?

The Bohr-Mollerup theorem (1922) gives the answer: Γ is the unique function f : (0, ∞) → (0, ∞) satisfying:

1. f(1) = 1,
2. f(z+1) = z·f(z) for all z > 0,
3. log f is convex on (0, ∞).

The third condition (log-convexity) is what singles out Γ. Without it, any function that satisfies (1) and (2) on the integers but wiggles between them would qualify. Log-convexity is a smoothness condition that says "no wiggles — the function curves up gently."

Γ vs factorial vs Beta — pick your tool

	Factorial n!	Gamma Γ(z)	Beta B(x, y)
Domain	Non-negative integers	ℂ ∖ {0, −1, −2, ...}	{(x, y) : Re x, Re y > 0}
Recurrence	n! = n·(n−1)!	Γ(z+1) = z·Γ(z)	B(x, y) = Γ(x)Γ(y)/Γ(x+y)
Where it appears	Combinatorics, basic probability	All special functions, complex analysis, QFT	Order statistics, beta distribution
Hard values	Trivial — multiply integers	Γ(1/2) = √π is famous	B(1/2, 1/2) = π
Closed form	Product of integers	Improper integral or limit	Beta integral on [0, 1]
Defined for negatives?	No	Yes (except non-positive integers)	Generally needs x, y > 0

Where Γ appears in the wild

• Probability distributions. The gamma distribution (waiting times in Poisson processes) uses Γ in its normalizing constant. Chi-squared, Student's t, beta, F, and Erlang distributions all involve Γ.
• The normal distribution. The normalizing constant 1/√(2π) comes from ∫e^(−x²/2) dx = √(2π) — and √(2π) is Γ(1/2)·√2 in disguise. The reason π appears in normality is the Gaussian's connection to Γ(1/2).
• Volume of an n-ball. The volume of the unit ball in n dimensions is πⁿ/² / Γ(n/2 + 1). For n = 1, 2, 3, 4 — 2, π, 4π/3, π²/2. The formula unifies all dimensions through Γ.
• Riemann zeta function. Riemann's functional equation has Γ(s/2) as a factor: ζ(s)·Γ(s/2)·π^(−s/2) is symmetric under s → 1 − s. Without Γ, the functional equation can't be stated.
• Quantum field theory regularization. Dimensional regularization replaces 4-dimensional integrals with d-dimensional ones; Γ provides the analytic continuation that lets you take d → 4 cleanly.
• Combinatorics. Binomial coefficients (a + b choose b) = Γ(a + b + 1)/(Γ(a + 1)·Γ(b + 1)) extend to non-integer arguments — useful in analytic combinatorics and generating-function asymptotics.
• Stirling's approximation. Γ(z) ≈ √(2π)·z^(z−1/2)·e^(−z) for large z — the smooth-curve version of Stirling's formula for factorial.

Common mistakes and misconceptions

• Confusing Γ(n) with n!. Γ(n) = (n−1)!, not n!. The offset is constant and easy to fix, but easy to forget mid-calculation.
• Trying to evaluate Γ at non-positive integers. Γ is undefined at 0, −1, −2, …. Computations involving Γ at these points must use limits or residues.
• Forgetting log-convexity as a defining property. Many alternative interpolations of factorial exist; only the log-convex one is Γ.
• Misapplying the Beta-Gamma identity. B(x, y) = Γ(x)Γ(y)/Γ(x + y) — not Γ(x + y)/(Γ(x)Γ(y)). The arrangement matters.
• Numerical instability. Direct evaluation of Γ(z) for very large z overflows; use Stirling, or lgamma (the log of |Γ|), which most numerical libraries provide.
• Sign confusion at negative values. Γ alternates sign between consecutive non-positive integers — Γ(−1/2) = −2√π is negative, Γ(−3/2) = (4/3)√π is positive. Track signs carefully.

Connections to other concepts

Beta function. Beta is Γ's two-variable cousin: B(x, y) = Γ(x)Γ(y)/Γ(x + y). It governs order statistics and integrals on the simplex.

Riemann zeta function. ζ(s) = ∑1/nˢ has a functional equation involving Γ(s/2). The connection links Γ to prime number distribution via analytic number theory.

Hypergeometric functions. ₁F₁, ₂F₁, and other hypergeometric functions have Γ in their power-series coefficients. They generalize the gamma function the same way the gamma function generalizes factorial.

Stirling's approximation. n! ≈ √(2πn)·(n/e)ⁿ extends to Γ(z) ≈ √(2π)·z^(z−1/2)·e^(−z). The smooth version of Stirling.

Volume of high-dimensional balls. Volume Vₙ = πⁿ/²/Γ(n/2+1) — the formula that says volume grows then shrinks with dimension, peaking around n = 5.