Gamma Function Calculator

The gamma function extends the factorial to non-integer and complex arguments. Leonhard Euler introduced it in the 1720s while looking for a smooth interpolation of the factorial. The defining property is the recurrence Γ(x+1) = x · Γ(x), which for positive integers gives Γ(n) = (n−1)!. So Γ(1) = 1, Γ(2) = 1, Γ(3) = 2, Γ(4) = 6, and so on. (The shift by one is a quirk of Euler’s original convention — it’s the price of admission.)

The integral definition for positive x is:

Γ(x) = ∫₀^∞ t^(x−1) · e^(−t) dt

Evaluating this integral at x = 1/2 gives the famously beautiful result Γ(1/2) = √π ≈ 1.77245. That identity shows up all over physics and statistics: it is essentially the reason the normalising constant of the Gaussian distribution is √(2π), and the reason the volume of an n-dimensional sphere has π in it. The half-integer values follow from there by the recurrence — Γ(3/2) = (1/2)·Γ(1/2) = √π / 2 ≈ 0.886, Γ(5/2) = (3/2)·Γ(3/2), and so on.

The gamma function has poles (infinite values) at 0 and all negative integers. Between the poles, it alternates sign for negative non-integer inputs. For positive x, it is always positive and grows extremely fast — Γ(171) already overflows double-precision floating point.

That overflow is why most numerical work uses the log-gamma function ln(Γ(x)) instead. Working in log space keeps the numbers manageable; you only exponentiate at the end if you actually need Γ rather than its logarithm.

The gamma function appears in:

• Probability distributions: the gamma, chi-squared, F, t, beta, and Dirichlet distributions all have Γ in their normalising constants.
• Physics: quantum mechanics, statistical mechanics, string theory.
• Number theory: the reflection formula Γ(x) · Γ(1−x) = π / sin(πx) ties Γ to trigonometry in a surprising way.
• Geometry: the volume of the unit n-ball is π^(n/2) / Γ(n/2 + 1) — Γ is what makes that formula work for all real n, not just integers.
• Engineering: signal processing, control theory.

The digamma function ψ(x) = Γ’(x) / Γ(x) is the logarithmic derivative and shows up in Bayesian statistics and special-function identities. For positive integers, ψ(n) = H(n−1) − γ, where H is the harmonic number and γ is the Euler-Mascheroni constant (≈ 0.5772).

This calculator uses the Lanczos approximation, which computes Γ(x) to about 15 significant digits by evaluating a rational approximation — the standard algorithm in scientific computing libraries.