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. That shift by one catches everybody the first time. It is a quirk of Euler’s original convention, not a piece of mathematics, and Legendre is usually blamed for making it stick.

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, so Γ(3/2) = (1/2)·Γ(1/2) = √π / 2 ≈ 0.886, then Γ(5/2) = (3/2)·Γ(3/2) ≈ 1.329, and so on up.

The gamma function has poles (infinite values) at 0 and all negative integers. Between the poles, it alternates sign for negative non-integer inputs, so Γ(−0.5) is negative, Γ(−1.5) is positive, Γ(−2.5) is negative again. This calculator handles those, reaching them through the reflection formula below. It cannot evaluate 0 or any negative whole number, because there is genuinely no value there.

For positive x the function is always positive and grows extremely fast. Γ(171) is about 7.26 × 10³⁰⁶, which a double still holds; Γ(172) is 171! and overflows to infinity. So the practical ceiling in a browser is x = 171.

That ceiling 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. ln(Γ(1000)) is about 5,905, a perfectly ordinary number, while Γ(1000) itself has 2,565 digits.

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), and Γ is what makes that formula work for all real n rather than just the 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, a rational approximation good to roughly 15 significant digits and the standard algorithm in scientific computing libraries. One place it is deliberately overridden: when you enter a whole number, the panel prints the factorial computed exactly with big integers instead. Exponentiating a floating-point logarithm is accurate to about eleven digits, which is fine for Γ(4.5) and not fine for Γ(20), where it would report 121645100410059440 for a 19! that is really 121645100408832000.