Gamma Function Calculator

What does this calculator do?

Enter a real number x and this tool computes Γ(x), the gamma function, along with its natural logarithm ln|Γ(x)|. The gamma function extends the factorial to non-integer and negative real numbers: for a positive integer n, Γ(n) = (n-1)!. It's evaluated with a Lanczos approximation accurate to roughly 14-15 significant digits.

How to use it

1. Enter a value for x, between -50 and 170 — outside that range the true value either underflows toward zero or overflows what a 64-bit floating-point number can represent.
2. Read Γ(x) and ln|Γ(x)| from the result table. The log column is useful once Γ(x) itself gets astronomically large (by x = 100, Γ(x) is already over 10¹⁵⁵) — its logarithm stays a normal-sized number you can actually compare.

The math

For positive real x, the gamma function is defined by the improper integral:

Γ(x) = ∫₀^∞ t^(x-1) · e^(-t) dt

which satisfies the recurrence Γ(x+1) = x · Γ(x), the same relationship the factorial follows — so Γ(1) = 1, Γ(2) = 1, Γ(3) = 2, Γ(4) = 6, and in general Γ(n) = (n-1)! for positive integers.

For non-integers, Γ(0.5) = √π ≈ 1.7725, and half-integer values follow Γ(n + 1/2) formulas built from that. For negative non-integers, Euler's reflection formula extends the function:

Γ(x) · Γ(1-x) = π / sin(πx)

which is how this calculator handles x < 0.5 — it computes Γ(1-x) with the same Lanczos approximation and solves for Γ(x). The function is undefined at zero and every negative integer (poles, where sin(πx) = 0 makes the denominator vanish) — the calculator rejects those inputs with an error rather than showing an incorrect result.

Example use cases

• Combinatorics and probability — the gamma function underlies the gamma distribution, beta distribution, and Student's t-distribution, all of which need Γ evaluated at non-integer arguments (like half-integer degrees of freedom).
• Combinatorial formulas with non-integer arguments — the generalized binomial coefficient and Stirling-type approximations extend naturally through Γ(x) rather than being restricted to whole-number factorials.
• Physics and engineering — Γ(x) shows up in normalization constants for Maxwell-Boltzmann and Bose-Einstein statistics, and in solutions to certain differential equations.
• Checking factorial values quickly — since Γ(n) = (n-1)!, entering x = 11 gives you 10! = 3628800 without needing a separate factorial calculator.

Privacy

All calculations run locally in your browser — your input is never sent to a server.

Related tools

For the closely related error function, see the Complementary Error Function Calculator. For general descriptive statistics, see the Statistics Calculator.