Gamma Function Γ(s) Calculator (Numerical Integration)

Compute the gamma function Γ(s), the integral of t^(s-1) e^(-t) from 0 to infinity. Enter a positive real s to get Γ(s), lnΓ(s), its link to factorials, and a direct double-exponential integration reference value.

Input

Enter a positive real number s to numerically compute the gamma function Γ(s), the integral of t^(s-1) e^(-t) for t from 0 to infinity.

s (positive real number)

Example: s = 5 gives Γ(5) = 4! = 24. Decimals such as 0.5 or 3.5 are allowed.

Result

Γ(5)

Γ(s) = integral of t^(s-1) e^(-t) dt from 0 to infinity

lnΓ(s) (natural log)

3.1780538303

4! = (s-1)!

Direct numerical integral

Accuracy check

The half-infinite range was integrated directly with a double-exponential (tanh-sinh) change of variables using 2,049 sample points. The relative difference from the Lanczos approximation is 2.960595e-16.

Since s = 5 is an integer, Γ(s) equals 4! (factorial).

How it works

The gamma function is the special function defined as Γ(s) = the integral of t^(s-1) e^(-t) over t from 0 to infinity. The integral converges for s ＞ 0.
For a positive integer n, Γ(n) = (n-1)! holds. For example Γ(5) = 4! = 24 and Γ(1) = 0! = 1. It generalizes the factorial to real and complex arguments.
This tool evaluates Γ(s) and lnΓ(s) with a high-accuracy Lanczos approximation. The log-gamma value lnΓ(s) helps avoid overflow when the result grows very large.
As a reference it also maps the half-infinite integral to a finite sum using a double-exponential (tanh-sinh) change of variables and integrates directly with the trapezoidal rule, then reports the relative difference between the two so you can check accuracy.
Very large s lets the tail stretch out and reduces integration accuracy, so the input s has an upper bound. Values of s at or below 0 are out of scope because this integral definition does not converge there.

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