Gamma Function Calculator

Work out gamma function instantly with clear inputs, formula shown and shareable results.

About Gamma Function Calculator

The gamma function extends the factorial to non-integers, with Γ(n) = (n − 1)! for positive integers. It satisfies the recurrence Γ(z + 1) = z·Γ(z) and has poles at zero and the negative integers. This uses the Lanczos approximation.

Formulas

Gamma recurrence

Γ(z + 1) = z·Γ(z), Γ(n) = (n − 1)! for whole n

Reflection

Γ(z)·Γ(1 − z) = π / sin(πz)

Worked example

With the default values below, Gamma Function Calculator returns Γ(z) of 52.34277778. Change any input above and every figure here is recalculated in your browser.

Inputs used

Argument z: 5.5
Also evaluate at: 0.5

Results

Γ(z): 52.34277778
Γ at the comparison value: 1.77245385
Γ(z + 1) = z·Γ(z): 287.88527782
ln Γ(z): 3.95781397

Step by step

1Values used
2Gamma recurrence
3Reflection
4Γ(z)= 52.34277778
5Γ at the comparison value= 1.77245385
6Γ(z + 1) = z·Γ(z)= 287.88527782
7ln Γ(z)= 3.95781397

Assumptions and limits

The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

What is Γ(0.5)?

√π ≈ 1.7724539, one of the most quoted values of the function.

Why are there poles at the negative integers?

The recurrence forces Γ(z) to blow up as z approaches 0, and stepping down by the recurrence propagates that to every negative integer.