What is the Gamma function?

The Gamma function Γ(x) is a smooth extension of the factorial to real (and complex) numbers, satisfying Γ(n) = (n − 1)! for every positive integer n, while also being defined for non-integer and negative non-integer values.

Why is Γ(n) = (n − 1)! instead of n!?

This offset is a historical convention from the function's integral definition. It means Γ(1) = 0! = 1, Γ(2) = 1! = 1, Γ(3) = 2! = 2, and so on — each shifted down by one from the factorial you might expect.

Why is the Gamma function undefined at zero and negative integers?

The Gamma function has poles (goes to infinity) at zero and every negative integer, so it has no finite value there — this calculator shows an error for those inputs instead of a misleading result.

How is the Gamma function computed for non-integer values?

This calculator uses the Lanczos approximation, a well-established numerical method that evaluates Γ(x) to high precision using a rational-function approximation, avoiding the need to compute the function's improper integral directly.

Can I compute the Gamma function for negative non-integer values?

Yes — the calculator uses the reflection formula Γ(x)Γ(1−x) = π / sin(πx) to correctly evaluate Γ(x) for negative non-integer x, where the direct approximation isn't accurate.

How is the Gamma function related to the Beta function?

The Beta function is defined directly in terms of the Gamma function: B(x, y) = Γ(x)Γ(y) / Γ(x + y). The site's Beta Function Calculator uses this same Gamma implementation internally.

How accurate is this calculator?

The Lanczos approximation used here is accurate to roughly 15 significant digits for typical inputs, matching what scientific computing libraries use internally for double-precision Gamma function evaluation.

Where is the Gamma function used in real life?

It appears throughout probability and statistics (e.g. the Beta, Chi-squared, and Student's t-distributions), physics, and combinatorics anywhere factorial-like growth needs to be extended to non-integer values.

Gamma Function Calculator — Evaluate Γ(x) for Any Real Number

Our Gamma Function Calculator evaluates Γ(x) for any real x (except zero and negative integers, where the function has poles) using the Lanczos approximation, the standard numerical method for computing the Gamma function to double-precision accuracy. For positive integers, it also shows the familiar Γ(n) = (n − 1)! identity as a cross-check, making the connection between the Gamma function and the factorial explicit.

Quick Answer

The Gamma function Γ(x) extends the factorial to real numbers, satisfying Γ(n) = (n − 1)! for positive integers. Enter any real x (except zero or a negative integer) to get Γ(x) computed via the Lanczos approximation, the standard numerical method for evaluating it accurately.

Enter a value for x, then click Calculate.

How to Use the Gamma Function Calculator — Γ(x) Online

1
Enter any real number x (except zero or a negative integer).
2
Click 'Calculate' to evaluate Γ(x).
3
For positive integers, review the Γ(n) = (n − 1)! cross-check shown alongside the result.

Why Use Gamma Function Calculator — Γ(x) Online?

The Gamma function extends the factorial to non-integer (and negative, non-integer) real numbers, satisfying Γ(n) = (n − 1)! for positive integers while also being defined smoothly in between and for negative non-integer values. Computing it directly from its integral definition is impractical; the Lanczos approximation is the standard numerical technique used by scientific computing libraries to evaluate Γ(x) accurately and quickly. This calculator applies that same method so you get a precise result instantly, without needing specialized math software.

Frequently Asked Questions

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