Enter a real number into the argument (x) field and the Gamma Function Calculator computes Γ(x) — the generalized factorial function. You also get the equivalent factorial value (when x is a positive integer), the natural log of Gamma ln(Γ(x)), and a note on whether the result is exact or approximated. Works for positive real numbers and negative non-integers.
Γ(x)
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ln(Γ(x))
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Factorial Equivalent (x−1)!
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Input x
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Result Type
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The Gamma function Γ(x) is a generalization of the factorial to real and complex numbers. For any positive integer n, Γ(n) = (n−1)!. It is defined by the integral Γ(x) = ∫₀^∞ t^(x−1) e^(−t) dt for positive real x, and extended analytically to most of the complex plane.
For positive integers, Γ(n) = (n−1)!. So Γ(1) = 0! = 1, Γ(2) = 1! = 1, Γ(3) = 2! = 2, Γ(5) = 4! = 24, and so on. This makes the Gamma function the natural continuous extension of the factorial operation.
Yes, Γ(x) can be evaluated for negative non-integer real numbers using the reflection or recurrence formula Γ(x+1) = x·Γ(x). However, Γ(x) is undefined (has poles) at zero and all negative integers (0, −1, −2, −3, …).
Γ(1/2) = √π ≈ 1.7724538509. This is one of the most famous special values of the Gamma function and arises frequently in probability theory, statistics, and physics.
For large x, the Stirling approximation is commonly used: Γ(x) ≈ √(2π/x) · (x/e)^x. This calculator uses the Lanczos approximation, which is highly accurate across a wide range of real inputs and is the method of choice in most scientific software.
For x greater than approximately 171.62, Γ(x) exceeds the maximum representable double-precision floating-point number (~1.8 × 10^308). In such cases, it is more practical to compute ln(Γ(x)) instead, which remains finite and manageable for much larger arguments.
The Gamma function satisfies the functional equation Γ(x+1) = x · Γ(x) for all valid x. This means you can shift the argument by 1 and scale by x to get the next value, which mirrors how n! = n · (n−1)! works for factorials.
Simply enter your desired argument x in the input field. The calculator will display Γ(x), its natural logarithm ln(Γ(x)), and the equivalent factorial value if x is a positive integer. A table and bar chart showing nearby Gamma values are also generated automatically.