Gamma Function Calculator
Compute Γ(x) for real values, including decimals and negative non-integers.
What is the gamma function?
The gamma function, written as Γ(x), is one of the most important extensions of the factorial function. For positive integers, it satisfies:
Γ(n) = (n-1)!
So while factorial is defined for whole numbers, the gamma function lets us compute factorial-like values for decimals and many negative values too. For example, Γ(1/2) = √π, which is about 1.77245.
How to use this gamma function calculator
- Enter any real number in the x field.
- Choose how many digits you want shown.
- Click Calculate Γ(x) (or press Enter).
- Read the main result plus supporting info like ln|Γ(x)| and sign.
Why the gamma function matters
You will see the gamma function across mathematics, statistics, physics, and engineering:
- Probability distributions: Gamma, Beta, Chi-square, and Student-related formulas.
- Combinatorics: Continuous extensions of discrete counting formulas.
- Calculus and analysis: Integrals and special functions.
- Physics: Normalization constants, partition functions, and analytic continuation.
Common examples
1) Integer input
If x = 6, then Γ(6) = 5! = 120.
2) Half-integer input
If x = 1/2, then Γ(1/2) = √π ≈ 1.77245.
3) Negative non-integer input
If x = -1/2, then Γ(-1/2) = -2√π ≈ -3.54491.
Numerical method used in this calculator
This tool uses a Lanczos approximation for stable, high-quality evaluation. For negative arguments, it applies the reflection formula:
Γ(x) = π / (sin(πx)Γ(1-x))
This approach is standard in scientific software and works very well for most practical inputs.
FAQ
Can I input very large values?
Yes, but Γ(x) grows extremely quickly. If the value exceeds standard floating-point range, the calculator will display a scientific-notation style value using logarithms.
Why do I get an undefined result at some negative numbers?
Because the gamma function has poles at non-positive integers: 0, -1, -2, -3, and so on.
Is this exact?
It is a numerical approximation with high practical accuracy for real inputs, not symbolic exact arithmetic.