Função Gama Γ(x) Stirling
- Created by
- Renato Passos, Eng. de Software
- Reviewed by
- Renato Passos, Eng. de Software
Last updated: Apr 18, 2026
About this calculator
The Stirling Gamma Function calculator estimates the gamma function Γ(x) using Stirling's approximation. This formula is useful for calculating Γ(x) for large numbers, where exact computation is impractical. The approximation uses √(2π/x)·(x/e)^x, providing a precise estimate for x greater than 1.
Stirling's approximation is based on the asymptotic expansion of the gamma function, simplifying calculations for large values. It is widely used in statistics, physics, and probability theory to approximate complex factorials. The formula considers the relationship between x and Euler's constant (e), adjusting the curve for higher accuracy.
Use this calculator when you need a quick estimate of the gamma function, especially in problems involving combinations, probabilities, or algorithm optimization. It is ideal for x values above 10, where approximation accuracy is higher. Avoid using it for very small x values, as relative error increases.
Be cautious when interpreting results, as Stirling's approximation is not exact. For critical values or rigorous scientific calculations, combine with additional numerical methods. Proper use depends on understanding the mathematical context of the problem being analyzed.
Frequently asked questions
What is Stirling's approximation used for?
It is used to estimate the gamma function for large numbers, where exact calculation is complex or computationally expensive.
How is the formula √(2π/x)·(x/e)^x calculated?
The formula combines the square root of 2π divided by the square root of x, multiplied by (x/e) raised to the x-th power.
When to use the approximation instead of exact calculation?
Use it when x is greater than 10 and relative precision is sufficient for the context. For smaller x, prefer exact methods.
What is the accuracy of Stirling's approximation?
Accuracy increases with larger x. For x = 10, the error is less than 1%. For x = 1, the error may exceed 100%.
Does the calculator work for non-integer values?
Yes, Stirling's approximation is valid for any x > 0, including real and complex numbers, as long as the real part is positive.