gamma(x)
Overview
The gamma function calculates the Gamma function \Gamma(x) using the Lanczos approximation method.
Parameters
| Parameter | Type | Description |
|---|---|---|
x |
Number | The value to find the Gamma function of. |
Returns
| Return | Type | Description |
|---|---|---|
gam |
Number | The calculated Gamma function value \Gamma(x) . |
Example
local x = 5
local result = StatBook.gamma(x)
print(result)
Mathematical Background
The function calculates \Gamma(x) using the Lanczos approximation, which is an efficient method to compute the Gamma function for complex numbers. The method approximates \Gamma(z) by:
\Gamma(z) \approx \sqrt{2\pi} \left( z + \frac{5}{6} \right)^{z+\frac{1}{2}} e^{-(z+\frac{5}{6})} \left( c_0 + \frac{c_1}{z+1} + \frac{c_2}{z+2} + \cdots + \frac{c_n}{z+n} \right)
Here c_0, c_1, \cdots, c_n are precomputed coefficients used in the approximation.