`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.

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