public static function gamma($n)
{
// Basic integer/factorial cases
if ($n == 0) {
return \INF;
}
// Negative integer, or negative int as a float (Ex: from beta(-0.1, -0.9) since it will call Γ(x + y))
if ((is_int($n) || is_numeric($n) && abs($n - round($n)) < 1.0E-5) && $n < 0) {
return -\INF;
}
// Positive integer, or postive int as a float (Ex: from beta(0.1, 0.9) since it will call Γ(x + y))
if ((is_int($n) || is_numeric($n) && abs($n - round($n)) < 1.0E-5) && $n > 0) {
return Combinatorics::factorial(round($n) - 1);
}
// Half integer cases (determine if int + 0.5)
if (round($n * 2) / 2 / $n == 1) {
// Compute parts of equation
$π = \M_PI;
$x = round($n - 0.5, 0);
$√π = sqrt($π);
if ($x == 0) {
return $√π;
}
$⟮2n−1⟯‼︎ = Combinatorics::doubleFactorial(2 * $x - 1);
/**
* Put it all together
* _ (2n-1)!!
* √π ---------
* 2ⁿ
*/
return $√π * ($⟮2n−1⟯‼︎ / 2 ** $x);
}
// Generic real number case
return self::gammaLanczos($n);
} /**
* Probability density function
*
* / ν + 1 \
* Γ | ----- |
* \ 2 / / x² \ ⁻⁽ᵛ⁺¹⁾/²
* ------------- | 1 + -- |
* __ / ν \ \ ν /
* √νπ Γ | - |
* \ 2 /
*
*
* @param number $x percentile
* @param int $ν degrees of freedom > 0
*/
public static function PDF($x, int $ν)
{
Support::checkLimits(self::LIMITS, ['x' => $x, 'ν' => $ν]);
$π = \M_PI;
// Numerator
$Γ⟮⟮ν＋1⟯∕2⟯ = Special::gamma(($ν + 1) / 2);
$⟮1＋x²∕ν⟯ = 1 + $x ** 2 / $ν;
$−⟮ν＋1⟯∕2 = -($ν + 1) / 2;
// Denominator
$√⟮νπ⟯ = sqrt($ν * $π);
$Γ⟮ν∕2⟯ = Special::gamma($ν / 2);
return $Γ⟮⟮ν＋1⟯∕2⟯ * $⟮1＋x²∕ν⟯ ** $−⟮ν＋1⟯∕2 / ($√⟮νπ⟯ * $Γ⟮ν∕2⟯);
}
