# ガンマ関数 Gamma Function
# 引数 値 ($X)
# 戻り値 ガンマ関数 ($GammaFunction)
sub GAMMAFUNCTION{
	my ($X) = @_;
	my $GammaFunction = 0;
	my $Factorial = 1;
	my $Diff = $X - int($X);
	my $Pi = atan2(1, 1) * 4;

	if($Diff == 0){
		for(my $i = $X - 1; $i >= 2; $i--){
			$Factorial *= $i;
		}

		# X-1の階乗
		$GammaFunction = $Factorial;
	}elsif($Diff == 0.5){
		for(my $i = (2 * int($X)) - 1; $i > 1; $i -= 2){
			$Factorial *= $i;
		}

		# 半整数
		$GammaFunction = $Factorial * sqrt($Pi) / (2 ** int($X));
	}else {
		my $X1 = 1 / (12 * $X);
		my $X2 = 1 / (288 * ($X * $X));
		my $X3 = 139 / (51840 * ($X * $X * $X));
		my $X4 = 571 / (2488320 * ($X * $X * $X * $X));
		my $X5 = 163879 / (209018880 * ($X * $X * $X * $X * $X));
		my $X6 = 5246819 / (75246796800  * ($X * $X * $X * $X * $X * $X));
		my $X7 = 534703531 / (902961561600  * ($X * $X * $X * $X * $X * $X * $X));
		my $X8 = 4483131259 / (86684309913600 * ($X * $X * $X * $X * $X * $X * $X * $X));
		my $X9 = 432261921612371 / (514904800886784000 * ($X * $X * $X * $X * $X * $X * $X * $X));

		# スターリングの近似 Stirling's Approximation
		$GammaFunction = sqrt(2 * $Pi) * ($X ** ($X - (1 / 2))) * exp(-$X) * (1 + $X1 + $X2 - $X3 - $X4 + $X5 + $X6 - $X7 - $X8 + $X9);
	}

	return $GammaFunction;
}