Task

Implement one algorithm (or more) to compute the Gamma (



Γ


{\displaystyle \Gamma }

) function (in the real field only).

If your language has the function as built-in or you know a library which has it, compare your implementation's results with the results of the built-in/library function.

The Gamma function can be defined as:

  



Γ
(
x
)
=


∫

0


∞



t

x
−
1



e

−
t


d
t



{\displaystyle \Gamma (x)=\displaystyle \int _{0}^{\infty }t^{x-1}e^{-t}dt}


This suggests a straightforward (but inefficient) way of computing the 



Γ


{\displaystyle \Gamma }

 through numerical integration.



Better suggested methods:

 Lanczos approximation
 Stirling's approximation


var a = [ 1.00000_00000_00000_00000,  0.57721_56649_01532_86061, -0.65587_80715_20253_88108,
         -0.04200_26350_34095_23553,  0.16653_86113_82291_48950, -0.04219_77345_55544_33675,
         -0.00962_19715_27876_97356,  0.00721_89432_46663_09954, -0.00116_51675_91859_06511,
         -0.00021_52416_74114_95097,  0.00012_80502_82388_11619, -0.00002_01348_54780_78824,
         -0.00000_12504_93482_14267,  0.00000_11330_27231_98170, -0.00000_02056_33841_69776,
          0.00000_00061_16095_10448,  0.00000_00050_02007_64447, -0.00000_00011_81274_57049,
          0.00000_00001_04342_67117,  0.00000_00000_07782_26344, -0.00000_00000_03696_80562,
          0.00000_00000_00510_03703, -0.00000_00000_00020_58326, -0.00000_00000_00005_34812,
          0.00000_00000_00001_22678, -0.00000_00000_00000_11813,  0.00000_00000_00000_00119,
          0.00000_00000_00000_00141, -0.00000_00000_00000_00023,  0.00000_00000_00000_00002 ]
 
func gamma(x) {
    var y = (x - 1)
    1 / a.reverse.reduce {|sum, an| sum*y + an}
}
 
for i in 1..10 {
    say ("%.14e" % gamma(i/3))
}