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


(scl 28)
 
(de *A
   ~(flip
      (1.00000000000000000000  0.57721566490153286061 -0.65587807152025388108
      -0.04200263503409523553  0.16653861138229148950 -0.04219773455554433675
      -0.00962197152787697356  0.00721894324666309954 -0.00116516759185906511
      -0.00021524167411495097  0.00012805028238811619 -0.00002013485478078824
      -0.00000125049348214267  0.00000113302723198170 -0.00000020563384169776
       0.00000000611609510448  0.00000000500200764447 -0.00000000118127457049
       0.00000000010434267117  0.00000000000778226344 -0.00000000000369680562
       0.00000000000051003703 -0.00000000000002058326 -0.00000000000000534812
       0.00000000000000122678 -0.00000000000000011813  0.00000000000000000119
       0.00000000000000000141 -0.00000000000000000023  0.00000000000000000002 ) ) )
 
(de gamma (X)
   (let (Y (- X 1.0)  Sum (car *A))
      (for A (cdr *A)
         (setq Sum (+ A (*/ Sum Y 1.0))) )
      (*/ 1.0 1.0 Sum) ) )