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


MODULE Gamma EXPORTS Main;
 
FROM IO IMPORT Put;
FROM Fmt IMPORT Extended, Style;
 
PROCEDURE Taylor(x: EXTENDED): EXTENDED =
  CONST a = ARRAY [0..29] OF EXTENDED {
    1.00000000000000000000X0, 0.57721566490153286061X0,
    -0.65587807152025388108X0, -0.04200263503409523553X0,
    0.16653861138229148950X0, -0.04219773455554433675X0,
    -0.00962197152787697356X0, 0.00721894324666309954X0,
    -0.00116516759185906511X0, -0.00021524167411495097X0,
    0.00012805028238811619X0, -0.00002013485478078824X0,
    -0.00000125049348214267X0, 0.00000113302723198170X0,
    -0.00000020563384169776X0, 0.00000000611609510448X0,
    0.00000000500200764447X0, -0.00000000118127457049X0,
    0.00000000010434267117X0, 0.00000000000778226344X0,
    -0.00000000000369680562X0, 0.00000000000051003703X0,
    -0.00000000000002058326X0, -0.00000000000000534812X0,
    0.00000000000000122678X0, -0.00000000000000011813X0,
    0.00000000000000000119X0, 0.00000000000000000141X0,
    -0.00000000000000000023X0, 0.00000000000000000002X0 };
  VAR y := x - 1.0X0;
      sum := a[LAST(a)];
 
  BEGIN
    FOR i := LAST(a) - 1 TO FIRST(a) BY -1 DO
      sum := sum * y + a[i];
    END;
    RETURN 1.0X0 / sum;
  END Taylor;
 
BEGIN
  FOR i := 1 TO 10 DO
    Put(Extended(Taylor(FLOAT(i, EXTENDED) / 3.0X0), style := Style.Sci) & "\n");
  END;
END Gamma.