Tridimensional display of the Z=Gamma(Z) iteration inside [-20.0,+20.0]x[-20.0,+20.0] [Visualisation tridimensionnelle de l'itération Z=Gamma(Z) dans [-20.0,+20.0]x[-20.0,+20.0]].




The Gamma function can be computed for all z with the following analytic continuation:
                    Gamma(z) = factorial(z-1)

                                                                     __
                                              1                log(2 ||)
                    log(factorial(z)) = (z + ---)log(z) - z + ------------
                                              2                    2

                                          k=V
                                        _______
                                        \             B
                                         \             2k
                                      +  /      ---------------
                                        /______           2k-1
                                                 2k(2k-1)z
                                          k=1

                                      + epsilon(z,N,V)


                                      factorial(z+n)
                    factorial(z) = --------------------
                                    (z+1)(z+2)...(z+n)


This picture displays the iteration of the Gamma function:
                    Z  = C (current point)
                     0

Z = Gamma(Z ) n n-1


(for each Z(0) inside a subset of the complex plane) as a surface in a tridimensional space (the two dimensions of the complex plane plus the number of iterations). The argument of the last computed Z(n) is displayed as colors painting the surface; the [0, 2.pi] segment is mapped on the {Blue,Red,Magenta,Green,Cyan,Yellow,White} set.

Here are more pictures about the iteration of the Gamma function:





(CMAP28 WWW site: this page was created on 03/17/2000 and last updated on 03/04/2019 12:36:34 -CET-)



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