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中文(臺灣):反伽瑪函數的色相環複變函數圖形
Date
Source Own work
Author A2569875

Mathematica code

based on this code:

but it doesn't support newer version of Mathematica.

The modification is as follows, it can run on Mathematica 11:

Options[PlotComplex] = {Mesh -> None, MeshStyle -> Automatic, 
   WorkingPrecision -> MachinePrecision, PlotPoints -> 240};
PlotComplex[fz_, {z_, min_, max_}, OptionsPattern[]] := 
  Module[{fn}, fn[gx_] := fz /. z -> gx;
   RegionPlot[x^2 y^2 > 0, {x, min, max}, {y, min, max}, 
    ColorFunction -> 
     Function[{x, y}, 
      Hue[Mod[Arg[
          fn[-max + max x - min x + I (-max + max y - min y)]], 
         2 \[Pi]]/(2 \[Pi]), 
       1/(1 + 0.3 Log[
            Abs[fn[-max + max x - min x + 
                I (-max + max y - min y)]] + 1]), 
       1 - 1/(1.1 + 
           5 Log[Abs[
               fn[-max + max x - min x + I (-max + max y - min y)]] + 
              1])]], PlotPoints -> OptionValue[PlotPoints], 
    Mesh -> OptionValue[Mesh], MeshStyle -> OptionValue[MeshStyle], 
    WorkingPrecision -> OptionValue[WorkingPrecision]]];

Inverse gamma function is modified from Afacc.cin, Fac.cin in mizugadro.mydns.jp, and changed into a Mathematica program as follows:

  facp[x_] := (\!\(
\*SubscriptBox[\(\[PartialD]\), \(y\)]\ \(Factorial[y]\)\)) /. y -> x;
  afacb[zl_] := Module[{
     z0 = 0.461632144968362341262659542325721328468196204,
     F0 = -0.12148629053584960809551455717769158215135617313,
     c2 = 0.483836122723810585213722380854825370205628608,
     p = 0.2090973242496979633924701135209125815611056,
     q = 0.0565790271828431799463572817754001404669620,
     A = 0.0008685913050832152753870514845664790993724,
     B = 0.0002046727298252365296379380008904113017495
     }, Module[{
      t = (Log[zl] - F0)/c2,
      }, Module[{
       v = Sqrt[t]
       }, v*(1.0 + v*(p + A*t))/(1. + v*(q + B*t)) + z0]]];
afacc[z_] := Module[{a, c, d},
  a = afacb[z];
  d = facp[a]; c = z - Factorial[a]; a = a + c/d; 
  If[Abs[c] < 10^(-16), a,
   d = facp[a]; c = z - Factorial[a]; a = a + c/d; 
   If[Abs[c] < 10^(-16), a,
    d = facp[a]; c = z - Factorial[a]; a = a + c/d; 
    If[Abs[c] < 10^(-16), a,
     d = facp[a]; c = z - Factorial[a]; a = a + c/d; 
     If[Abs[c] < 10^(-16), a,a]]]]
  
  ];

(Note that this program call by Plot may require more than 64GB of RAM.)

This function is the inverse of factorial. We can use the following relationship between the factorial function and the Gamma function:

to plot the inverse Gamma function.

plot = PlotComplex[afacc[z]+1, {z, -8, 8}, PlotPoints -> 4000];
Show[plot, ImageSize -> {4000,4000}]

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w:en:Creative Commons
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