Gallery of mathematical images

Author

Bernd Thaller
Institute for Mathematics and Scientific Computing
University of Graz
Austria

Source

http://www.uni-graz.at/imawww/vqm/

Abstract

Mathematica source code for the images at  http://www.uni-graz.at/imawww/vqm/pages/colorgallery/index.html

In this notebook, the value of the option PlotPoints has been reduced for performance reasons.

Load package

<< VQM`ComplexPlot`;

01: alien1

Remove[myColorMap];

myColorMap = 
        Function[{ab, ar}, 
          Hue[Sin[ab], (1/(1 + (ar^2))), (1/(1 + (ar^2)))]];

QComplexDensityPlot[
        Exp[Tan[y + I x]], {x, -(π/3), (π/3)}, {y, 0, π}, 
        QComplexToColorMap -> myColorMap, PlotPoints -> 100, 
        Mesh -> False, Frame -> False];

02: alien2

Remove[myColorMap];

myColorMap = 
      Function[{ab, ar}, Hue[Sin[ab], (1/(1 + (ar^2))), (1/(1 + (ar^2)))]]; 
    QComplexDensityPlot[
      Zeta[Zeta[y + I x]], {x, - .5,  .5}, {y,  .5, 1.2}, 
      QComplexToColorMap -> myColorMap, PlotPoints -> 100, 
      Mesh -> False, Frame -> False];

03: alien3

Remove[myColorMap];

myColorMap = 
        Function[{ab, ar}, 
          Hue[Sin[ab], (1/(1 + (ar^2))), (1/(1 + (ar^2)))]];

QComplexDensityPlot[
        Exp[x + I y]^Tan[y + I x], {x, -(π/8), (π/4)}, {y, π + 0.0001, ((15 π)/8)}, QComplexToColorMap -> myColorMap, 
        PlotPoints -> 100, Mesh -> False, 
        Frame -> False];

04: alien4

Remove[myColorMap];

myColorMap = 
        Function[{ab, ar}, 
          Hue[Sin[(Sqrt[ab])], (1/(1 + (ar^2))), (1/(1 + (ar^2)))]];

QComplexDensityPlot[
        Exp[x + I y]^Tan[y + I x], {x, -(π/9), ((5 π)/12)}, {y, π + 0.0001, ((14 π)/8)}, 
        QComplexToColorMap -> myColorMap, PlotPoints -> 100, 
        Mesh -> False, Frame -> False];

05: butterfly

Remove[myColorMap];

myColorMap = 
        Function[{ab, ar}, 
          Hue[(1/(1 + (ab^2))), (1/(1 + (ar^2))), 
            (1/(1 + (ar^2)))]];

QComplexDensityPlot[
        Sin[Sin[(1/(y + I x))]], {x, -1, 1}, {y, - .5,  .5}, 
        QComplexToColorMap -> myColorMap, PlotPoints -> 100, 
        Mesh -> False, Frame -> False];

06: crossing1

Remove[myColorMap];

myColorMap = 
        Function[{ab, ar}, 
          Hue[(1/(1 + (ab^2))), (1/(1 + (ar^2))), 
            (1/(1 + (ar^2)))]];

QComplexDensityPlot[
        Cot[x + I y]^Tan[y + I x], {x, -((5 π)/4), 
          3 π - ((5 π)/4)}, {y, -π, 2 π}, 
        QComplexToColorMap -> myColorMap, PlotPoints -> 100, 
        Mesh -> False, Frame -> False];

07: crossing2

Remove[myColorMap];

myColorMap = 
        Function[{ab, ar}, 
          Hue[Sin[ab], (1/(1 + (ar^2))), (1/(1 + (ar^2)))]];

QComplexDensityPlot[
        (E^(Tan[x + I y] Tan[y + I x])), {x, -2 π, 
          2 π}, {y, -2 π, π}, 
        QComplexToColorMap -> myColorMap, PlotPoints -> 100, 
        Mesh -> False, Frame -> False];

08: crossing3

QComplexDensityPlot[
        (E^(Tan[x + I y] Tan[y + I x])), {x, -2 π, 
          2 π}, {y, -2 π, π}, PlotPoints -> 100, 
        Mesh -> False, Frame -> False];

09: flames1

Remove[myColorMap];

myColorMap = 
        Function[{ab, ar}, 
          Hue[(1/(1 + (ab^2))), (1/(1 + (ar^2))), 
            1 -  .(8/(1 + (ab^2)))]];

QComplexDensityPlot[
        Sin[Sin[Tan[-y + I x]]], {x, -2, 2}, {y, -2.5, 1}, 
        QComplexToColorMap -> myColorMap, PlotPoints -> 100, 
        Mesh -> False, Frame -> False];

10: flames2

Remove[myColorMap];

myColorMap = 
        Function[{ab, ar}, 
          Hue[(1/(1 + (ab^2))), (1/(1 + (ar^2))), 
            1 -  .(8/(1 + (ab^2)))]];

QComplexDensityPlot[
        Sin[Sin[Log[-y + I x]]], {x, -20, 20}, {y, -20, 20}, 
        QComplexToColorMap -> myColorMap, PlotPoints -> 100, 
        Mesh -> False, Frame -> False];

11: flowers1

Remove[myColorMap];

myColorMap = 
        Function[{ab, ar}, 
          Hue[Cos[ar]^4, Sin[ar]^2, Cos[ab]^2]];

QComplexDensityPlot[
        (1/(((1 - (1/(((1 - Sin[x + I y]))^2))))^2)), {x, -(π/2), ((3 π)/2)}, {y, -((3 π)/4), ((3 π)/4)}, 
        QComplexToColorMap -> myColorMap, PlotPoints -> 100, 
        Mesh -> False, Frame -> False];

12: flowers2

Remove[myColorMap];

myColorMap = 
        Function[{ab, ar}, 
          Hue[(1/(1 + (ar^2))), (1/(1 + (ab^2))), 
            (1/(1 + (ab^2)))]];

QComplexDensityPlot[
        (1/(((1 - (1/(((1 - Sin[x + I y]))^2))))^2)), {x, -(π/2), ((11 π)/2)}, {y, -π, π}, 
        QComplexToColorMap -> myColorMap, PlotPoints -> 100, 
        Mesh -> False, Frame -> False];

13: flowers3

Remove[myColorMap];

myColorMap = 
        Function[{ab, ar}, 
          Hue[Mod[ar, 1], Mod[ab, 1], Mod[ar, 1]]];

QComplexDensityPlot[
        (1/(((1 - (1/(((1 - Sin[x + I y]))^2))))^2)), {x, -1.5, 
          2.5}, {y, -1.5, 2.5}, QComplexToColorMap -> myColorMap, 
        PlotPoints -> 100, Mesh -> False, 
        Frame -> False];

14: flowers4

Remove[myColorMap];

myColorMap = 
        Function[{ab, ar}, 
          Hue[Mod[ab, 1], Mod[ab, 1], Mod[ar, 1]]];

QComplexDensityPlot[
        (1/(((1 - (1/(((1 - Sin[x + I y]))^2))))^2)), {x, -2, 
          6}, {y, -2.2, 3.3}, QComplexToColorMap -> myColorMap, 
        PlotPoints -> 100, Mesh -> False, 
        Frame -> False];

15: flowers5

Remove[myColorMap];

myColorMap = 
        Function[{ab, ar}, 
          RGBColor[ .(6/(1 + (ar^2))) +  .(4/(1 + (ab^2))),  .(4/(1 + (ar^2))) +  .(6/(1 + (ab^2))),  .(1/(1 + (ar^2))) +  .(9/(1 + (ab^2)))]];

QComplexDensityPlot[
        Sin[(1/(((1 - (1/(((1 - Sin[x + I y]))^2))))^2))]^2, {x, -(π/2), ((3 π)/2)}, {y, -((3 π)/4), ((3 π)/4)}, 
        QComplexToColorMap -> myColorMap, PlotPoints -> 100, 
        Mesh -> False, Frame -> False];

16: flowers6

Remove[myColorMap];

myColorMap[ab_, ar_] := 
        Hue[(1/(1 + (ab^2))), Sin[Log[ab]]^2, 
          (1/(1 + (ar^2)))];

QComplexDensityPlot[
        Log[Sin[(1/(((y + I x))^5))]], {x, -1.5, 1.5}, {y, -1.5, 
          1.5}, QComplexToColorMap -> myColorMap, 
        PlotPoints -> 100, Mesh -> False, 
        Frame -> False];

17: flowers7

Remove[myColorMap];

myColorMap[ab_, ar_] := 
        Hue[(1/(1 + (ab^2))), 
          Sin[Log[ab]]^2,  .(3/(1 + (ar^2))) +  .7 Cos[Log[Abs[ar]]]^2];

QComplexDensityPlot[
        Log[Sin[Log[(((x + I y))^7)]]], {x, -π, π}, {y, -π, π}, QComplexToColorMap -> myColorMap, 
        PlotPoints -> 100, Mesh -> False, 
        Frame -> False];

18: flowers8

Remove[myColorMap];

myColorMap[ab_, ar_] := 
        Hue[(1/(1 + (ab^2))), 
          Sin[Log[ab]]^2,  .(1/(1 + (ar^2))) +  .9 Cos[Log[Abs[ar]]]^2];

QComplexDensityPlot[
        Log[Sin[(1/(((x + I y))^3))]], {x, -(π/2), (π/2)}, {y, -(π/2), (π/2)}, QComplexToColorMap -> myColorMap, 
        PlotPoints -> 100, Mesh -> False, 
        Frame -> False];

19: flowers9

Remove[myColorMap];

myColorMap[ab_, ar_] := 
        Hue[(1/(1 + (ab^2))), 
          Sin[Log[ab]]^2,  .(1/(1 + (ar^2))) +  .9 Cos[Log[Abs[ar]]]^2];

QComplexDensityPlot[
        Log[Sin[(1/(((x + I y))^11))]], {x, -(π/2), (π/2)}, {y, -(π/2), (π/2)}, QComplexToColorMap -> myColorMap, 
        PlotPoints -> 100, Mesh -> False, 
        Frame -> False];

20: fractal1

Remove[mandelbrot];

mandelbrot = 
        Compile[{{k, _Complex}}, 
          Module[{val = 0 + 0 I, cnt = 0}, 
            While[Re[val]^2 + Im[val]^2 < 15000 && cnt < 100, 
              val = (val^2) + k; cnt++]; 
            cnt Exp[I Arg[val]]]];

Timing[
      QComplexDensityPlot[
        mandelbrot[x + I y], {x, -2.6, 1.4}, {y, -2, 2}, 
        PlotPoints -> 100, Mesh -> False, Compiled -> False, 
        QSphereRadius -> 10, 
        QLightnessRange -> {0, 0.9}]]

21: fractal2

Remove[mandelbrot];

mandelbrot = 
        Compile[{{k, _Complex}}, 
          Module[{val = 0 + 0 I, cnt = 0}, 
            While[Re[val]^2 + Im[val]^2 < 15000 && cnt < 100, 
              val = (val^2) + k; cnt++]; 
            cnt Exp[I Arg[val]]]];

QComplexDensityPlot[
        mandelbrot[x + I y], {x, -1.49, -1.47}, {y, -0.01, 
          0.01}, Compiled -> False, PlotPoints -> 100, 
        Mesh -> False, QSphereRadius -> 50, 
        QValueRange -> {15, 110}];

22: kitchen1

Remove[f, g, myColorMap];

f[x_] := 1 - (x^2);

g = Compile[{{x, _Complex}}, Simplify[Nest[f, x, 5]]];

myColorMap = 
        Function[{ab, ar}, 
          RGBColor[Mod[Log[ab + 1. ], 1], Mod[Log[Log[ab + 1. ]], 1], 
            Mod[ .8 ar, 1]]];

QComplexDensityPlot[
        g[x + I y], {x, -1.4, 1.4}, {y, - .8,  .8}, 
        Compiled -> False, QComplexToColorMap -> myColorMap, 
        PlotPoints -> 100, Mesh -> False, 
        Frame -> False];

23: kitchen2

Remove[f, g, h, myColorMap];

f[x_, y_] = (1/((((x + I y))^5) - 1));

g[x_, y_] = Simplify[f[f[x, y], (1/f)[x, y]]];

h = Compile[{{x, _Complex}, y}, g[x, y]];

myColorMap[ab_, ar_] = 
        Hue[5 ab, (1/2) (1 + Sin[(ar/π)]), 
          (1/2) (1 + Cos[(ar/π)])];

QComplexDensityPlot[h[x, y], {x, -1.4, 1.4}, {y, -1.4, 1.4}, 
        Compiled -> False, QComplexToColorMap -> myColorMap, 
        PlotPoints -> 100, Mesh -> False, 
        Frame -> False];

24: quantum1

Remove[Gauss, p0, f, doplot];

Gauss[x_, t_, a_, x0_, 
          p0_] := (((Sqrt[(a/π)]) Exp[-((((a x . x)/2) - I p0 . x + (1/2) I t p0 . p0)/(1 + I t a))])/(1 + I t a));

p0 = {2, 2. };

f[x_, y_, t_] := Gauss[{x, y}, t, (1/2), x0, p0];

doplot[t_, opts___] := 
        QComplexDensityPlot[
          Exp[I 10 Sin[3 x] Cos[3 y]] f[x, y, t], {x, -5.5, 
            5.5}, {y, -5.5, 5.5}, opts, Mesh -> False, 
          Frame -> False, QSphereRadius ->  .5, 
          PlotPoints -> 100];

doplot[- .8, QLightnessRange -> { .4,  .4}];

25: quantum2

Remove[Gauss, p0, f, myColorMap, doplot];

Gauss[x_, t_, a_, x0_, 
          p0_] := (((Sqrt[(a/π)]) Exp[-((((a x . x)/2) - I p0 . x + (1/2) I t p0 . p0)/(1 + I t a))])/(1 + I t a));

p0 = {2, 2. };

f[x_, y_, t_] := Gauss[{x, y}, t, (1/2), x0, p0];

myColorMap = 
        Function[{ab, ar}, 
          Hue[Sin[1 - ab]/(1 + (ar^2)), (1/(1 + (ab^2))), 
            (1/(1 + (ar^2)))]];

doplot[t_] := 
        QComplexDensityPlot[
          Exp[I 12 Sin[3 x] Cos[3 y]] f[x, y, t], {x, -4, 
            3}, {y, -4, 3}, Mesh -> False, Frame -> False, 
          QComplexToColorMap -> myColorMap, 
          PlotPoints -> 100];

doplot[- .3];

26: satin1

Remove[myColorMap];

myColorMap = 
        Function[{ab, ar}, 
          Hue[Sin[ab], 1 -  .(8/(1 + (ar^2))), 
            (1/(1 + (ar^2)))]];

QComplexDensityPlot[
        LegendreP[5, LegendreP[5, x + I y]], {x, -1, 
          1}, {y, - .5,  .5}, QComplexToColorMap -> myColorMap, 
        PlotPoints -> 100, Mesh -> False, 
        Frame -> False];

27: satin2

Remove[myColorMap];

myColorMap = 
        Function[{ab, ar}, 
          Hue[Cos[ab +  .7],  .(5/(1 + (ar^2))) +  .(5/(1 + (ab^2))),  .(7/(1 + (ar^2))) +  .(3/(1 + (ab^2)))]];

QComplexDensityPlot[
        LegendreP[5, LegendreP[4, x + I y]], {x, -1, 
          1}, {y, - .3,  .3}, QComplexToColorMap -> myColorMap, 
        PlotPoints -> 100, Mesh -> False, 
        Frame -> False];

28: satin3

Remove[myColorMap];

myColorMap = 
        Function[{ab, ar}, 
          Hue[1 - (1/(1 + (ab^2))), 1 -  .(8/(1 + (ar^2))), 
            (1/(1 + (ar^2)))]];

QComplexDensityPlot[
        Zeta[Zeta[y + I x]], {x, - .15,  .15}, {y,  .84, 1.01}, 
        QComplexToColorMap -> myColorMap, PlotPoints -> 100, 
        Mesh -> False, Frame -> False];

29: sinsin1

Remove[myColorMap];

myColorMap = 
        Function[{ab, ar}, 
          Hue[Sin[ar]^2, Mod[ab, 1], Mod[ar, 1]]];

QComplexDensityPlot[
        Sin[Sin[x + I y]], {x, -π - 0.6, 
          0.6}, {y, -(π/2) - 0.6, (π/2) + 0.6}, 
        QComplexToColorMap -> myColorMap, PlotPoints -> 100, 
        Mesh -> False, Frame -> False];

30: sinsin2

Remove[myColorMap];

myColorMap = 
        Function[{ab, ar}, 
          Hue[Mod[ab, 1], Mod[ar, 1], Mod[ar, 1]]];

QComplexDensityPlot[
        Sin[Sin[x + I y]], {x, -π, π}, {y, -π, π}, 
        QComplexToColorMap -> myColorMap, PlotPoints -> 100, 
        Mesh -> False, Frame -> False];

31: sinsin3

Remove[myColorMap];

myColorMap = 
        Function[{ab, ar}, 
          RGBColor[
            Cos[ab]^2,  .(5/(1 + (ar^2))) +  .(5/(1 + (ab^2))),  .(7/(1 + (ar^2))) +  .(3/(1 + (ab^2)))]];

QComplexDensityPlot[
        Sin[2 Sin[
              2 (x + I y)]], {x, -(π/2), π}, {y, -1, 
          1}, QComplexToColorMap -> myColorMap, PlotPoints -> 100, 
        Mesh -> False, Frame -> False];

32: spider1

Remove[myColorMap];

myColorMap = 
        Function[{ab, ar}, 
          Hue[(1/((ab^2) + 1)), (1/((ar^2) + 1)), 
            (1/((ar^2) + 1))]];

QComplexDensityPlot[
        Sin[x + I y]^Sin[x + I y], {x, -(π/2), ((3 π)/2)}, {y, -π, π}, QComplexToColorMap -> myColorMap, 
        PlotPoints -> 100, Mesh -> False, 
        Frame -> False];

33: spider2

Remove[myColorMap];

myColorMap = 
        Function[{ab, ar}, 
          RGBColor[ .(8/(1 + (ar^2))) +  .(2/(1 + (ab^2))),  .(2/(1 + (ar^2))) +  .(8/(1 + (ab^2))),  .(5/(1 + (ar^2))) +  .(5/(1 + (ab^2)))]];

QComplexDensityPlot[
        Sin[2 Sin[
              2 (x + I y)]], {x, -(π/2), π}, {y, -2, 
          2}, QComplexToColorMap -> myColorMap, PlotPoints -> 100, 
        Mesh -> False, Frame -> False];

34: spider3

Remove[myColorMap];

myColorMap[ab_, ar_] := 
        Hue[(1/(1 + (ab^2))), 
          Sin[Log[ab]]^2,  .(1/(1 + (ar^2))) +  .9 Cos[Log[Abs[ar]]]^2];

QComplexDensityPlot[
        Log[Log[Log[
              Sin[(1/(x + I y))]]]], {y, - .3,  .3}, {x, - .2,  .2}, QComplexToColorMap -> myColorMap, PlotPoints -> 100, 
        Mesh -> False, Frame -> False];

35: spider4

Remove[myColorMap];

myColorMap[ab_, ar_] := 
        Hue[(1/(1 + (ab^2))), 
          Sin[Log[ab]]^2,  .(1/(1 + (ar^2))) +  .9 Cos[Log[Abs[ar]]]^2];

QComplexDensityPlot[
        Log[Log[Log[
              Sin[(1/(x + I y))]]]], {y, - .5,  .5}, {x, - .3,  .7}, QComplexToColorMap -> myColorMap, PlotPoints -> 100, 
        Mesh -> False, Frame -> False];

36: spider5

Remove[myColorMap];

myColorMap[ab_, ar_] := 
        Hue[ar,  .8 Sin[Log[ab]]^2 +  .2 Cos[Log[ab]]^2,  .(1/(1 + (ab^2))) +  .9 Cos[Log[Abs[ar]]]^2];

QComplexDensityPlot[
        Log[Log[Log[
              Sin[(1/(x + I y))]]]], {y, - .5,  .5}, {x, - .3,  .7}, QComplexToColorMap -> myColorMap, PlotPoints -> 100, 
        Mesh -> False, Frame -> False];

37: spider6

Remove[myColorMap];

myColorMap[ab_, ar_] := 
        Hue[Sin[ar], Sin[ab]^2, Cos[ar]^2];

QComplexDensityPlot[
        Log[Log[Log[
              Sin[(1/(x + I y))]]]], {y, - .5,  .5}, {x, - .3,  .4}, QComplexToColorMap -> myColorMap, PlotPoints -> 100, 
        Mesh -> False, Frame -> False];

38: tantan1

Remove[myColorMap];

myColorMap = 
        Function[{ab, ar}, 
          Hue[Mod[ab, 1], Mod[ab, 1], 
            1.  - (3/4) Mod[(ar/2), 1]]];

QComplexDensityPlot[
        Tan[Tan[x + 
              I y]], {x, -(π/2), π}, {y, -(π/2), π}, QComplexToColorMap -> myColorMap, PlotPoints -> 100, 
        Mesh -> False, Frame -> False];

39: tantan2

Remove[myColorMap];

myColorMap = 
        Function[{ab, ar}, 
          Hue[1 - ab, Mod[ab, 1], 
            1.  - (1/4) 3.5 Mod[(ar/2), 1]]];

QComplexDensityPlot[
        Tan[Tan[x + I y]], {x, (π/8), π + 
            0.01}, {y, -(π/3), (π/2)}, 
        QComplexToColorMap -> myColorMap, PlotPoints -> 100, 
        Mesh -> False, Frame -> False];

40 tantan3

Remove[myColorMap];

myColorMap = 
        Function[{ab, ar}, 
          Hue[1 - ab, Mod[ab 2, 1], 
            1.  - (1/4) 3.5 Mod[(ar/2), 1]]];

QComplexDensityPlot[
        Tan[Tan[x + 
              I y]], {x, (π/2), (π/2) + (π/8)}, {y, -(π/16), (π/16)}, QComplexToColorMap -> myColorMap, 
        PlotPoints -> 100, Mesh -> False, 
        Frame -> False];

41 tantan4

Remove[myColorMap];

myColorMap = 
        Function[{ab, ar}, 
          Hue[Cos[ab]^2, Mod[ab 3, 1], 
            1.  - (3/4) Mod[ar 2, 1]]];

QComplexDensityPlot[
        Tan[Tan[x + 
              I y]], {x, (π/2), π}, {y, -(π/5), (π/5)}, 
        QComplexToColorMap -> myColorMap, PlotPoints -> 100, 
        Mesh -> False, Frame -> False];

42 waves1

Remove[myColorMap];

myColorMap = 
        Function[{ab, ar}, 
          Hue[(1/(1 + (ab^2))), (1/(1 + (ar^2))), 
            (1/(1 + (ar^2)))]];

QComplexDensityPlot[
        Sin[x + I y]^Tan[y + I x], {x, -(π/2), π}, {y, -2 π, 6 π}, QComplexToColorMap -> myColorMap, 
        PlotPoints -> 100, Mesh -> False, 
        Frame -> False];

43: logsine

Remove[myColorMap];

\(\(myColorMap[ab_, ar_] := 
		Hue[ .52 + 1/\((1 + ab^2)\), 
          Sin[Log[ab]]^2, 1/\((1 + ar^2)\)];\)\)

QComplexDensityPlot[Log[ Sin[Exp[y+I x]]], {x,-2.5,4.5}, {y,-0.5,4},
QComplexToColorMap->myColorMap, PlotPoints->100, Mesh->False, Frame->False];

44: etantan

Remove[myColorMap];

myColorMap[ab_, ar_] := 
        Hue[ .5 + (1/(1 + (ab^2))), 
          Sin[Log[ab]]^2,  .(1/(1 + (ar^2))) +  .9 Cos[Log[Abs[ar]]]^2];

QComplexDensityPlot[
        (E^(Tan[x + I y] Tan[y + I x])),  {x, - π, 
          2 π},  {y, -2 π, π},  
        QComplexToColorMap -> myColorMap,  PlotPoints -> 100, 
        Mesh -> False,  Frame -> False];

45: etan

Remove[myColorMap];

myColorMap[ab_, ar_] := 
        Hue[ .65 + (1/(1 + (ab^2))), 
          Sin[Log[ab]]^2,  .(1/(1 + (ar^2))) +  .9 Cos[Log[Abs[ar]]]^2];

QComplexDensityPlot[
        Exp[x + I y]^Tan[y + I x], {x, -(π/3), (π/3)}, {y, π + 0.0001, ((13 π)/8)}, QComplexToColorMap -> myColorMap, 
        PlotPoints -> 100, Mesh -> False, 
        Frame -> False];

46: flowder1

Remove[myColorMap];

myColorMap[ab_, ar_] := 
        Hue[Cos[Log[Abs[ar]]]^2 + (1/(1 + (ab^2))), Sin[Log[ab]]^2, 
          Cos[Log[Abs[ar]]]^2];

QComplexDensityPlot[
        (1/(((1 - (1/((1 - 1/(((1 + Sin[x + I y]))^2))))^2))^2)), {x, -((11  π)/8), ((3  π)/8)}, {y, -((7 π)/8), ((7 π)/8)}, QComplexToColorMap -> myColorMap, 
        PlotPoints -> 100, Mesh -> False, 
        Frame -> False];

47: sin

Remove[myColorMap];

myColorMap[ab_, ar_] := 
        Hue[ .85 + Cos[ab]^2 + Sin[ar], Cos[Log[Abs[ar]]]^2, 
          Cos[ar]^2/2 + Sin[Log[Cos[ar + ab]^2]]^2/2];

QComplexDensityPlot[
        Sin[x + I y], {x, -Pi/2, Pi}, {y, -2, 2}, 
        QComplexToColorMap -> myColorMap, PlotPoints -> 100, 
        Mesh -> False, Frame -> False];

48: xmas1

Remove[myColorMap];

myColorMap[ab_, ar_] := 
        Hue[ .85 + Cos[ab]^2 + Sin[ar], Cos[Log[Abs[ar]]]^2, 
          Cos[ar]^2/2 + Sin[Log[Cos[ar + ab]^2]]^2/2];

QComplexDensityPlot[
        (1/(((1 - (1/((1 - 1/(((1 + Sin[x + I y]))^2))))^2))^2)), {x, -π -  .2, 0.2}, {y, -(π/2) -  .2, (π/2) +  .2}, 
        QComplexToColorMap -> myColorMap, PlotPoints -> 100, 
        Mesh -> False, Frame -> False];