Gallery of complex functions

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/complex/index.html

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

Initialize notebook

<<VQM`;

If you do not own the VQM packages, you can download them from http://www.uni-graz.at/imawww/vqm/software.html

01: f(z)=z

QComplexDensityPlot[
        x + I y, {x, -π, π}, {y, -π, π}, 
        PlotPoints -> 50, Mesh -> False, 
        Background -> GrayLevel[0]];

02: 1/z

QComplexDensityPlot[
        (1/(x + I y)), {x, -π, π}, {y, -π, π}, 
        PlotPoints -> 50, Mesh -> False, 
        Background -> GrayLevel[0]];

03: 1/(z^3)

QComplexDensityPlot[
        (1/(((x + I y))^3)), {x, -π, π}, {y, -π, π}, 
        PlotPoints -> 50, Mesh -> False, 
        Background -> GrayLevel[0]];

04: (1/((z^3) - 1))

QComplexDensityPlot[
        (1/((((x + I y))^3) - 1)), {x, -π, π}, {y, -π, π}, PlotPoints -> 50, Mesh -> False, 
        Background -> GrayLevel[0]];

QComplexContourPlot[
        (1/((((x + I y))^3) - 1)), {x, -π, π}, {y, -π, π}, PlotPoints -> 50, Mesh -> False, 
        Background -> GrayLevel[0], 
        Contours -> {0.05, 0.1, 0.2, 0.4, 0.8, 1.6}, 
        ContourStyle -> GrayLevel[0.9]];

05: exp

QComplexContourPlot[
        Exp[x + I y], {x, -π, π}, {y, -π, π}, 
        PlotPoints -> 50, Mesh -> False, 
        Contours -> Table[n, {n, 0, 20, 2}], 
        ContourStyle -> GrayLevel[ .5], 
        Background -> GrayLevel[0]];

QComplexPlot3D[Exp[x + I y], {x, -2, 2}, {y, -4, 4}, 
        PlotPoints -> 50, Mesh -> False, 
        Background -> GrayLevel[0]];

06: sin

\(\(QComplexContourPlot[
        Sin[x + I\ y], {x, \(-\(\(3\ π\)\/2\)\), \(3\ π\)\/2}, {y, \(-2.5\), 2.5}, PlotPoints -> 50, Mesh -> False, 
        FrameTicks -> {{{\(-π\), "-π"}, {0, "0"}, {π, "π"}}, Automatic, Automatic, Automatic}, 
        Contours -> {0.1,  .5,  .9, 1.3, 1.7, 2.1}, 
        Background -> GrayLevel[0]];\)\)

QComplexPlot3D[
        Sin[x + I y], {x, -((3 π)/2), ((3 π)/2)}, {y, -2, 2}, PlotPoints -> 50, Mesh -> False, 
        Background -> GrayLevel[0]];

07: cos

\(\(QComplexContourPlot[
        Cos[x + I\ y], {x, \(-\(\(3\ π\)\/2\)\), \(3\ π\)\/2}, {y, \(-2.5\), 2.5}, PlotPoints -> 50, Mesh -> False, 
        FrameTicks -> {{{\(-π\), "-π"}, {0, "0"}, {π, "π"}}, Automatic, Automatic, Automatic}, 
        Contours -> {0.1,  .5,  .9, 1.3, 1.7, 2.1}, 
        Background -> GrayLevel[0]];\)\)

08: tan

QComplexContourPlot[
        Tan[x + I y], {x, -(π/8), ((9 π)/8)}, {y, -((5 π)/8), ((5 π)/8)}, PlotPoints -> 50, Mesh -> False, 
        FrameTicks -> {{0, (π/2), π}, Automatic, Automatic, 
            Automatic}, Contours -> {0.2, 0.5, 0.7, 1, 1.4, 2.2, 5}, 
        Background -> GrayLevel[0]];

QComplexPlot3D[
        Tan[x + I y], {x, -2 π, 2 π}, {y, -2 π, 
          2 π}, PlotPoints -> 50, Mesh -> False, 
        Ticks -> {{-π, 0, π}, {-π, 0, π}, 
            Automatic}, PlotRange -> {0, 8}, 
        Background -> GrayLevel[0]];

09: cot

QComplexContourPlot[
        Cot[x + I y], {x, -π/8, 9  π/8}, {y, -5 π/8, 
          5  π/8}, PlotPoints -> 50, Mesh -> False, 
        FrameTicks -> {{0, π/2, π}, Automatic, Automatic, 
            Automatic}, Contours -> {0.2, 0.5, 0.7, 1, 1.4, 2.2, 5}, 
        Background -> GrayLevel[0]];

10: sinh

QComplexContourPlot[
        Sinh[x + I y], {x, -2.5, 
          2.5}, {y, -((3 π)/2), ((3 π)/2)}, 
        PlotPoints -> 50, Mesh -> False, 
        FrameTicks -> {Automatic, {-π, 0, π}, Automatic, 
            Automatic}, Contours -> {0.1,  .5,  .9, 1.3, 1.7, 2.1}, 
        Background -> GrayLevel[0]];

QComplexPlot3D[
        Sinh[x + I y], {x, -2, 
          2}, {y, -((3 π)/2), ((3 π)/2)}, 
        PlotPoints -> 50, Mesh -> False, 
        Background -> GrayLevel[0]];

11: cosh

QComplexContourPlot[
        Cosh[x + I y], {x, -2.5, 
          2.5}, {y, -((3 π)/2), ((3 π)/2)}, 
        PlotPoints -> 50, Mesh -> False, 
        FrameTicks -> {Automatic, {-π, 0, π}, Automatic, 
            Automatic}, Contours -> {0.1,  .5,  .9, 1.3, 1.7, 2.1}, 
        Background -> GrayLevel[0]];

QComplexPlot3D[
        Cosh[x + I y], {x, -2, 
          2}, {y, -((3 π)/2), ((3 π)/2)}, 
        PlotPoints -> 50, Mesh -> False, 
        Background -> GrayLevel[0]];

12: tanh

QComplexContourPlot[
        Tanh[x + 
            I y], {x, -((5 π)/8), ((5 π)/8)}, {y, -((5 π)/8), ((5 π)/8)}, PlotPoints -> 50, 
        Mesh -> False, 
        FrameTicks -> {{-(π/2), 
              0, (π/2)}, {-(π/2), 0, (π/2)}, Automatic, 
            Automatic}, Contours -> {0.2, 0.5, 0.7, 1, 1.4, 2.2, 5}, 
        Background -> GrayLevel[0]];

QComplexPlot3D[
        Tanh[x + 
            I y], {x, -((5 π)/8), ((5 π)/8)}, {y, -((5 π)/8), ((5 π)/8)}, PlotPoints -> 50, 
        Mesh -> False, 
        Ticks -> {{-(π/2), 0, (π/2)}, {-(π/2), 
              0, (π/2)}, Automatic}, PlotRange -> {0, 8}, 
        Background -> GrayLevel[0]];

13: coth

QComplexContourPlot[
        Coth[x + 
            I y], {x, -((5 π)/8), ((5 π)/8)}, {y, -((5 π)/8), ((5 π)/8)}, PlotPoints -> 50, 
        Mesh -> False, 
        FrameTicks -> {{-(π/2), 
              0, (π/2)}, {-(π/2), 0, (π/2)}, Automatic, 
            Automatic}, Contours -> {0.2, 0.5, 0.7, 1, 1.4, 2.2, 5}, 
        Background -> GrayLevel[0]];

14: sqrt

QComplexContourPlot[(Sqrt[x + I y]), {x, -π, π}, {y, -π, π}, PlotPoints -> 50, Mesh -> False, 
        ContourStyle -> GrayLevel[ .5], 
        Background -> GrayLevel[0]];

QComplexPlot3D[(Sqrt[x + I y]), {x, -π, π}, {y, -π, π}, PlotPoints -> 50, Mesh -> False, 
        Background -> GrayLevel[0]];

15: log

QComplexContourPlot[
        Log[x + I y], {x, -π, π}, {y, -π, π}, 
        PlotPoints -> 50, Mesh -> False, PlotRange -> {0, 3}, 
        ContourStyle -> GrayLevel[ .5], 
        Background -> GrayLevel[0]];

QComplexPlot3D[
        Log[x + I y], {x, -π, π}, {y, -π, π}, 
        PlotPoints -> 50, Mesh -> False, PlotRange -> {0, 5}, 
        Background -> GrayLevel[0]];

16: arcsin

QComplexContourPlot[
        ArcSin[x + I y], {x, -π, π}, {y, -π, π}, 
        PlotPoints -> 50, Mesh -> False, 
        ContourStyle -> GrayLevel[ .3], 
        Background -> GrayLevel[0]];

QComplexPlot3D[
        ArcSin[x + I y], {x, -π, π}, {y, -π, π}, 
        PlotPoints -> 50, Mesh -> False, 
        Background -> GrayLevel[0]];

17: arccos

QComplexContourPlot[
        ArcCos[x + I y], {x, -π, π}, {y, -π, π}, 
        PlotPoints -> 50, Mesh -> False, 
        ContourStyle -> GrayLevel[ .3], 
        Background -> GrayLevel[0]];

QComplexPlot3D[
        ArcCos[x + I y], {x, -π, π}, {y, -π, π}, 
        PlotPoints -> 50, Mesh -> False, 
        Background -> GrayLevel[0]];

18: arctan

QComplexContourPlot[
        ArcTan[x + I y], {x, -π, π}, {y, -π, π}, 
        PlotPoints -> 50, Mesh -> False, PlotRange -> {0, 2}, 
        ContourStyle -> GrayLevel[ .5], 
        Background -> GrayLevel[0]];

QComplexPlot3D[
        ArcTan[x + I y], {x, -π, π}, {y, -π, π}, 
        PlotPoints -> 50, Mesh -> False, 
        PlotRange -> {0, 2.8}, 
        Background -> GrayLevel[0]];

19: arccot

QComplexContourPlot[
        ArcCot[x + I y], {x, -π, π}, {y, -π, π}, 
        PlotPoints -> 50, Mesh -> False, 
        PlotRange -> {0, 1.8}, Contours -> 8, 
        ContourStyle -> GrayLevel[ .8], 
        Background -> GrayLevel[0]];

QComplexPlot3D[
        ArcCot[x + I y], {x, -π, π}, {y, -π, π}, 
        PlotPoints -> 50, Mesh -> False, PlotRange -> {0, 3}, 
        Background -> GrayLevel[0]];

20: arcsinh

QComplexContourPlot[
        ArcSinh[x + I y], {x, -π, π}, {y, -π, π}, 
        PlotPoints -> 50, Mesh -> False, 
        ContourStyle -> GrayLevel[ .3], 
        Background -> GrayLevel[0]];

QComplexPlot3D[
        ArcSinh[x + I y], {x, -π, π}, {y, -π, π}, 
        PlotPoints -> 50, Mesh -> False, 
        Background -> GrayLevel[0]];

21: arccosh

QComplexContourPlot[
        ArcCosh[x + I y], {x, -π, π}, {y, -π, π}, 
        PlotPoints -> 50, Mesh -> False, 
        ContourStyle -> GrayLevel[ .3], 
        Background -> GrayLevel[0]];

QComplexPlot3D[
        ArcCosh[x + I y], {x, -π, π}, {y, -π, π}, 
        PlotPoints -> 50, Mesh -> False, 
        Background -> GrayLevel[0]];

22: arctanh

QComplexContourPlot[
        ArcTanh[x + I y], {x, -π, π}, {y, -π, π}, 
        PlotPoints -> 50, Mesh -> False, PlotRange -> {0, 2}, 
        ContourStyle -> GrayLevel[ .5], 
        Background -> GrayLevel[0]];

QComplexPlot3D[
        ArcTanh[x + I y], {x, -π, π}, {y, -π, π}, 
        PlotPoints -> 50, Mesh -> False, PlotRange -> {0, 3}, 
        Background -> GrayLevel[0]];

23: arccoth

QComplexContourPlot[
        ArcCoth[x + I y], {x, -π, π}, {y, -π, π}, 
        PlotPoints -> 50, Mesh -> False, 
        PlotRange -> {0, 1.8}, Contours -> 8, 
        ContourStyle -> GrayLevel[ .8], 
        Background -> GrayLevel[0]];