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Re: Complex diagram

• To: mathgroup at smc.vnet.net
• Subject: [mg123521] Re: Complex diagram
• From: DrMajorBob <btreat1 at austin.rr.com>
• Date: Sat, 10 Dec 2011 07:27:38 -0500 (EST)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <jbni9o$48v$1@smc.vnet.net> <201112091056.FAA03855@smc.vnet.net>
• Reply-to: drmajorbob at yahoo.com

I would use Set, not SetDelayed, and pre-compute Arg and Abs where  
possible:

Clear[f, abs, arg, simple]
simple[expr_] :=
   ComplexExpand@FullSimplify[expr, {-Pi < x < Pi, -Pi > y < Pi}];
f[1][x_, y_] = x + I y;
f[2][x_, y_] = 3 x + 2 y I;
f[3][x_, y_] = Cos[x + I y] // simple;
f[4][x_, y_] = Sin[x + I y] // simple;
f[5][x_, y_] = (x + I y)^2 // simple;
f[6][x_, y_] = (3 x + 2 y I)^2 // simple;
f[7][x_, y_] = Cosh[x + I y] // simple;
f[8][x_, y_] = Sinh[x + I y] // simple;
Do[abs[k][x_, y_] = f[k][x, y] // Abs // simple;
  arg[k][x_, y_] = f[k][x, y] // Arg // simple, {k, 1, 8}]

GraphicsGrid[
  Partition[
   Table[Plot3D[abs[k][x, y], {x, -\[Pi], \[Pi]}, {y, -\[Pi], \[Pi]},
     AxesLabel -> {"x", "y", "z"},
     PlotRange -> {{-\[Pi], \[Pi]}, {-\[Pi], \[Pi]}, {0, 2 \[Pi]}},
     PlotPoints -> 80, MaxRecursion -> 2, BoxRatios -> 1, Mesh -> 11,
     MeshFunctions -> {({x, y} \[Function]
         abs[k][x, y]), ({x, y} \[Function] arg[k][x, y])},
     ColorFunctionScaling -> False,
     ColorFunction -> ({x, y} \[Function]
        Hue[\[LeftFloor]12 (arg[k][x,
                y] + \[Pi])/(2 \[Pi])\[RightFloor]/12, 0.5, 1])], {k,
     1, 8}], 2], ImageSize -> 72*16]

I can't say the speed improvement is tremendous, however!

Bobby

On Fri, 09 Dec 2011 04:56:35 -0600, Chris Young <cy56 at comcast.net> wrote:

> On 2011-12-07 11:22:32 +0000, é�â?? 厚 said:
>
>> Can we draw complex funtions's diagram in Mathematica?
>>
>> For example, draw the picture of z=2*x+I*3*y
>
> Finally got something fast and sharp for complex functions, using
> Plot3D and a stepped color function. The trick is to remember that the
> number of mesh lines is one less than the number of contours. Only then
> will the colors be correspond to the mesh, and, for some reason,
> they're more sharply discretized.
>
> http://home.comcast.net/~cy56/Sharp3DComplexPlots.nb
> http://home.comcast.net/~cy56/Sharp3DComplexPlots.png
>
> f[1][z_] := z;
> f[2][z_] := 3 Re[z] + 2 Im[z] I;
> f[3][z_] := Cos[z];
> f[4][z_] := Sin[z];
>
> f[5][z_] := z^2;
> f[6][z_] := (3 Re[z] + 2 Im[z] I)^2;
> f[7][z_] := Cosh[z];
> f[8][z_] := Sinh[z];
>
> GraphicsGrid[
>  Partition[
>   Table[
>    Plot3D[
>     Abs[f[k][x + y I]],
>
>     {x, -\[Pi], \[Pi]}, {y, -\[Pi], \[Pi]},
>
>     AxesLabel -> {"x", "y", "z"},
>     PlotRange -> {{-\[Pi], \[Pi]}, {-\[Pi], \[Pi]}, {0, 2 \[Pi]}},
>     PlotPoints -> 80,
>     MaxRecursion -> 2,
>     BoxRatios -> 1,
>     Mesh -> 11,
>     MeshFunctions -> {
>       ( {x, y} \[Function] Abs[f[k][x + y I]]),
>       ( {x, y} \[Function] Arg[f[k][x + y I]])
>       },
>     ColorFunctionScaling -> False,
>     ColorFunction -> ({x, y} \[Function]
>        Hue[\[LeftFloor]12 (Arg[f[k][x + y I]] + \[Pi])/(
>           2 \[Pi])\[RightFloor]/12, 0.5, 1])
>     ],
>    {k, 1, 8}
>    ],
>   4
>   ],
>  ImageSize -> 72*16
>  ]
>
>
> For 2D, ContourPlot works best, with the modulus contours drawn in via
> a mesh function. Only one problem, the Sinh function is coming in at
> lower resolution than the others for some reason.
>
> http://home.comcast.net/~cy56/Sharp2DComplexPlots.nb
> http://home.comcast.net/~cy56/Sharp2DComplexPlots.png
>
> f[1][z_] := z;
> f[2][z_] := 3 Re[z] + 2 Im[z] I;
> f[3][z_] := Cos[z];
> f[4][z_] := Sin[z];
>
> f[5][z_] := z^2;
> f[6][z_] := (3 Re[z] + 2 Im[z] I)^2;
> f[7][z_] := Cosh[z];
> f[8][z_] := Sinh[z];
>
> GraphicsGrid[
>  Partition[
>   Table[
>    ContourPlot[
>     Arg[f[k][x + y I]],  {x, -\[Pi], \[Pi]}, {y, -\[Pi], \[Pi]},
>
>     Contours -> 11,
>
>     ColorFunctionScaling -> False,
>     ColorFunction -> (arg \[Function]
>        Hue[0.9 (arg + \[Pi])/(2 \[Pi]), 0.5, 1]),
>
>     Mesh -> 11,
>     MeshFunctions -> ( {x, y} \[Function] Abs[f[k][x + y I]]),
>
>     PlotRange -> {{-\[Pi], \[Pi]}, {-\[Pi], \[Pi]}},
>     PlotPoints -> 40,
>     MaxRecursion -> 2,
>     BoxRatios -> 1
>     ],
>    {k, 1, 8}
>    ],
>   4
>   ],
>  ImageSize -> 72*16
>  ]
>
>


-- 
DrMajorBob at yahoo.com

• References:
  • Re: Complex diagram
    • From: Chris Young <cy56@comcast.net>