Math 421  •  Fall 2006

Exponential and Log functions visualized

Copyright © 2004–2006 by Murray Eisenberg.  All rights reserved.

Initializations

Load David Park's Cardano3 application packages:

Needs["Cardano3`ComplexGraphics`"]
Needs["Cardano3`ComplexRoutines`"]

Turn off some annoying warning messages that might otherwise appear about similarly-named objects.  (If you want to see these messages so as to be sure to avoid harm, evaluate expressions like the ones below but replace Off there by On.)

Off[General::spell]
Off[General::spell1]

The exponential function

How exp maps vertical and horizontal lines

First look at how exp maps the its fundamental period set {z  :  -π  <  
            Im(z)  ≤  π} onto the punctured complex plane:

grid  =  {{{LightBlue},   {-8,  8,  
              96},  {-π,  π,  
              96}}};
TwoPanelPlot[
{{# &},  
          CartesianGrid,  {0,  
            8},  {grid},   {}}, 
{{Exp[#] &},  
          CartesianGrid,  {0,  20},  {grid},  {}}, 
 
        ImageSize  ->  9.5*72];

Next, use different colors for the horizontal and vertical lines in the grid (for printing on a black-only printer, use different Thickness or Dashing instead of different colors):

grids  =  {{{Blue},   {-8,  8,  
              36},  {-π,  π,  
              0}}, 
{{VenetianRed},   {-8,  8,  
              0},  {-π,  π,  
              36}}};
TwoPanelPlot[
{{# &},  
          CartesianGrid,  {0,  
            8},  {grids},   {}}, 
{{Exp[#] &},  
          CartesianGrid,  {0,  20},  {grids},  {}}, 
 
        ImageSize  ->  9.5*72];

Now look at the image under exp of one vertical line segment.  (The option PlotPoints -> 100 was used in the ComplexCurve construction for the vertical segment in order to make the image circle smooth and not jagged.)

grid  =  {{{LightBlue},   {-5,  5,  16},  {-5,  5,  
              16}}};
vertical  =  {CobaltGreen,  Thickness[0.012],  
          ComplexCurve[(1  -  
                    s) (2.33  -  π I)  +  
              s (2.33  +  π I),  {s,  0,  
              1}, 
PlotPoints  ->  
              100]};
pts  =  {PointSize[0.025],  Sepia,  
          ComplexPoint[2.33  -  (π/2) I],  
          DarkOrchid,  ComplexPoint[2.33],  GeraniumLake,  
          ComplexPoint[2.33  +  (π/2) I],  
          CadmiumOrange,  
          ComplexPoint[
            2.33  +  π I] };
TwoPanelPlot[
{{# &},  
          CartesianGrid,  {0,  4},  {grid},   {vertical, 
            pts}}, 
{{Exp[#] &},  
          CartesianGrid,  {0,  12},  {grid},  {vertical,  
            pts}}, 
 
        ImageSize  ->  9.5*72];

Next look at the image under exp of one horizontal line segment.

grid  =  {{{LightBlue},   {-5,  5,  16},  {-5,  5,  
              16}}};
horizontal  =  {CobaltGreen,  Thickness[0.012],  
          ComplexCurve[(1  -  s) (-2 π   +  
                    2.33 I)  +  
              s (2  π  +  2.33 I),  {s,  0,  
              1}]};
pts  =  {PointSize[0.025],  Sepia,  
          ComplexPoint[-1.25 π  +  2.33  I],  
          DarkOrchid,  ComplexPoint[2.33  I],  GeraniumLake,  
          ComplexPoint[1.5  +  2.33  I ],  CadmiumOrange,  
          ComplexPoint[
            2.5 +  2.33 I] };
TwoPanelPlot[
{{# &},  
          CartesianGrid,  {0,  4},  {grid},   {horizontal, 
            pts}}, 
{{Exp[#] &},  
          CartesianGrid,  {0,  12},  {grid},  {horizontal,  
            pts}}, 
 
        ImageSize  ->  9.5*72];

The real and imaginary parts of the exponential function

An arbitrary complex function f can be written in Cartesian form 
	f(z)  =  f(x  +  i y)  =  
          u(x,  y)  +  i v(x,  y).  
In other words,
	u(x,  y)   =  
        Re (f(x  +  i y)), 	v(x,  y)  =  
        Im (f(x  +  i y)).
Since the inputs to u(x,  y) and v(x,  y) are ordered pairs of real numbers, each of the functions u and v may be regarded as real-valued functions whose domains are (subsets of) the complex plane.  Then u and v are just the composite functions
	Re ∘ f, 	Im ∘ f,
each of which is a real-valued function with domain (a subset of) the complex plane.

To visualize u and v with the Cardano2 application, use a new form of TwoPanelPlot as shown below.  The first argument for each panel is now not just the underlying complex function f, but rather the list consisting of the two functions whose composite is the real part u and the imaginary part v of f, respectively.

Here is such a plot where the domain is restricted so as not to extend beyond the fundamental period strip:

\(\(TwoPanelPlot[
{{Exp, Re}, 
          CartesianSurface, \ {0, π}, {LightSkyBlue}}, 
{{Exp, Im}, 
          CartesianSurface, \ \ {0, π}, {LightCoral}}];\)\)

Here is another such plot where now the domain is extended beyond the fundamental period strip:

\(\(TwoPanelPlot[
{{Exp, Re}, 
          CartesianSurface, \ {0, 2  π}, {LightSkyBlue}}, 
{{Exp, Im}, 
          CartesianSurface, \ \ {0, 2  π}, {LightCoral}}];\)\)

See if you can "see" the periodicity of exp—the property that exp(z  +  2  π i)  =  
        exp(z)—by looking at the preceding plot of the real and imaginary parts of exp.

The principal logarithm function Log

Using TwoPanelPlot to visualize how Log maps subsets of the punctured complex plane \(TraditionalForm\`\(ℂ\^*\)\) to subsets of the complex plane is left as an exercise.

The real and imaginary parts of the principal logarithm

According to the definition
	Log(z)  =  
        ln  | z |  + i Arg(z)
of the principal logarithm function, its real and imaginary parts u and v are given by
	u(
          z)   =  (Re ∘ Log) (z)  =  
          ln  | z | , 	v  =  Im ∘ Log  =  Arg.
The TwoPanelPlot below visualizes these two composites as functions from \(TraditionalForm\`\(ℂ\^*\)\) to ℝ.

\(\(TwoPanelPlot[
{{Log, Re}, 
          CartesianSurface, \ {0, π}, {LightSkyBlue}}, 
{{Log, Im}, 
          CartesianSurface, \ \ {0, π}, {LightCoral}}];\)\)

The downward-descending "funnel" in the graph of the real part of Log indicates the singularity of Log at the origin.

The plot of the graph of the imaginary part Arg of Log is a bit misleading: The true graph does not include the steep wall from below the negative real axis to above the negative real axis; Mathematica is just "connecting the dots".

Exercise.  In terms of properties of Log, what does that steep wall indicate?