Math 421  •  Fall 2006

Complex sine function

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]

Trig identities

The key Mathematica function to use to express trig and hyperbolic functions in terms of the complex exponential is the built-in TrigToExp:

?TrigToExp

TrigToExp[Sin[z]]

TrigToExp[Cos[z]]

TrigToExp[Sinh[z]]

To convert back from exponential form to trig and hyperbolic form, you may use the built-in function ExpToTrig.  For example:

ExpToTrig[Exp[I z]]

Proof of additon formula for the complex sin

Sin[(Subscript[z, 1])  +  (Subscript[z, 2])]  //  TrigToExp

Sin[(Subscript[z, 1])] Cos[(Subscript[z, 2])]  +  Cos[(Subscript[z, 1])] Sin[(Subscript[z, 2])]  //  
      TrigToExp

Periodicity of sin

Mathematica already knows the basic identities for the trig and hyperbolic functions.  For example:

Sin[z  +  2  π]

The following illustrates that sin is periodic of real period 2  π:

grid = {{{DimGray},  {-π/2, π/2, 
              14},  {-π/2, π/2, 
              14}}};
points = {Table[{Hue[0.1677  j/20], 
            ComplexPoint[-π + j π/10 + 
                0.95  I]}, {j, 0, 20}]}; 
    positiveIAxis  =  {MediumSeaGreen, AbsoluteThickness[1],  
        ComplexCurve[I y,  {y,  
            0,  π}]};
primitives = 
{positiveIAxis,  
          AbsolutePointSize[5], points, 
          First[CartesianGrid[{# &}, {0, π/2}, {grid}]]};

TwoPanelPlot[
{{# &}, 
          CartesianGrid, {0, π}, {grid}, {primitives}}, 
{{Sin[#] &},  
          CartesianGrid,  {0,  π},  {grid},  {primitives}}
];

The following also illustrates the periodicity, but with the image lifted to the Riemann sphere.

TwoPanelPlot[
{{# &}, 
            CartesianGrid, {0, π}, {grid}, {Take[primitives, 3]}},
{{Sin[#] &}, RiemannSphere, {0, 1.1}, {ColoredRiemannSphere[{None},{},{}],primitives},{},{Background -> LightSkyBlue, 
                ViewPoint -> {1.737, -2.813, 0.713}, 
                PlotRegion -> {{-0.15, 1.1}, {-0.25, 1}}}}];

Cartesian form of sin (x  +  i y)

ComplexExpand[Sin[x  +  I y]]

Derivative of complex sin

Of course, Mathematica knows the formula for the derivative of sin:

Sin'[z]

Zeros of complex sin

Solve[Sin[z]  ==  0,  z]

Reduce[Sin[z]  ==  0,  z]

TraditionalForm[%]

Mapping properties of complex sin

grid  =  {{{LightBlue},   {-π,  π,  
              12},  {-π,  π,  
              12}}};
TwoPanelPlot[
{{# &},  
          CartesianGrid,  {0,  π},  {grid},   {}}, 
{{Sin[#] &},  
          CartesianGrid,  {0,  4},  {grid},  {}}, 
 
        ImageSize  ->  9.5*72];

It looks like horizontal lines and vertical lines, respectively, map to arcs of hyperbolas and ellipses centered at the origin—or vice versa.  Which?  Use a two-color grid to see:

grids = {{{MediumBlue}, {-π,  π,  
              12}, {-π,  π, 0}}, {{AbsoluteDashing[{5}], 
              VenetianRed}, {-π , π,  0}, {-π,  π, 
              12}}};
TwoPanelPlot[
{{# &},  
          CartesianGrid,  {0,  π},  {grids},   {}}, 
{{Sin[#] &},  
          CartesianGrid,  {0,  4},  {grids},  {}}, 
 
        ImageSize  ->  9.5*72];

Examine more closely the way a horizontal lines map maps to an arc of a hyperbola with center at the origin:

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

And examine more closely the way a vertical line maps to an ellipse centered at the origin:

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

Principal value of arcsin

ArcSin[5/4]

ComplexExpand[ArcSin[5/4]]

Mapping properties of complex Arcsin

Try to visualize the image of the open upper half-plane {z  :  Im z  >  0}.  To do so, start the vertical lines in the grid at a small positive value rather than at 0:

grid  =  {{{LightBlue},   {-20,  20,  24},  {0.01,  40,  
              24}}};
TwoPanelPlot[
{{# &},  
          CartesianGrid,  {0,  
            20},  {grid},   {}}, 
{{ArcSin[#] &},  
          CartesianGrid,  {0,  5},  {grid},  {}}, 
 
        ImageSize  ->  9.5*72];