Isolated Singularities and Series Expansions

Part 1. Taylor Series Expansions

Enter the following command to generate a Taylor series for the hyperbolic sine function.  The parameters specify the function, the complex value about which the expansion should take place, and the order of the expansion.  (Some of these terms may turn out to be zero.)  The expression O[z]^9 can be translated as "terms involving the ninth and higher powers of  z."

Series[Sinh[z], {z, 0, 8}]

Enter your reponses and answers to Part 1 in the input cell below:

Answer:

Part 2. Isolated Singularities

Enter the definitions of the first sample function.

f1[z_] :=  ?;

Enter the series command for Step 1 in the input cell below.

Series[f1[z], {z, 0, 8}]

The following sequence of commands will display the graph of Abs[f1[z]] over each of the circles in turn.  You will need to change the definition of f as you work the various steps in Part 2.  Watch for changes in the vertical scale.

f1[z_] :=  Sin[z]/z;
f2[z_] :=  Cosh[z]/z;
f3[z_] :=  Exp[1/z];
X[t_] :=  r*Cos[t];
Y[t_] :=  r*Sin[t];

f[t_] :=  f1[t];

r  =  1;
plot1  =  
        ParametricPlot3D[{X[t], Y[t], Abs[f[X[t] + I*Y[t]]], 
            RGBColor[1, 0, 0]}, {t, 0, 2*Pi}, 
          BoxRatios -> {1, 1, 1}];

r  =  0.8;
plot2  =  
        ParametricPlot3D[{X[t], Y[t], Abs[f[X[t] + I*Y[t]]], 
            RGBColor[0, 0, 1]}, {t, 0, 2*Pi}, BoxRatios -> {1, 1, 1}, 
          DisplayFunction -> Identity];
Show[plot1, plot2, 
        DisplayFunction -> $DisplayFunction];

r  = 0.6;
plot3  =  
        ParametricPlot3D[{X[t], Y[t], Abs[f[X[t] + I*Y[t]]], 
            RGBColor[0, 1, 0]}, {t, 0, 2*Pi}, BoxRatios -> {1, 1, 1}, 
          DisplayFunction -> Identity];
Show[plot1, plot2, plot3, 
        DisplayFunction -> $DisplayFunction];

r  =  0.4;
plot4  =  
        ParametricPlot3D[{X[t], Y[t], Abs[f[X[t] + I*Y[t]]], 
            CMYKColor[1, 0, 0, 0]}, {t, 0, 2*Pi}, 
          BoxRatios -> {1, 1, 1}, 
          DisplayFunction -> Identity];
Show[plot1, plot2, plot3, plot4, 
        DisplayFunction -> $DisplayFunction];

r  = 0.2;
plot5  =  
        ParametricPlot3D[{X[t], Y[t], Abs[f[X[t] + I*Y[t]]],  
            CMYKColor[0, 1, 0, 0]}, {t, 0, 2*Pi}, 
          BoxRatios -> {1, 1, 1}, 
          DisplayFunction -> Identity];
Show[plot1, plot2, plot3, plot4, plot5, 
        DisplayFunction -> $DisplayFunction];

r  =  0.1;
plot6  =  
        ParametricPlot3D[{X[t], Y[t], Abs[f[X[t] + I*Y[t]]]}, {t, 0, 2*Pi}, 
          BoxRatios -> {1, 1, 1}, 
          DisplayFunction -> Identity];
Show[plot1, plot2, plot3, plot4, plot5, plot6, 
        DisplayFunction -> $DisplayFunction];

Use the input cell below to record your responses to Steps 2-9 in Part 2. You may copy and edit the commands above as needed.

Part 3. Conjectures

Record your conjectures and the new functions used to test them in the text cell below.

Part 4. Residues

Here is a sample residue command.

Residue[Exp[3*z]/z^3, {z, 0}]

Series[Exp[3*z]/z^3, {z, 0, 8}]

Record your responses for steps 1-5 in the input cell below:

Answer:

Part 5. Summary

Answers to the summary questions: