Complex Transcendental Functions

Load the necessary package.

<< Graphics`ComplexMap`

Part 1. The Exponential Function

1. Increase n one step at a time to see how the polynomial approximation compares with the actual function.

n  =  1;
P[z_] :=  Sum[(z)^k/k!, {k, 0, n}];
P[z]

graph1a  =  
        CartesianMap[Exp, {-2, 2}, {-2, 2}, 
          PlotStyle -> RGBColor[1, 0, 0], 
          DisplayFunction -> Identity];
graph1b  =  
        CartesianMap[P, {-2, 2}, {-2, 2}, 
          PlotStyle -> RGBColor[0, 0, 1], 
          DisplayFunction -> Identity];
Show[graph1a, graph1b, PlotRange -> {{-4, 8}, {-8, 8}}, 
        DisplayFunction -> $DisplayFunction];

Using the input cell below, repeat the above over a square with lower left corner at -4 - 4i and upper right corner at 4 + 4i.  Now how large must n be for a good approximation?

n  =  1;
graph1a  =  
        CartesianMap[Exp, {-4, 4}, {-4, 4}, 
          PlotStyle -> RGBColor[1, 0, 0], 
          DisplayFunction -> Identity];
graph1b  =  
        CartesianMap[P, {-4, 4}, {-4, 4}, 
          PlotStyle -> RGBColor[0, 0, 1], 
          DisplayFunction -> Identity];
Show[graph1a, graph1b, PlotRange -> {{-4, 8}, {-8, 8}}, 
        DisplayFunction -> $DisplayFunction];

2. Look at the conformal plot of the exponential function alone.  Watch the scale on the plot.  Which curves are the images of horizontal lines in the Cartesian plane? which of the vertical?

d  =  0.5;
CartesianMap[Exp, {-d, d}, {-d, d}, 
        PlotRange -> {{0, 5}, {-5, 5}}, 
        PlotStyle -> RGBColor[1, 0, 0]];

3. Now look at the graph of the magnitude of the exponential function.

Plot3D[
        Abs[Exp[x + I*y]], {x, -1, 1}, {y, -1, 1}];

4. The real and imaginary parts of the exponential function

u[x_, y_] :=  Re[Exp[x + I*y]];
v[x_, y_] :=  Im[Exp[x + I*y]];

Plot3D[u[x, y], {x, -6, 6}, {y, -6, 6}, 
        PlotPoints -> 20, 
        PlotRange -> {-400, 400}];

Plot3D[v[x, y], {x, -6, 6}, {y, -6, 6}, 
        PlotPoints -> 20, 
        PlotRange -> {-400, 400}];

We can change the viewpoint by specifying it as an option using the command Viewpoint->{xscale, yscale, zscale}.  Note that these values do not refer to a Cartesian point.  Think of the center of the box as {0,0,0} with the longest side scaled to one unit.  The ordered triple for ViewPoint is the location of the viewpoint in this scaled coordinate system.
	There is quick way to find the viewpoint you want.  Go to the toolbar, and click on Input->3D ViewPoint Selector.  Select Cartesian Form, and experiment with the viewpoint until you are satisfied.  Then, click Paste, and the expression Viewpoint->{xvalue, yvalue, zvalue} will automatically be pasted into your open notebook at the location of your cursor.  Try this on the preceding plots.

5. The exponential of a sum

z1  =  1 + I
z2  =  2 - 3*I

n  =  50;
Sum[(z1 + z2)^k/k!, {k, 0, n}]  -  
        Sum[z1^k/k!, {k, 0, n}]*Sum[z2^k/k!, {k, 0, n}];
N[Abs[%]]

6. Write your explanation below.  You do not need Mathematica's symbolic capability for this step.

Answer:

7. Write your explanation below.  You do not need Mathematica's symbolic capability for this step.

Answer:

Part 2. The Complex Sine and Cosine Functions

1& 2. Look at the conformal plot of the sine function.  For comparison we give the conformal plot for the identity function.  Which curves are the images of horizontal lines in the Cartesian plane? which of the vertical?

d  =  0.5;

CartesianMap[Identity, {-d, d}, {-d, d}, 
        PlotStyle -> RGBColor[1, 0, 0]];

CartesianMap[Sin, {-d, d}, {-d, d}, 
        PlotStyle -> RGBColor[1, 0, 0]];

3 & 4. The graph of the magnitude of the sine function

Plot3D[Abs[Sin[x + I*y]], {x, -3, 3}, {y, -1, 1}, 
        PlotPoints -> 20];

5 & 6. The real and imaginary parts of the sine and cosine functions

u[z_] :=  Re[Sin[z]];
v[z_] :=  Im[Sin[z]];

Plot3D[u[x + y*I], {x, -6, 6}, {y, -2, 2}, 
        PlotPoints -> 20];

Plot3D[v[x + y*I], {x, -6, 6}, {y, -2, 2}, 
        PlotPoints -> 20];

7. Symbolic descriptions of the real and imaginary parts of the sine and cosine functions

ComplexExpand[Sin[x + I*y]]

ComplexExpand[Cos[x + I*y]]

8. Here are plots of the real hyperbolic functions.

Plot[Sinh[t], {t, -3, 3}];
Plot[Cosh[t], {t, -3, 3}];

Answer to question in step 8:

Part 3. Summary