Complex Line Integrals I

Part 1. The Definition of the Complex Line Integral

1. First we define four functions.

f1[z_] :=  1/z;
f2[z_] :=  z^2;
f3[z_] :=  (Conjugate[z])^2;
f4[z_] :=  Exp[z];

Our first curve (Subscript[C, 1]) will be the piece of the parabola y = x^2 between 0 and 1 + i.  We designate the parametrization as Z1.  To plot a complex valued function in Mathematica, we need to separate the real and complex parts and plot using ParameticPlot.

Z1[t_] :=  t  +  t^2*I;
ParametricPlot[{Re[Z1[t]], Im[Z1[t]]}, {t, 0, 1}];

The complex line integral of the square function, f2,over (Subscript[C, 1]) is calculated below.

Integrate[f2[Z1[t]]*D[Z1[t], t], {t, 0, 1}]

We check that the value of the integral is independent of the parametric representation of the curve (Subscript[C, 1]).  Enter (as Gamma1) another parametric representation for this curve.

\(Gamma1[t_] := \ \(?;\)
 (*\ 
      Fill\ in\ values\ for\ start\ and\ end\ below\ *) \)
ParametricPlot[{Re[Gamma1[t]], Im[Gamma1[t]]}, {t, start, 
        end}]

2. Calculate the values of the integral of f4 over each of the parametric representations of (Subscript[C, 1]) using the input cell below.

3. Here are the parametric representations of the two curves (Subscript[C, 2]) (red) and (Subscript[C, 3]) (blue), designated as Z2 and Z3 respectively.

Z2[t_] :=  t  +  t*I;
Z3[t_] :=  Cos[5*t + 3*Pi/2] - Sin[3*t]*I;
curve2  =  
        ParametricPlot[{Re[Z2[t]], Im[Z2[t]]}, {t, 0, 1}, 
          PlotStyle -> RGBColor[1, 0, 0], 
          DisplayFunction -> Identity];
curve3  =  
        ParametricPlot[{Re[Z3[t]], Im[Z3[t]]}, {t, 0, Pi/2}, 
          PlotStyle -> RGBColor[0, 0, 1], 
          DisplayFunction -> Identity];
Show[curve2, curve3, 
        DisplayFunction -> $DisplayFunction];

Record your results and observations in the text cell below:

Observations:

4. The curve (Subscript[C, 4]) may be parametrized as follows:

Z4[t_] :=  Cos[t]  +  Sin[t]*I;
curve4  =  
        ParametricPlot[{Re[Z4[t]], Im[Z4[t]]}, {t, 0, Pi}, 
          PlotStyle -> RGBColor[0, 0, 1]];

Enter a parametric representation the curve (Subscript[C, 5]) and evaluate the integrals.

Z5[t_] :=  ?;
curve5  =  
      ParametricPlot[{Re[Z5[t]], Im[Z5[t]]}, {t, 0, Pi}, 
        PlotStyle -> RGBColor[0, 0, 1]]

Calculate the line integrals of f1 through f4 and record the results.  The code here calculates the integrals numerically-- to accommodate f3.  Pay attention to the size of the results.

f[t_] :=  f1[t];
N[Integrate[f[Z4[t]]*D[Z4[t], t], {t, 0, Pi}]]
N[Integrate[f[Z5[t]]*D[Z5[t], t], {t, 0, Pi}]]

Record your results here in the text cell below:

Answer:

Part 2. Experimentation

Record the results of your applet experiments in the text cell below:

Answer:

Part 3: Calculation

1. Enter your comparisons in cell below:

Modify the instructions below to carry out the calculations requested in steps 2 and 3.

Z6[t_] :=  Cos[t]  +  Sin[t]*I;
ParametricPlot[{Re[Z6[t]], Im[Z6[t]]}, {t, 0, 2*Pi}, 
        PlotStyle -> RGBColor[0, 0, 1], 
        AspectRatio -> 1];

Integrate[f1[Z6[t]]*D[Z6[t], t], {t, 0, 2*Pi}]

2. Record the results of your computations and the comparisons in the cell below.

Answer:

3. Record the results of your computations and the comparisons in the cell below.

Answer:

4. Enter your explanation in the cell below.

Answer:

Part 4. Summary

1. Write your summary below.

Summary:

2. Make your conjecture below.

Conjecture:

3. Define your functions and carry out your test using the input cell below:

4. State your final conjecture below.

Answer: