Complex Line Integrals I
Part 2: Experimentation
The following Java applet will let
you experiment with complex line integrals over curves that you draw out with
your mouse. Follow the steps listed below for each line integral you want to evaluate.
Complex Line Integral Evaluator
Directions
for use
- Select the function you want from the list on the right.
- Move the mouse over the green dot, depress the left mouse key, and hold it down while you move the green dot to the point where you want the curve to start.
- Use the same approach to move
the red dot to the point where you want the curve to stop. The coordinates
of the initial point (Point 1) and the terminal point (Point 2) are displayed
in the form (real part, imaginary part).
- Bring the mouse near the green dot, but not on it. Depress and hold the left mouse button down while you draw a curve to the point marked by the red dot. At any time you let up on the mouse button, the curve will be completed by a straight line.
- Click on the "Integral" button
to display the value of the integral. (The integral is calculated by a numerical
method and so will be only approximate.)
Now carry out the following experiments. For each integral, record on your worksheet the function used, the initial and terminal points, and the value of the integral.
- Select the z2 function and any pair of initial and terminal points. Draw the curve and display the value of the integral. Repeat for two other curves connecting the same initial and terminal points.
- Repeat Step 1 for the function ez.
- Repeat Step 1 for the function (conjugate(z))^2.
- Select the 1/z function. Place the green initial point on the right side of the origin near the real axis and the red terminal point on the left side of the origin near the real axis. Draw a curve connecting the two points such that the curve passes above the origin and display the value of the integral. Repeat with an additional curve with the same initial and terminal points -- each curve passing above the origin.
- Keep the same initial and terminal points as in Step 4. This time display the values of the line integrals for two curves that pass below the origin.
- Place the terminal point on top of the initial point. (Thus, when you draw the curve it will be closed.)
- Draw a closed curve counterclockwise around the origin. Evaluate the line integral for each of the four functions.
- Repeat the step above for a closed curve that is traced out clockwise around the origin.
