## Three applets that may be of interest to M337 students

The first applet allows you to plot a complex number and then plot the roots.

- Plot the original number by clicking the mouse on the complex plane.
- Choose the number of roots wanted and then
press "Find Roots".
- Move the number around with the mouse and watch how the roots behave. The
principal root is the dark blue one.

The second applet is intended to enable you to explore some
complex functions.

- Leave the box saying 'Free draw' as it is and choose a function from the list.
- Click on the left hand plane.
- Drag the mouse around the left hand plane and observe the image
of your path on the right hand plane. The green circles are the unit circles. Listen for the
discontinuity chimes!
- Now choose a style of path from the 'Free draw' box. If you choose a circle, select a radius from the given list.
- Click on the left hand plane. This point will be the centre of the circle or a point on the line, according to which style you have chosen. The path together with its image will appear.

The third applet demonstrates the Argument Principle.
- Enter the coefficients of a polynomial (up to degree 4) in the form a+bi in the places on the bottom panel.
- Enter a radius in the top box on the right and press 'Apply'. The applet will plot the image of a circle, centre 0 with the entered radius, under the given polynomial.
- Change the radius, press 'Clear' and 'Apply' again.
- Try the polynomial z
^{3}-z^{2} with a circle of radius 2. You should get the curve used on the cover of the Block C units!
- If you wish to see an animated change of radius enter two radii in the top two boxes on the right and press 'Animate'. Have fun!