Mapping Diagrams: Complex Analysis-functions with lines or circles
A simple but still interesting example is presented with the complex function .
In this GeoGebra activity, The parameters and are controlled by the complex numbers (points) a and b in the domain frame. The slider marked L1orC2 has value 2. This slider controls the choice of a circle or line for the domain curve.
The value 2 indicates the choice is a circle. The circle has radius , controlled by the slider, and center at the complex number (point) is used to locate n complex numbers (points) on the circle where n is controlled by another slider.
Changing the L1orC2 slider to a value of 1 will choose the domain curve as a line with slope m controlled by the m slider through the complex number (point)
The 3-dimensional frame has the mapping diagram for the function visualized using the points on the circle or line in the domain frame along with the point . For comparison the graph of an associated real quadratic function is shown in the third 2-dimensional frame.
Changes can be made by moving , m, a, b or , or by entering a new complex function using parameters a or b. This example can be removed from sight in the activity by unchecking the box labelled "Show function g with parameters a and b".
A more subtle example can be shown by checking the box labelled
"Show function h with parameters a, b, c and d".
This will show the function using parameters a,b,c, and d controlled by the appropriate complex numbers (points) in the domain frame. The other two frames are as with the quadratic example.
Again the diagram can be controlled by changing the parameters.