# Complex Function Iterator

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This applet iterates any complex function, and displays the orbit as a numerical list and as points plotted in the complex plane. It allows the user to choose up to two seed values at a time and view their orbits. It is very similar in use to the Real Function Iterator Applet and so we explain only the differences here.

This applet allows for either Polar or Euclidean inputs of seed value and independently allows for either Polar or Euclidean computation of functions. Thus in Polar computation mode all seed values will be converted to polar form FOR THE PURPOSE OF COMPUTING THE FUNCTION VALUES and evaluation of the function will be made through the polar form of the map. The Polar computation mode allows only maps of the form z^n.

For example, evaluating the map f(z)=z^2 in Polar computation mode means it will be evaluated as (r, θ) -> (r^2, 2θ). Hence using the seed z_0 = 0.6 + 0.8i in Euclidean seed form and iterating f(z)=z^2 in Polar computation mode will cause the applet to convert z_0 to polar form (r, θ) = (1, arctan (4/3)) and so all computed iterates will be on the unit circle. However, starting with the same seed and iterating f(z)=z^2 in Euclidean computation mode will, due to round off error, cause the orbit to leave the unit circle.

The orbit data is displayed in the same form as the seed value and can be toggled between Euclidean and Polar forms by clicking on the Euclidean seed form and Polar seed form buttons. This applet allows for up to two seed values and includes a checkbox for a thin plot of the unit circle to appear (as reference).

Note about computational equivalency: Though mathematically equivalent, the expressions z^2 and z*z are not computationally equivalent. The former is evaluated as exp(2 Log z), where Log z is the principle logarithm, and the latter is evaluated through usual complex multiplication. Thus each will incorporate different rounding errors at times. The end result of this difference is quite evident when iterating the seed 0.6 + 0.8i (on the unit circle) in Euclidean computation mode under each of these maps.

This material is based upon work supported by the National Science Foundation under Grant No. 0632976.