We started today with a brief recap of the work we’ve done with complex linear functions using the back of the Complex Paint worksheet we’ve been working off of.
From there, we started work with complex quadratic functions. We will be replicating the process we did with real numbers: analyzing a family of quadratics where we only adjust a single parameter value and investigate the behavior of iterations for specific values of the parameter. Eventually, we will find a way to categorize and catalog these behaviors.
The family of functions we will be analyzing is z² + C, and the first parameter we will look at is C = 0. We found that squaring a complex number in polar form resulted simply in squaring the value of R and doubling the value of θ, so an initial seed like [2,10°] becomes [4,20°], then [16,40°], then [256,80°] and so on, for larger and larger values of R, suggesting that the seed [2,10°] diverges to infinity under iterations of z².
If you weren’t in class, pick two of the seeds below to iterate. Iterate until you’re convinced what the long-term destination might be (diverging? converging? limit cycle?). Note also that θ should never exceed 360°. If the pattern for [2,10°] from above were continued, we would get 160°, then 320°, then 540°. But 540° is larger than 360°, and is equivalent to 540-360=280°. And that’s the angle we would record.