Question

Find the first four iterates of the function f(z)=z²-2-2t with an initial value of z base 0 = 2 i

Answer 1

To find the first four iterates of the function f(z)=z²-2-2t with an initial value z_0 = 2i, we continually apply the function to each successive iterate, starting with the initial value. By computing and simplifying, we derive z_1, z_2, z_3, and z_4 in that order.

Step-by-step explanation:

The question involves finding the first four iterates of the function f(z)=z²-2-2t with an initial value of z_0 = 2i (where i is the imaginary unit). To find these iterates, we will successively apply the function to the initial value.

1. First iterate (z_1): Apply the function to the initial value z_0. Since z_0 is 2i, we plug it into the function to get f(2i) = (2i)² - 2 - 2t. This simplifies to -4 - 2 - 2t. Thus, z_1 = -6 - 2t.
2. Second iterate (z_2): Apply the function to z_1. Substituting z_1 into the function gives f(-6 - 2t) = (-6 - 2t)² - 2 - 2t. After simplifying, we find the value for z_2.
3. Third iterate (z_3): Similar to the previous steps, we substitute z_2 into the function to get f(z_2), and after simplifying, we determine z_3.
4. Fourth iterate (z_4): Finally, we calculate f(z_3) and simplify to get the value for z_4.