Question

Find the first four iterates of the function f(z)=z^2-2-2i with an initial value of z0=2+i

Answer 1

Obtain the next iterate by plugging the previous iterate into the function.

First iterate:

$z_0=2+i\implies z_1=f(z_0)=(2+i)^2-2-2i=1+2i$

Second iterate:

$z_2=f(z_1)=(1+2i)^2-2-2i=-5+2i$

Third iterate:

$z_3=f(z_2)=(-5+2i)^2-2-2i=19-22i$

Fourth iterate:

$z_4=f(z_3)=(19-22i)^2-2-2i=-125-838i$