Question: Ploting Newton's Raphson (basin of attraction)

Here is the question:Prior to this question I was given f(z)=z^2+1, N(z)=(z^2-1)/(2z), T(z)=z-I/z+I such that T(N^k (z))=(T(z))^2^k. And L is a set of number on the real axis. Now the question is that given we have two regions of the complext plane as follow:

R+ = {z : Nk{z) -> i as k -> ∞}; R- = {z : Nk(z) -> -i as k -> ∞}.

Draw a diagram to illustrate these regions, the line L and the roots i and -i. We call R+ the basin of attractionfor the root +i, and similarly R-is the basin of attraction for the root -i.

 Show that if z is on the set L (the common boundary of the two regions R+ and R_, then Nk(z) stays on L for all values of k. (This is easy once you identify what L is.) So in this case iteration does not produce a root at all.

So basically my problem is that the fact I'm not very familar with the commands to draw such diagram, and I don't know much about Newton's method to compute complex roots. It would be appreciated if anyone can help me how to get start with the question. Thanks.


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