# How to apply Taylor in this equation?

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Aisha Mohamed on 25 Sep 2022
Edited: Torsten on 26 Sep 2022
Hi all
I am really sorry because I have just now found (Bruno Luong answered your question what does the positive and negative values concave mean?.) (I do not know why I could not see it before now? maybe an error in my device).
The expert Bruno said,
((I'm bet you plot the colormap of cos(angle(P(z)) with P degree 2 and centered about z0 the root of P(z ).
A taylor expansion dz := z-z0 is
P(dz) = 0 + P'(z0)*dz + P''(z0)/2 * dz^2 .
So the phase of P(dz) make 2 turns when going around z0, so the cosine make 2 periods : you get 2 blues opposite and 2 yellows opposite. That's all normal and nothing to discover beside the fact that taylor expansion explains all that.)).
I understood from this answer and previuos answers from Torsten and David Goodmanson that the phase of P(dz) make 2 turns when going around z0. My questions are,
1-How to apply Taylor expansion in the case of this function
f_b(1/z) = (0.10 − 0.3i)z −1 + (0.2121 − 0.0008i)z −2 + (0.9 + 0.001i)z −3 and I wrote it in MATLAB as p1,
p1=[(0.9000 + 0.0010i) (0.2121 - 0.0008i) (0.1000 - 0.3000i) (0)] ;
when I plotted it I got 2-Does here the phase of P(dz) make 3 turns when going around z0,
3-what happened in the center of the figure(I put it inside circle ) , why the positive (yellow) and negative (blue) values overlap by this way?
I appreciate any help
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Torsten on 26 Sep 2022
Edited: Torsten on 26 Sep 2022
Since this seems a private communication with Bruno Luong, you should ask him, not the forum.
We don't know the context of his remark.