How to apply Taylor in this equation?

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

[Aisha Mohamed]

Hi all

From the answer of the expert Bruno, using Taylor expansion to explain the colormap of cos(angle(P(z)) with P degree 2.

I am sure that the expert want to show important point here, but my humble understanding could not reach it all what I know that

(the turn will be around the two roots of this function and we can calculate these roots by using polyval(p,z)),

so why we use Taylor expansion?

I appreciate any help may help me to understand the above figure also.

[Torsten]

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.