Plotting roots of a predefined complex polynomial

Asked Jan 16 '17 at 15:53

Active Jan 16 '17 at 16:59

Viewed 45 times

I have the following :

 w[n_] := E^((2*I*\[Pi])/n)
 Phi[n_, k_, x_] := (1/n)*Sum[(w[n]^(-j*k)*E^(x*w[n]^j)), {j, 0, n - 1}]
 NEuler[n_, x_] := (1/Phi[n, 0, x])
 NEulerPoly[n_, x_, z_] := (E^(x*(z - 1/n)))/Phi[n, 0, x/n]
 NEPoly[n_, z_, r_, M_] := Expand[FullSimplify[Coefficient[r!Normal[Series[NEulerPoly[n, x, z], {x, 0, M}]], x,  r]]]

The polynomials in z are indexed by r and form a sequence $\{z\}_{r\in\mathbb{N}}$. I would like to look at the roots of these complex polynomials by extracting the real and imaginary parts and plotting them as an ordered pair, or by some other method. HOwever, I have no idea how to do this. I don't know mathematica very good and am trying to learn the commands and functions that can handle this type of procedure.

edited Jan 16 '17 at 16:59

J. M.'s missing motivation

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asked Jan 16 '17 at 15:53

AveryJessup

Plot[ReIm@(your function),(your domain)]? – Feyre Jan 16 '17 at 16:33
That seems to plot the actual polynomial, not the complex roots of the polynomial... – AveryJessup Jan 16 '17 at 16:36
You can find appropriate ideas here: Finding real roots of negative numbers – Artes Jan 16 '17 at 16:39
When I try to evaluate NEPoly, I get all zeroes: NEPoly[n, z, r, #] & /@ Range[20] – bill s Jan 16 '17 at 16:44
I used a term $M$...it must be greater than or equal to r. So it you try and print when $r=5$, $M\ge 5$ – AveryJessup Jan 16 '17 at 16:46
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Here's a start: ListPlot[{Re[z], Im[z]} /. Table[Solve[0 == NEPoly[2, z, r, 20], z], {r, 1, 10}] // N]. – march Jan 16 '17 at 16:51
If the answers in the linked thread were not useful, please edit your question to mention why the solutions there could not be adapted to your situation. – J. M.'s missing motivation Jan 16 '17 at 17:02
@march, is there any way to just get a single value for r instead of a list plot for all r up to a particular value? – AveryJessup Jan 16 '17 at 17:14
Just replace the Table[]: Solve[0 == NEPoly[2, z, 10, 20], z], for instance. – J. M.'s missing motivation Jan 17 '17 at 09:25

Plotting roots of a predefined complex polynomial

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