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I'm studying for an exam and professor gave us to create a little program that automatically does a transformation for a polynomial with complex coefficients, I don't have many problems doing the transformation but I don't know what this transformation is useful to (i have only my notes) and i hope you can help, here's what i have:

Given a $P_n(S) = s^5+c_4s^4+c_3s^3+c_2s^2+c_1s+c_0$ where $c_i \in C$

As $S = R(cos(t)+i sin(t))$

now $P_n(S)= R^5(cos(5t)+isin(5t)) + c_4R^4(cos(4t)+isin(4t)) +\ldots+ c_0$

Let $X(t)=Real(P_n)(t)$ and $Y(t)=Img(P_n)(t)$

draw $\Gamma_R(t) = \begin{cases} x= X(t) \\ y=Y(t) \end{cases}$

Now he want that the program shows the function $P_n$ in the imaginary plane and then shows the funciont in a plane where $x=X(t)$ and $y=Y(t)$. He said something about counting how many loops the second graph has...

The questions are:

  1. What is this? (Is just transforming to polar form?)
  2. What the second graph tell me about the polynomial?
  3. How I calculate R?
  4. Is that working only with 5?

Thank you very much, I hope someone can help me!

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I'm confused. As I read it, $P(S) = $ expression in $s$. Do you mean a polynomial in uppercase $S$ or lowercase $s$?! – user2468 Feb 25 '12 at 20:54
Good question.. i wrote it in my paper twice and is $P(S)$ so uppercase.. – Fabio F. Feb 25 '12 at 20:57

Is "$R = \frac{Real(S)}{cos(t)}+\frac{img(S)}{isin(t)}$" your own work? It doesn't follow from the definition of S that precedes it. You can't divide complex numbers by dividing the real and complex parts separately. What you want there must be $$R=\sqrt{(\operatorname{\mathfrak{Re}} S)^2+(\operatorname{\mathfrak{Im}} S)^2}$$ though perhaps it's even more probable that you just want to consider $R$ a constant that goes into defining what $S$ is for each $t$. If so, you're not supposed to compute $R$, but to choose a sequence of different $R$'s and draw a graph for each.

The transformation is not just going to polar coordinates, but also computing the image of a circle around the origin of radius $R$ when transformed by the polynomial function.

As for what this is used for: The remark about counting loops strongly suggests that the point is to illustrate the central idea in the winding-number based proof of the Fundamental Theorem of Algebra. Then the degree being 5 is just an example -- the same principle will work for any degree $\ge 1$.

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Oh.. now i see that i've wrote with pen over the pencil some later notes. That stupid equation about R is now cancelled thank you. – Fabio F. Feb 25 '12 at 21:21
up vote 1 down vote accepted

Well, asking to some people i've found that, that was the introduction to something related to this

I should take more notes..

Thank you all..

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