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:
- What is this? (Is just transforming to polar form?)
- What the second graph tell me about the polynomial?
- How I calculate R?
- Is that working only with 5?
Thank you very much, I hope someone can help me!