I was reading the proof for the Fundamental Theorem of Algebra in Ian Stewart's book Galois Theory. The author defines $p(w+h)=p_0+p_1h+p_2h^2+\dots+p_nh^n$ for $h\in\Bbb{C}$, then he lets $h=\varepsilon e^{i\theta}$ for small $\varepsilon>0$. Let $m$ be the smallest integer $\ge1$ from which $p_m\neq0$. Then he claims that "$p(w+\varepsilon e^{i\theta})=p_0+p_m\varepsilon^me^{mi\theta}+O(\varepsilon^{m+1})$, where $O(\varepsilon^{n})$ indicates terms of order $n$ or more in $\varepsilon$, and therefore
$|p(w+\varepsilon e^{i\theta})|^2=|p_0+p_m\varepsilon^me^{mi\theta}|^2+O(\varepsilon^{m+1})$."
My question: why do we have $|p(w+\varepsilon e^{i\theta})|^2=|p_0+p_m\varepsilon^me^{mi\theta}|^2+O(\varepsilon^{m+1})$?
I know little to nothing about complex numbers and complex analysis; but using a theorem without understanding the proof is against my religion. Please help if you can – many thanks in advance!
Best Answer
I assume that was meant to be $\,p_m\color{red}{\varepsilon^m}\varepsilon^{mi\theta}\,$ instead. Then:
$$ \begin{align} \left|p(w+\varepsilon e^{i\theta})\right|^2 &= p(w+\varepsilon e^{i\theta})\,\overline{p(w+\varepsilon e^{i\theta})} \tag{1} \\ &= \left(p_0+p_m\varepsilon^m\varepsilon^{mi\theta}+O(\varepsilon^{m+1})\right)\,\left(\overline{p_0+p_m\varepsilon^m\varepsilon^{mi\theta}}+O(\varepsilon^{m+1})\right) \tag{2} \\ &= \left|p_0+p_m\varepsilon^me^{mi\theta}\right|^2+O(\varepsilon^{m+1})\cdot\left(O\left(\varepsilon^0\right)+O\left(\varepsilon^0\right)+O\left(\varepsilon^{m+1}\right)\right) \tag{3} \\ &= \left|p_0+p_m\varepsilon^me^{mi\theta}\right|^2+O(\varepsilon^{m+1}) \tag{4} \end{align} $$
[ EDIT ] $\;$ Step by step:
$\;\;{(1)}\;$ using that $\,|z|^2 = z \,\overline{z}\,$ for all complex $\,z\,$;
$\;\;{(2)}\;$ using that $\,\overline{z_1+z_2}=\overline{z_1}+\overline{z_2}\,$ and $\,\overline {\,O(\varepsilon^k)\,} = O(\varepsilon^k)\,$ for real $\,\varepsilon\,$;
$\;\;{(3)}\;$ using that $\,a + b \cdot \varepsilon^k+\dots = a \color{red}{\cdot \varepsilon^0} + b \cdot \varepsilon^k+\dots=O\left(\varepsilon^0\right)\,$;
$\;\;{(4)}\;$ using that $\,O(n_1)\,\pm\,O(n_2) = O\left(\min (n_1,n_2)\right)\,$ and $\,O(n_1)\,O(n_2) = O(n_1+n_2)\,$.