1. ## Complex help

Q: For fixed $\displaystyle a\in{\mathbb{C}}$ find the value of $\displaystyle |z^{n}-a|$ that is maximized when $\displaystyle |z|<1$.

A: Well, I would think I need to find the the roots to the equation $\displaystyle z^{n}=a$ firs. After that I am not sure. Since this is a max question, I feel that I need to take the derivative at some point with respect to z.

Any help would be appreciated.

Thanks

2. Setting $\displaystyle z= e^{\rho + i\ \theta}$ and $\displaystyle a= e^{\alpha + i\ \beta}$ is...

$\displaystyle \displaystyle |z^{n} - a|^{2} = (e^{n\ \rho + i\ n\ \theta} - e^{\alpha + i\ \beta})\ (e^{n\ \rho - i\ n\ \theta} - e^{\alpha - i\ \beta})$ (1)

Now what You have to do is to find the values of $\displaystyle \rho$ and $\displaystyle \theta$ [with the condition $\displaystyle \rho <0$, $\displaystyle \implies$ Lagrange's multipliers ...] so that (1) is maximized...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

3. $\displaystyle If \; |z|<1 \; then \; f(z)=z^n \; |f(z)|<1$

$\displaystyle max |z^n-a| \; is \; the \; same \; as \; max|z-a|$

$\displaystyle max|z-a|=|a|+1$