# Math Help - Complex help

1. ## Complex help

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

A: Well, I would think I need to find the the roots to the equation $z^{n}=a$ 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 $z= e^{\rho + i\ \theta}$ and $a= e^{\alpha + i\ \beta}$ is...

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

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

Kind regards

$\chi$ $\sigma$

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

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

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