Proof that f(z) = $z^{n}(z^*)^{m}$ is not analytic in any point.
If i look at the limit of a more simple function of this form: f(z) = $\frac{z}{z^*}$
I would say that the limit does not exist, because if look at $\lim\limits_{z \to 0} \frac{z}{z^*} = \lim\limits_{z \to 0} \frac{x+iy}{x-iy}$
approach along real axis: $\lim\limits_{y \to 0} \frac{x+i0}{x-i0} = 1$
approach along im axis: $\lim\limits_{x \to 0} \frac{0+iy}{0-iy} = -1$
these two values are not the same, so the limit does not exist, from which we cam conclude not analytical.
For the original problem there are the exponentials m and n.
$\lim\limits_{x,y \to 0} \frac{(x+iy)^n}{(x-iy)^{-m}}$
As a hint someone said i could use the binomial function(i believe to solve this limit), but am not sure how.
can some one explain this to me?
thanx