Math Help - Two question from complex analysis

1. 1. Find the general form of bilinear
mapping that maps the upper halfplane $H^+=\{z : Imz >0\}$
to yourself.

2. Prove that for all $R: R>0$, $n \in N$ (large enough),
all zeros of polynom
$p_n(z)=\frac{1}{n!}z^n + ... + \frac{1}{2!}z^2 + 1$
are in $\{z \in \mathbb{C}: |z| > R\}$

Coment: For 2. $p_n -> e^z$, and... ?

New question:

$H^+ = \{z : Im z>0\}.$
$f(z)=z^5.$

Now, $f(H^+) = \{ z= r e^{i \theta} : r >0, 0 < \theta < 5\pi\} = \mathbb{C} \ \{x \leq 0\}$
or $f(H^+) = \{z= r e^{i\theta} : r >0, 0 < \theta < \pi\}$ ?

Is needed $mod 2\pi$ ?