Here's the question:
Let $r>0$; let $f:\Delta^\ast(\infty, r):=\{z\in\mathbb C;\mid z\mid > r^{-1}\}\rightarrow\mathbb C$ be the function given by $f(z)=(1+\cos(\sqrt{z}))e^z$. Find $f(\Delta^\ast(\infty, r))$.
My attempt: By Picard's theorem, the set $M:=\mathbb{C}-f(\Delta^\ast(\infty, r))$ contains at most one point. Since $f$ satisfies $f(z)=\overline {f(\bar z)}$ and $f(\Delta^\ast(\infty, r))$ contains all positive reals, if there is a point in $M$, it has to be a negative real number. But I have no idea how to tell if $M$ even contains a point, and if it does, which negative real number it is.
This is as far as I could get. Can anyone help me? Thanks in advance!