$z\exp(z)$ surjectivity with the Little Picard Theorem

Question

I would like to prove the surjectivity of this function :

\begin{align*} f\colon\mathbb{C}&\to\mathbb{C}\\ z&\mapsto z\exp(z) \end{align*}

You can use the Little Picard Theorem: If a function $f\colon\mathbb{C}\to\mathbb{C}$ is entire and non-constant, then the set of values that $f(z)$ assumes is either the whole complex plane or the plane minus a single point.

Thanks.

Answer 1

Suppose $w$ is not in the image of $f$. Since $f(z) - w$ is entire and never zero, it can be written as $f(z) - w = \exp(g(z))$ for some entire function $g$. Notice that $f(z) - w = -w$ only when $z = 0$. Apply the little Picard theorem to to $g(z)$ to get a contradiction.