The question is quite short: is the analytic function $w(z)=e^z+z$ surjective in $\mathbb{C}$? The motivation is that my professor mentioned this function in class, but he did not give the answer.
Since $z=\infty$ is an essential singularity, by Weierstrass Theorem, the range of $w(z)$ is dense in $\mathbb{C}$.
However, we can even know more about $w(z)$. By Little Picard Theorem, we know the range of $w(z)$ is either the whole complex plane or the plane minus a single point.
Can we learn more about $w(z)$? It looks innocent but I have no idea how to go further. By the way, I haven't learned the proof of Picard's Theorem yet, so I am temporarily accepting it as a fact.
The answer is yes. Suppose there exists $y\in\mathbb C$ such that $e^z + z\ne y$ for any $z\in\mathbb C$. By little Picard then, there must be some $z_0\in\mathbb C$ such that $e^{z_0} + z_0 = y + 2\pi i$.
But then $w(z_0-2\pi i) = y$, a contradiction.