[math-fun] Is this entire function surjective?

A complex-valued function f : C —> C on the complex plane C is called "entire" if it's analytic for all z ∊ C. Or in other words it can be defined by a power series f(z) = ∑ c_n z^n with an infinite radius of convergence. The "little theorem" of Picard states that an entire function is surjective or omits just one complex value, or else it is a constant. Clearly the function f(z) = z exp(-z) is not a constant. Does it omit a value? In more generality, what about entire functions of the form g(z) = P(z) exp(Q(z)) where P and Q are polynomials? —Dan