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?
An entire function that's never zero can be written as exp F where F is entire. So an entire function that omits the value u can be written as u + exp F where F is entire. If z exp -z = u + exp F then exp F = z exp -z - u equals -u for _exactly one_ choice of z, namely z=0. Hence there is only one z for which F is _any_ logarithm of -u. But there are lots of logarithms of -u, so F omits almost all of them, which is impossible for entire F. (What if u=0? Impossible, because z exp z definitely doesn't omit the value 0.) If we have P exp Q (P,Q polynomials) instead, we can still do more or less the same. Suppose P exp Q omits the value u (again u can't be zero because P has roots); then P exp Q - u is only -u at finitely many points, so exp F is only -u at finitely many points, but there are infinitely many logarithms of -u, so F omits lots of points, contradiction. -- g