I think the user Av's answer (using Liouville's theorem) is correct, and a very simple argument, though it does use the fairly hefty machinery that goes into "a

I think the user Av's answer (using Liouville's theorem) is correct, and a very simple argument, though it does use the fairly hefty machinery that goes into "a bounded analytic function is constant". EDIT: as tome kindly points out, e^x f(x) need not remain bounded as x goes to -infinity, or indeed any other direction of infinity on the complex plane.