Given a non-integral real $\alpha$, is there an entire (see http://en.wikipedia.org/wiki/Entire_function) function $h(x)$ such that $x^{-\alpha}h(x)\longrightarrow 1$ for $x\rightarrow+\infty$ (with $x$ real non-negative)?

Clearly, such a function if it exists is not unique since $h(x)+e^{-x}$ and similar functions work also.