(asked by Nathaniel Hellerstein on the Q&A board at JMM)

Is there a "half-exponential" function h(x) such that h(h(x))=e^x? Is it unique? Is it analytic?

Related question: Is there an invertible smooth function E such that E(x+1)=e^E(x)? Is it unique? If so, then we can take h(x)=E(E^-1(x)+1/2).

[Ed: please retag appropriately]

(asked by Nathaniel Hellerstein on the Q&A board at JMM)

Is there a "half-exponential" function $h(x)$ such that $h(h(x))=e^x$? Is it unique? Is it analytic?

Related question: Is there an invertible smooth function $E$ such that $E(x+1)=e^{E(x)}$? Is it unique? If so, then we can take $h(x)=E(E^{-1}(x)+1/2)$.