Does the exponential function have a square root?
(asked by Nathaniel Hellerstein on the Q&A board at JMM)
Is there a "half-exponential" function h(x) such that h(h(x))=ex? Is it unique? Is it analytic?
Related question: Is there an invertible smooth function E such that E(x+1)=eE(x)? Is it unique? If so, then we can take h(x)=E(E-1(x)+1/2).
[Ed: please retag appropriately]