On the complexity of computing the logarithm and square root functions on a complex domain

The problems of computing single-valued, analytic branches of the logarithm and square root functions on a bounded, simply connected domain S   are studied. If the boundary ∂S$\partial S$ of S   is a polynomial-time computable Jordan curve, the complexity of these problems can be characterized by counting classes #P$\#P$, MP (or MidBitP  ), and ⊕P$\oplus P$: The logarithm problem is polynomial-time solvable if and only if FP=#P$\mathit{FP}=\#P$. For the square root problem, it has been shown to have the upper bound P^MP$P^{\mathit{MP}}$ and lower bound P^⊕P$P^{\oplus P}$. That is, if P=MP$P=\mathit{MP}$ then the square root problem is polynomial-time solvable, and if P≠⊕P$P\ne \oplus P$ then the square root problem is not polynomial-time solvable.