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 axe studied. If the boundary partial derivative S of S is a polynomial-time computable Jordan curve, the complexity of these problems can be characterized by counting classes #P, MP (or MidBitP), and circle plus P: The logarithm problem is polynomial-time solvable if and only if FP = #P. For the square root problem, it has been shown to have the upper bound P-MP and lower bound P-circle plus P. That is, if P = MP then the square root problem is polynomial-time solvable, and if P not equal circle plus P then the square root problem is not polynomial-time solvable.