∂Ω with precision of at most In particular, if ∂Ω is polynomial space computable in space for some constant and time then A can be used to compute the uniformizing map in space and time There is an algorithm A′ that computes the uniformizing map in the following sense. Let be a bounded simply-connected domain, and Suppose that for some ∂Ω is given to A′ with precision by pixels. Then A′ computes the absolute values of the uniformizing map within an error of in randomized space bounded by and time polynomial in (that is, by a BPL(n)-machine). Furthermore, the algorithm computes the value of with precision as long as Negative results: Suppose there is an algorithm A that given a simply-connected domain with a linear-time computable boundary and an inner radius > 1/2 and a number computes the first digits of the conformal radius then we can use one call to A to solve any instance of a #SAT(n) with a linear time overhead. In other words, #P is poly-time reducible to computing the conformal radius of a set. Consider the problem of computing the conformal radius of a simply-connected domain where the boundary of is given with precision by an explicit collection of pixels. Denote the problem of computing the conformal radius with precision by Then, is AC0 reducible to for any See also Measurable Riemann mapping theorem Schwarz–Christoffel mapping – a conformal transformation of the upper half-plane onto the interior of a simple polygon. Conformal radius Notes References External links Theorems in complex analysis Bernhard Riemann