Convergence of Shafer quadratic approximants

Let [functionof] [setmembership]M([scriptR] sub(3)) be a (single-valued) meromorphic function on a Riemann surface with three sheets [pi]:[scriptR] sub(3) arrow right [dbl-struck R] (z [mapsto] z) which has only second-order branch points and does not have z= [infinity] as a branch point. The set [pi] super(-1)([infinity]) consists of three distinct points. We identify one of them, [infinity] super((1))[setmembership] [dbl-struck R] sub(3), with the point [infinity] [setmembership] [dbl-struck C] and assume that [functionof] [setmembership] [scriptH]([infinity]). Let [scriptU] be an Abelian integral of the third kind on [scriptR] sub(3) with purely imaginary periods that has singularities only at points in [pi] super(-1)([infinity]) and satisfies [scriptU](z) = -2log z+ (regular terms) as z arrow right [infinity] super((1)) and [scriptU](z) = log z+ (regular terms) as z tends to any other point in [pi] super(-1)([infinity]). The inequalities Re [scriptU](z super((1))) < Re [scriptU](z super((2))) < Re [scriptU](z super((3))) partition the Riemann surface [scriptR] sub(3) into three open 'sheets' [scriptR] super((1))[suchthat] [infinity] super((1)), [scriptR] super((2)), and [scriptR] super((3)).