Multisingularity and scaling in partial differential approximants. I

Abstract

A partial differential approximant (or PDA),F(x, y) approximates a functionf(x, y), specified by its truncated power series, in terms of a solution of a defining linear partial differential equation with polynomial coefficients. The intrinsic multisingularities of a PDA, which may approximate corresponding singularities off(x, y), are analysed formally and shown to obey, in general, asymptotic scaling (as familiar in the theory of critical phenomena), i. e.F(x, y) ≈C|x͠|-γZ(y͠/|x͠|ϕ) +B, wherex͠andy͠are linear combinations of the deviations, (x-xc) and (y-yc), from the multisingular point (xc,yc). Explicit formulae, suitable for numerical computation, are derived for the characteristic exponents,γandϕ, for the scaling functionZ( • ), for its expansion coefficients and for the related coefficientsCandB, in the case when the crossover exponent,ϕ, lies in the interval (½, 2). (Part II extends these results to general values ofϕ, which requires the introduction of the nonlinear scaling fields associated with the PDA.)