Padéapproximants

Padé approximants are a class of rational approximants to a function, constructed to match as many terms as possible of a function’s power series at a point, typically the origin. Given a function f with a Taylor series f(z) = sum_{k=0}^\infty a_k z^k, the [m/n] Padé approximant is the rational function P_m(z)/Q_n(z) where P_m is a polynomial of degree at most m and Q_n is a polynomial of degree at most n, with Q_n(0) = 1. The approximant is chosen so that the power series of P_m(z)/Q_n(z) agrees with f(z) up to the order z^{m+n}; equivalently, f(z) Q_n(z) - P_m(z) = O(z^{m+n+1}).

Construction begins by writing Q_n(z) = 1 + q_1 z + ... + q_n z^n and P_m(z) = p_0 + p_1 z + ... + p_m z^m. The coefficients are determined by equating coefficients of z^k in the identity f(z) Q_n(z) - P_m(z) = O(z^{m+n+1}) for k = 0, ..., m+n. This yields a linear system for the unknowns p_0, ..., p_m and q_1, ..., q_n. The solution is unique up to an overall nonzero factor; normalization Q_n(0) = 1 fixes the scale.

Variants and properties include diagonal and near-diagonal approximants, collectively forming the Padé table. Padé approximants can provide analytic continuation beyond the radius of convergence of the original series and can represent functions with nearby poles more accurately than polynomials. They may, however, introduce spurious poles (Froissart doublets) and do not always converge to f.

History and use: Padé approximants are named after Henri Padé, who introduced them in the late 19th century. They are employed in numerical analysis, physics, and engineering to approximate functions from series, evaluate integrals, and study analytic structure, with connections to continued fractions and other rational approximation methods.