Padé approximant

A rational function whose Taylor series agrees with a given power series through a specified number of terms. It can represent behavior, such as poles, that a polynomial truncation cannot.

Taylor series: A power series representing a function through its derivatives at a point. Its coefficients supply the data that the rational function is built to match.
Analytic continuation: The extension of an analytic function beyond its initial domain while preserving its local values. Padé approximants can estimate function values outside a power series' disk of convergence.
Taylor polynomial: A finite polynomial formed by truncating a function's Taylor series. Unlike a Padé approximant, it has no denominator to represent poles or other rational behavior.
Henri Padé: A French mathematician known for systematic study of rational approximation and continued fractions. His name became attached to the approximants he developed and organized.
Rational function: A quotient of two polynomials, with a nonzero denominator. A Padé approximant belongs to this broader class rather than the class of polynomials.
Borel summation: A method for assigning values to certain divergent series through a transformed integral. Padé approximants often help evaluate the transformed series used in this resummation method.
Chebyshev approximation: Polynomial approximation that minimizes maximum error over a specified interval. It optimizes error over an interval, while Padé matching prioritizes local series agreement.
Charles Hermite: A French mathematician whose work included approximation theory and transcendental number theory. His rational approximation work helped shape the mathematical setting in which Padé worked.
Padé table: An array of Padé approximants indexed by numerator and denominator degrees. It organizes approximants formed from the same power series at different degree choices.
Perturbation theory: Methods that approximate a problem's solution using expansions in a small parameter. Rational approximants can improve predictions from finite perturbative expansions.

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