---
tag: 1F16G4V64X
kind: concept
name: Padé approximant
msc: ["41A21", "30E10", "30B70"]
specializes: [0XDK9CGYKJ]
sources:
  - url: https://en.wikipedia.org/wiki/Pad%C3%A9_approximant
same_as:
  wikidata: Q2452504
  wikipedia: Padé approximant
---

## Summary

A Padé approximant of a power series is a [[rational function|0WCZH4AFMT]] with prescribed degrees.
Its own power series agrees with the given one to as high an order as these degrees allow.
For numerator degree at most $m$ and denominator degree at most $n$, the agreement is up to the power $z^{m+n}$.
Padé approximants often approximate a function far beyond the disc of convergence of its Taylor series.
They can also show the location of poles.
They are closely linked to [[continued fractions|0BK6JBK58W]] and to orthogonal polynomials.

## Explanations

### L1 human

A [[Taylor polynomial|0HFSFVMWQB]] approximates a function by a polynomial, which cannot have poles and tends to infinity far away.
A Padé approximant uses a quotient of two polynomials instead, so it can imitate poles and limits at infinity.
The coefficients are chosen so that the quotient matches as many Taylor coefficients as possible.
For example, $(1 + z/2)/(1 - z/2)$ matches $e^z = 1 + z + z^2/2 + \cdots$ up to the term $z^2$.
The approximants for all pairs of degrees form a table, the Padé table.