The Padé approximation is a rational function that can be thought of as a generalization of a Taylor polynomial. A rational function is the ratio of polynomials. Because these functions only use the elementary arithmetic operations, they are very easy to evaluate numerically. The polynomial in the denominator allows you to approximate functions that have rational singularities.
More precisely, a Padé approximation of order
(n, m) to an analytic function
f (x) at a regular point or pole
x0 is the rational function
where
p (x) is a polynomial of degree
n,
q (x) is a polynomial of degree
m, and the formal power series of
f (x)q (x)-p (x) about the point
x0 begins with the term
(x-x0)n+m+1. If
m is equal to
n, the approximation is called a diagonal Padé approximation of order
n.
In
Mathematica PadeApproximant is generalized to allow expansion about branch points.
