Padé approximant

Definition

About a general point and for a given order

Suppose $f$ is a function, $x_0$ is a point in the domain of $f$, and $m,n$ are (possibly equal, possibly distinct) nonnegative integers. The Padé approximant to $f$ of order $[m/n]$ at $x_0$ is a rational function of the form:

$R(x)=\frac{a_0+a_1(x-x_0)+a_2(x-x_0{)}^2+\dots +a_m(x-x_0{)}^m}{1+b_1(x-x_0)+\dots +b_n(x-x_0{)}^n}$

where $a_i,b_j$ are all real numbers, and where $f^{(j)}(x_0)=R^{(j)}(x_0)$ for $j\in \{0,1,2,\dots ,m+n\}$.