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. Suppose further that $f$ is at least $m+n$ times differentiable at $x_0$.

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 $j\in \{0,1,2,\dots ,m+n\}$.

About the point 0 and for a given order

This definition adapts the previous one for the case $x_0=0$.

Suppose $f$ is a function and $m,n$ are (possibly equal, possibly distinct) nonnegative integers. Suppose further that $f$ is at least $m+n$ times differentiable at 0.

The Padé approximant to $f$ of order $[m/n]$ at 0 is a rational function of the form:

$R(x)=\frac{a_0+a_{1}x+a_2{x}^2+\dots +a_m{x}^m}{1+b_{1}x+\dots +b_n{x}^n}$

where $a_0,a_1,\dots ,a_m,b_1,b_2,\dots ,b_n$ are all real numbers, and where $f^{(j)}(0)=R^{(j)}(0)$ for $j\in \{0,1,2,\dots ,m+n\}$.

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