The function pade computes a Pade approximation of degree $m,n$ for the function f with respect to the variable x.

Specifically, f is expanded in a Taylor (or Laurent) series about the point $x=a$ (if a is not specified then the expansion is about the point $x=0$), to order $m+n+1$, and then the Pade rational approximation is computed.

The $m,n$ Pade approximation is defined to be the rational function $\frac{p\left(x\right)}{q\left(x\right)}$ with $\mathrm{deg}\left(p\left(x\right)\right)\le m$ and $\mathrm{deg}\left(q\left(x\right)\right)\le n$ such that the Taylor (or Laurent) series expansion of $\frac{p\left(x\right)}{q\left(x\right)}$ has maximal initial agreement with the series expansion of f. In normal cases, the series expansion agrees through the term of degree $m+n$.

If $n=0$ or if the third argument is simply an integer m then the Taylor (or Laurent) polynomial of degree m is computed.

Various levels of user information will be displayed during the computation if infolevel[pade] is assigned values between 1 and 3.

The command with(numapprox,pade) allows the use of the abbreviated form of this command.

$\mathrm{with}\left(\mathrm{numapprox}\right)\:$

$\mathrm{pade}\left({\ⅇ}^x\,x\,\left[3\,3\right]\right)$

$\frac{\frac{1}{10}{x}^2+\frac{1}{2}x+1+\frac{1}{120}{x}^3}{\frac{1}{10}{x}^2-\frac{1}{2}x+1-\frac{1}{120}{x}^3}$

$\mathrm{pade}\left(\frac{1}{x\sin \left(x\right)}\,x\=0\,\left[4\,6\right]\right)$

$\frac{75x^4+5460x^2+166320}{551x^6-22260x^4+166320x^2}$

$\mathrm{pade}\left(\mathrm{GAMMA}\left(x\right)\,x\=1\,\left[1\,1\right]\right)$

$\frac{\mathrm{\gamma }+\left(-\frac{{\mathrm{\gamma }}^2}{2}+\frac{{\mathrm{\pi }}^2}{12}\right)\left(x-1\right)}{\mathrm{\gamma }+\left(\frac{{\mathrm{\pi }}^2}{12}+\frac{{\mathrm{\gamma }}^2}{2}\right)\left(x-1\right)}$

$\mathrm{pade}\left(\cos \left(x\right)\,x\,\left[3\,4\right]\right)$

$\frac{1-\frac{61x^2}{150}}{\frac{7}{75}{x}^2+1+\frac{1}{200}{x}^4}$

$\mathrm{pade}\left(\cos \left(x\right)\,x\,7\right)$

$1-\frac{1}{2}{x}^2+\frac{1}{24}{x}^4-\frac{1}{720}{x}^6$