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 the order of the lowest order term in the Laurent series is a negative integer $v$ and $n+v<0$, then no rational approximation with a denominator of degree at most $n$ can exist, and an error is raised. If $v\&gt;m\ge 0$, the return value is $0$.

If the third argument is simply an integer $m$, then the Taylor (or Laurent) polynomial of (relative) 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.

$\mathrm{with}\left(\mathrm{numapprox}\right)\&colon;$

$\mathrm{pade}\left(\exp \left(x\right)\&comma;x\&comma;\left[3\&comma;3\right]\right)$

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

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

$\frac{1+\frac{13}{396}{x}^2+\frac{5}{11088}{x}^4}{\frac{551}{166320}{x}^6-\frac{53}{396}{x}^4+x^2}$

$\mathrm{pade}\left(\mathrm{\Gamma }\left(x\right)\&comma;x=1\&comma;\left[1\&comma;1\right]\right)$

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

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

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

$\mathrm{pade}\left(\cos \left(x\right)\&comma;x\&comma;7\right)$

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

$\mathrm{pade}\left(\frac{\exp \left(x\right)}{x^3}\&comma;x\&comma;\left[4\&comma;0\right]\right)$

Error, (in `convert/ratpoly`) no rational approximation with denominator degree <= 0

$\mathrm{pade}\left(\frac{\exp \left(x\right)}{x^3}\&comma;x\&comma;4\right)$

$\frac{1}{x^3}+\frac{1}{x^2}+\frac{1}{2x}+\frac{1}{6}+\frac{x}{24}$

$\mathrm{pade}\left(\exp \left(x^3\right)-1\&comma;x\&comma;\left[2\&comma;5\right]\right)$

$0$