# Preface

This is a
tutorial made solely for the purpose of education and it was designed
for students taking Applied Math 0330. It is primarily for students who
have very little experience or have never used
*Mathematica* before and would like to learn more of the basics for this computer algebra system. As a friendly reminder, don't forget to clear variables in use and/or the kernel.

Finally, the commands in this tutorial are all written in bold black font,
while *Mathematica* output is in normal font. This means that you can
copy and paste all commands into *Mathematica*, change the parameters and
run them. You, as the user, are free to use the scripts for your needs to learn the *Mathematica* program, and have
the right to distribute this tutorial and refer to this tutorial as long as
this tutorial is accredited appropriately.

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# Padé Approximation

A **Padé approximant** is the "best" approximation of a function by a rational function of given
order -- under this technique, the approximant's power series agrees with the power series of the function it is
approximating. The technique was developed around 1890 by the French mathematician Henri Padé (1863--1953), but goes
back to the German mathematician Georg Frobenius (1849--1917) who introduced the idea and investigated the
features of rational approximations of power series. Henri Eugène Padé, while preparing his doctorate under Charles Hermite,
he introduced what is now known as the Padé approximant.

Given a function *f* and two integers \( m \ge 0 \quad\mbox{and}\quad n \ge 1, \)
the Padé approximant of order [m/n] is the rational function

*f(x)*to the highest possible order, which amounts to

*R(x)*is expanded in a Maclaurin (or Taylor) series its first

*m + n*terms would cancel the first

*m + n*terms of

*f(x)*, and as such:

Example: Consider Padé approximants for cosine functions

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