This post was inspired by a lecture series of Carl Bender [1].

## The case

I will consider the function on the complex plane, defined by

Have a look at a complex plot of it

- has a a logarithmic singularity at .
- is analytic in
- has a logarithmic cut on the line

For the following series representation holds

Claim: The series sucks in practice!

## The study (i)

Once upon a time some guy called Padé came along, being unnerved by the wackness of Taylor-like series, and proposed what is now known as Padé approximants. His idea was to find polynomials and of degree and , such that

.

In a similar way of Taylor polynomials approximating , he simply tried rational expressions.

Q: How does one find these these polynomials? A: By comparing to the Taylor series.

**Example 1:** Find . Start with an ansatz:

.

where I already put the lowest order coefficient in the denominator to , which is a convention that removes the ambiguousness of the polynomial parameters when considering ratios. So we have four parameters, which we can match to a Taylor polynomial of order three. To make progress, we expand our ansatz in powers of

where I already dropped terms that will be of fourth power or higher in . This expression should be matched to order by order in . A little algebra yields

.

This set of equations has a simple solution, resulting in

A quick fireprobe: this approximant suggests the approximate value for . This isn’t too bad, in fact you can add it to your stash of quick and dirty calculation library. Students are always impressed by this sort of witchcraft. I dare you to find an easier-to-remember value from the series representation…

**Example 2:** The Taylor polynomials are contained within the Padé approximants as

.

To be continued… I’m lacking blogging time atm…

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