This post was inspired by a lecture series of Carl Bender .
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…