From: cet1@cus.cam.ac.uk (Chris Thompson)
Newsgroups: sci.math
Subject: Re: Bailey/Borwein/Plouffe method (was: Re: HELP--> Pi? Rational?)
Date: 28 Oct 1995 18:59:24 GMT

In article <46oird$rn5@news-rocq.inria.fr>, harley@pauillac.inria.fr (Robert Harley)
writes:
|> > cet1@cus.cam.ac.uk (Chris Thompson) writes:
[...]
|> >I rediscovered your formula after reading the BBP paper, 

In case it wasn't obvious, the "paper" in question is the PostScript document
fetched from the URL that has been posted on this group:
http://www.cecm.sfu.ca/~pborwein/PAPERS/P123.ps

Of course, I wasn't claiming any great merit in said "rediscovery"; just 
that it's easy to find once you have seen BBP's.

|> >which is unnecessarily obfusticating on the details here.
|> >
|> >[... some more obfuscation removed ;) ...]

I still claim BBP obfusticate more! Of course, it was them saying "That the 
integral (1.4) evaluates to \pi is an exercise in partial fractions most
easily done in Maple or Mathematica" that really set me off. I should need
Maple or Mathematica to do this? Ooh, what an insult! :-)

|> Huh?
|> 
|> ----------------------------------------------------------------------
|> Express ln((1+i)/2) using Taylor's series for ln(1+x) with x=(i-1)/2.
|> ----------------------------------------------------------------------
|> 
|> There, pi in base 2.  You even get ln(2) for free.  I believe I first
|> noticed this in a boring maths class in Terminale C (French
|> high-school), as I'm sure many people have in the last 280 years.

Indeed, that is a much cleaner and more concise way of doing things. Looking
at ln(1+x) for x each of the 8'th roots of 1/16 [cf. Joe Keane's shifted octagaon]
covers the space of linear combinations of s_{1..8} mentioned in my post. The 
imaginary part of the above gives the combination (2,2,1,0,-1/2,-1/2,-1/4,0) for
\pi which I did say was the "natural" one, but I had better withdraw the 
"suprising". To realise that there is a zero combination, one observes that

   (1 - 1/\sqrt2)(1 + 1/\sqrt2) = (1 - (1+i)/2)(1 - (1-i)/2)

which might still be worth a single twitch of the eyebrows.

Chris Thompson
Email: cet1@cam.ac.uk