Departing from our usual programming, this will not be a blog post about anything related (at least directly) to magnetic resonance or quantum dynamics. It will be a post about mathematics, particularly continued fractions.
How I met continued fractions
After graduating high school at 16, I discovered continued fractions, having been intrigued by much of the mythology surrounding the Indian mathematician Srinavasa Ramanujan. Continued fractions were one (of several) areas of mathematics Ramanujan made incredible contributions to.
[It goes without saying that Ramanujan was one of the most naturally gifted mathematicians in history, despite never having received any formal training, with a signature habit of providing extraordinary results without proof.]
So I spent a lot of my time on random number theory problems, and published a couple of my continued fractions in the Online Encyclopedia of Integer Sequences (OEIS) you can find here.
Later, as an undergrad, I enjoyed the work of the Soviet mathematician Alexander Khinchin – yes physicists, the same Khinchin of Wiener-Khinchin theorem fame – since the master of the Soviet school of probability theory also published the classic book on continued fractions.
I then became a scientist and unfortunately lost a lot of the freedom/time to pursue this little side-gig.
So here’s a collection of cool continued fractions I have encountered over the last 10 years.
1. Powers of the Golden Ratio and the Lucas numbers
This is the most trivial example of a simple continued fraction:
Which is the golden ratio:
Continued fractions can be more neatly expressed in the “K notation” commonly ascribed to Carl Friedrich Gauss (from Kettenbruch, the German name for a continued fraction):
One of the first cool simple patterns I stumbled upon in mathematics was the following relationship relating to powers of the golden ratio:
Which has the general form:
In which Ln denotes the Lucas numbers.
2. Good ol’ π
The simple continued fraction for π may appear to show no consistent pattern:
But there are incredible non-simple continued fractions for Pi. The most famous is Lord Brouncker‘s continued fraction:
Lord Brouncker had a fascinating biography which included being the first president of the Royal Society and having a prolonged affair with the wife of the first cousin of Oliver Cromwell.
Other contfracs include:
- “An Elegant Continued Fraction for Pi” by Leo Jerome Lange, American Mathematical Monthly 106 (1999)
- “Another Continued Fraction for Pi” by Thomas J .Pickett, Ann Coleman, American Mathematical Monthly 115 (2008)
3. Ratios of the Modified Bessel Function
In 1990, Stanley Rabowitz posed a fascinating problem:
Which was formally proven by Neville Robbins in 1993 in the Fibonacci Quarterly, 33(4); pages 311-312.
Incredibly, there is a general identity which may be found in the Handbook of Continued Fractions for Special Functions:
[I may add some more contfracs in the future. For now I have to get back to doing my job.]