{"id":282,"date":"2024-06-24T17:39:34","date_gmt":"2024-06-24T17:39:34","guid":{"rendered":"https:\/\/sabba.me\/nmr\/?p=282"},"modified":"2024-06-24T17:48:53","modified_gmt":"2024-06-24T17:48:53","slug":"cfihkal-continued-fractions-i-have-known-and-loved","status":"publish","type":"post","link":"https:\/\/sabba.me\/nmr\/2024\/06\/24\/cfihkal-continued-fractions-i-have-known-and-loved\/","title":{"rendered":"CFiHKAL: Continued Fractions I have Known and Loved"},"content":{"rendered":"\n

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<\/a>.<\/p>\n\n\n\n

How I met continued fractions<\/strong><\/p>\n\n\n\n

After graduating high school at 16, I discovered continued fractions, having been intrigued by much of the mythology surrounding the Indian mathematician Srinavasa Ramanujan<\/a>. Continued fractions were one (of several) areas of mathematics Ramanujan made incredible contributions to.<\/p>\n\n\n\n

[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.]<\/p>\n\n\n\n

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<\/a>) you can find here<\/a>. <\/p>\n\n\n\n

Later, as an undergrad, I enjoyed the work of the Soviet mathematician Alexander Khinchin<\/a> – yes physicists, the same Khinchin of Wiener-Khinchin theorem<\/a> fame – since the master of the Soviet school of probability theory also published the classic book on continued fractions<\/a>.<\/p>\n\n\n\n

I then became a scientist and unfortunately lost a lot of the freedom\/time to pursue this little side-gig.<\/p>\n\n\n\n

So here’s a collection of cool continued fractions I have encountered over the last 10 years.<\/p>\n\n\n\n

1. Powers of the Golden Ratio and the Lucas numbers<\/strong><\/p>\n\n\n\n

This is the most trivial example of a simple continued fraction:<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Which is the golden ratio<\/a>:<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Continued fractions can be more neatly expressed in the “K notation” commonly ascribed to Carl Friedrich Gauss (from Kettenbruch<\/em>, the German name for a continued fraction):<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

One of the first cool simple patterns I stumbled upon in mathematics was the following relationship relating to powers of the golden ratio:<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Which has the general form:<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

In which Ln<\/sub><\/em> denotes the Lucas numbers<\/a>.<\/p>\n\n\n\n

2. Good ol’ \u03c0<\/em><\/strong><\/p>\n\n\n\n

The simple <\/em>continued fraction for \u03c0 may appear to show no consistent pattern:<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

But there are incredible <\/em>non-simple continued fractions for Pi. The most famous is Lord Brouncker<\/a>‘s continued fraction:<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

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<\/a>. <\/p>\n\n\n\n

Other contfracs include:<\/p>\n\n\n\n

    \n
  1. “An Elegant Continued Fraction for Pi”<\/a> by Leo Jerome Lange, American Mathematical Monthly 106 (1999)


    \"\"<\/figure>
    <\/li>\n\n\n\n
  2. “Another Continued Fraction for Pi”<\/a> by Thomas J .Pickett, Ann Coleman, American Mathematical Monthly 115 (2008)


    \"\"<\/figure>
    <\/li>\n<\/ol>\n\n\n\n

    3. Ratios of the Modified Bessel Function<\/strong><\/p>\n\n\n\n

    In 1990, Stanley Rabowitz posed a fascinating problem:<\/p>\n\n\n\n

    \"\"<\/figure>\n\n\n\n

    Which was formally proven by Neville Robbins in 1993 in the Fibonacci Quarterly, 33(4); pages 311-312<\/a>.<\/p>\n\n\n\n

    Incredibly, there is a general identity which may be found<\/a> in the Handbook of Continued Fractions for Special Functions<\/a>:<\/p>\n\n\n\n

    \"\"<\/figure>\n\n\n\n

    [I may add some more contfracs in the future. For now I have to get back to doing my job.]<\/p>\n","protected":false},"excerpt":{"rendered":"

    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 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[21],"tags":[23,22,24],"class_list":["post-282","post","type-post","status-publish","format-standard","hentry","category-math","tag-continued-fractions","tag-math","tag-ramanujan"],"_links":{"self":[{"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/posts\/282","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/comments?post=282"}],"version-history":[{"count":5,"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/posts\/282\/revisions"}],"predecessor-version":[{"id":310,"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/posts\/282\/revisions\/310"}],"wp:attachment":[{"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/media?parent=282"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/categories?post=282"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/tags?post=282"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}