The digits of pi go on forever apparently with no discernible pattern. However, there are beautifully simple patterns in its (ironically) *non-simple continued fraction* expansions. Examples are,

known by Lord Brouncker (1620-1684), and,

One can see the affinity between the two. They in fact belong to the same family. Given complex numbers {*n, x*} with , then,

where is the *gamma function*. This is Entry 25 in *Ramanujan’s Second Notebook* (Chapter 12) though this result was also known by Euler. For the case *n *= 0, the continued fraction assumes the form of the examples and the function simplifies as,

For x an *odd integer*, then *F *(0, x) is a rational multiple of or . Specifically, for *x* = {1, 3, 5, 7, 9}, we have,

More generally,

See p. 178 of Annie Cuyt’s *HandBook of Continued Fractions for Special Functions*. For more examples, see also this article *Ramanujan’s Continued Fractions, Apery’s Constant, and more*.

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Posted by nicco1mnisi on October 7, 2015 at 4:27 pm

That’s a beautiful continued fraction