Continued fraction
Regular patterns in continued fractions

While there is no discernible pattern in the simple continued fraction expansion of π, there is one for e, the base of the natural logarithm:

which is a special case of this general expression for positive integer n:

Another, more complex pattern appears in this continued fraction expansion for positive odd n:

with a special case for n = 1:

Other continued fractions of this sort are

where n is a positive integer; also, for integer n:

with a special case for n = 1:

If I (x) is the modified, or hyperbolic, Bessel function of the first kind, we may define a function on the rationals p / q by

which is defined for all rational numbers, with p and q in lowest terms. Then for all nonnegative rationals, we have

with similar formulas for negative rationals; in particular we have

Many of the formulas can be proved using Gauss's continued fraction.
length_constraints:number_paragraphs[3], detectable_content:postscript