Fundamental recurrence formulas
In complex analysis, a branch of mathematics, a generalized continued fraction is a generalization of regular continued fractions in canonical form, in which the partial numerators and partial denominators can assume arbitrary real or complex values.
A generalized continued fraction is an expression of the form
where the an (n > 0) are the partial numerators, the bn are the partial denominators, and the leading term b0 is called the integer part of the continued fraction.
The successive convergents of the continued fraction are formed by applying the fundamental recurrence formulas:
and in general
where An is the numerator and Bn is the denominator, called continuants, of the nth convergent.
If the sequence of convergents {xn} approaches a limit the continued fraction is convergent and has a definite value. If the sequence of convergents never approaches a limit the continued fraction is divergent. It may diverge by oscillation (for example, the odd and even convergents may approach two different limits), or it may produce an infinite number of zero denominators Bn.
The story of continued fractions begins with the Euclidean algorithm, a procedure for finding the greatest common divisor of two natural numbers m and n. That algorithm introduced the idea of dividing to extract a new remainder – and then dividing by the new remainder repeatedly.
Nearly two thousand years passed before Rafael Bombelli devised a technique for approximating the roots of quadratic equations with continued fractions. Now the pace of development quickened. Just 24 years later Pietro Cataldi introduced the first formal notation for the generalized continued fraction. Cataldi represented a continued fraction as
with the dots indicating where the next fraction goes, and each & representing a modern plus sign.
Late in the seventeenth century John Wallis introduced the term "continued fraction" into mathematical literature. New techniques for mathematical analysis (Newton's and Leibniz's calculus) had recently come onto the scene, and a generation of Wallis' contemporaries put the new phrase to use.
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