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 complex values. A generalized continued fraction is an expression of the form where the () are the partial numerators, the are the partial denominators, and the leading term is called the integer part of the continued fraction. The successive convergents of the continued fraction are formed by applying the fundamental recurrence formulas: where is the numerator and is the denominator, called continuants, of the th convergent. They are given by the recursion with initial values If the sequence of convergents 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 . History The story of continued fractions begins with the Euclidean algorithm, a procedure for finding the greatest common divisor of two natural numbers and . 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 devised a technique for approximating the roots of quadratic equations with continued fractions in the mid-sixteenth century. Now the pace of development quickened. Just 24 years later, in 1613, 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. In 1748 Euler published a theorem showing that a particular kind of continued fraction is equivalent to a certain very general infinite series. Euler's continued fraction formula is still the basis of many modern proofs of convergence of continued fractions. In 1761, Johann Heinrich Lambert gave the first proof that is irrational, by using the following continued fraction for : Continued fractions can also be applied to problems in number theory, and are especially useful in