Continuous fraction

The same as continued fraction, i.e. an expression of the form

where

are finite or infinite sequences of complex numbers or functions. For continued fractions one uses the notation

(*)

As a rule, it is assumed that the sequences and are such that for all , ,

(The are defined recursively, with , .)

Comments

In translations the phrase continuous fraction occurs occasionally instead of continued fraction.

The notation (*) is due to A. Pringsheim, cf. [a1] for other notations.

The number or function obtained by evaluation of the truncation

is called the -th convergent of the continued fraction. It equals where satisfy the recursion given above (with ) and starting values , ; , . If the convergents tend to a limit as , one says that the continued fraction converges.

Continued fractions play an important role in number theory, in particular in Diophantine approximation, where , , in ergodic theory, again with , and in numerical mathematics, where are rational functions. Famous examples of explicit continued fractions are those for hypergeometric functions such as

whose corresponding 's are orthogonal polynomials. In number theory the most famous is the golden ratio

Good introductions to continued fractions in number theory are [a1], Chapt. X, [a2], in numerical mathematics [a3], and generally [a4].

References

[a1]	G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Chapts. 5; 7; 8
[a2]	A.Ya. [A.Ya. Khinchin] Khintchine, "Kettenbrüche" , Teubner (1956) (Translated from Russian)
[a3]	H.S. Wall, "Continued fractions" , v. Nostrand (1948)
[a4]	O. Perron, "Die Lehre von den Kettenbrüche" , 1–2 , Teubner (1954)