Continuous fraction
The same as continued fraction, i.e. an expression of the form
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where
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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
,
,
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(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
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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
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whose corresponding 's are orthogonal polynomials. In number theory the most famous is the golden ratio
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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) |
Continuous fraction. O.A. Ivanova (originator), Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Continuous_fraction&oldid=13285