View source for Continuous fraction

The same as [[Continued fraction|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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025640/c0256403.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

As a rule, it is assumed that the sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025640/c0256404.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025640/c0256405.png" /> are such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025640/c0256406.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025640/c0256407.png" />,

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(The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025640/c0256409.png" /> are defined recursively, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025640/c02564010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025640/c02564011.png" />.)



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

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

The number or function obtained by evaluation of the truncation

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is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025640/c02564013.png" />-th convergent of the continued fraction. It equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025640/c02564014.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025640/c02564015.png" /> satisfy the recursion given above (with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025640/c02564016.png" />) and starting values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025640/c02564017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025640/c02564018.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025640/c02564019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025640/c02564020.png" />. If the convergents <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025640/c02564021.png" /> tend to a limit as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025640/c02564022.png" />, one says that the continued fraction converges.

Continued fractions play an important role in number theory, in particular in Diophantine approximation, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025640/c02564023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025640/c02564024.png" />, in ergodic theory, again with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025640/c02564025.png" />, and in numerical mathematics, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025640/c02564026.png" /> are rational functions. Famous examples of explicit continued fractions are those for hypergeometric functions such as

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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025640/c02564028.png" /></td> </tr></table>

whose corresponding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025640/c02564029.png" />'s are orthogonal polynomials. In number theory the most famous is the golden ratio

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025640/c02564030.png" /></td> </tr></table>

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

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