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| In [[complex analysis]], a branch of mathematics, a '''generalized continued fraction''' is a generalization of regular [[continued fraction]]s in canonical form, in which the partial numerators and partial denominators can assume arbitrary real or complex values.
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| A generalized continued fraction is an expression of the form
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| :<math>x = b_0 + \cfrac{a_1}{b_1 + \cfrac{a_2}{b_2 + \cfrac{a_3}{b_3 + \cfrac{a_4}{b_4 + \ddots\,}}}}</math>
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| where the ''a''<sub>''n''</sub> (''n'' > 0) are the partial numerators, the ''b''<sub>''n''</sub> are the partial denominators, and the leading term ''b''<sub>0</sub> is called the ''integer'' part of the continued fraction.
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| The successive '''convergents''' of the continued fraction are formed by applying the '''fundamental recurrence formulas''':
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| :<math>
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| x_0 = \frac{A_0}{B_0} = b_0, \qquad
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| x_1 = \frac{A_1}{B_1} = \frac{b_1b_0+a_1}{b_1},\qquad
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| x_2 = \frac{A_2}{B_2} = \frac{b_2(b_1b_0+a_1) + a_2b_0}{b_2b_1 + a_2},\qquad\cdots\,
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| </math>
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| and in general<ref name=JT20>Jones & Thron (1980) p.20</ref>
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| :<math>
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| A_n = b_n A_{n-1} + a_n A_{n-2}, \qquad
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| B_n = b_n B_{n-1} + a_n B_{n-2}, \,
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| </math>
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| where ''A''<sub>''n''</sub> is the ''numerator'' and ''B''<sub>''n''</sub> is the ''denominator'', called [[Continuant (mathematics)|continuant]]s,<ref>{{cite book | title=The Markoff and Lagrange Spectra | author=Thomas W. Cusick | coauthors=Mary E. Flahive | publisher=American Mathematical Society | year=1989 | isbn=0-8218-1531-8 | pages=89 }}</ref><ref>{{cite book | title=Algebra, an Elementary Text-book for the Higher Classes of Secondary Schools and for Colleges: Pt. 1 | author=George Chrystal | authorlink=George Chrystal | publisher=American Mathematical Society | year=1999 | isbn=0-8218-1649-7 | pages=500 }}</ref> of the ''n''th convergent.
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| If the sequence of convergents {''x''<sub>''n''</sub>} 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 ''B''<sub>''n''</sub>.
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| == History of continued fractions ==
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| The story of continued fractions begins with the [[Euclidean algorithm]],<ref>300 BC [[Euclid]], ''Elements'' - The Euclidean algorithm generates a continued fraction as a by-product.</ref> 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 again, and again, and ''again''.
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| Nearly two thousand years passed before [[Rafael Bombelli]]<ref>1579 [[Rafael Bombelli]], ''L'Algebra Opera''</ref> devised a [[Solving quadratic equations with continued fractions|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<ref>1613 [[Pietro Cataldi]], ''Trattato del modo brevissimo di trovar la radice quadra delli numeri''; roughly translated, ''A treatise on a quick way to find square roots of numbers''.</ref> for the generalized continued fraction. Cataldi represented a continued fraction as
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| :<math>a_0.\,</math> &<math> n_1 \over d_1. </math> &<math> n_2 \over d_2. </math> &<math> {n_3 \over d_3},</math>
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| with the dots indicating where the next fraction goes, and each & representing a modern plus sign. | |
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| Late in the seventeenth century [[John Wallis]]<ref>1695 [[John Wallis]], ''Opera Mathematica'', Latin for ''Mathematical Works''.</ref> introduced the term "continued fraction" into the mathematical literature. New techniques for mathematical analysis ([[Isaac Newton|Newton's]] and [[Gottfried Wilhelm Leibniz|Leibniz's]] [[calculus]]) had recently exploded onto the scene, and a generation of Wallis' contemporaries put the new word to use right away.
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| In 1748 [[Leonhard Euler|Euler]] published a very important theorem showing that a particular kind of continued fraction is equivalent to a certain very general [[infinite series]].<ref name="Euler">1748 [[Leonhard Euler]], ''Introductio in analysin infinitorum'', Vol. I, Chapter 18.</ref> Euler's continued fraction theorem is still of central importance in modern attempts to whittle away at the [[convergence problem]].
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| Continued fractions can also be applied to problems in [[number theory]], and are especially useful in the study of [[Diophantine equation]]s. In the late eighteenth century [[Joseph Louis Lagrange|Lagrange]] used continued fractions to construct the general solution of [[Pell's equation]], thus answering a question that had fascinated mathematicians for more than a thousand years.<ref>[[Brahmagupta]] (598 - 670) was the first mathematician to make a systematic study of Pell's equation.</ref> Amazingly, Lagrange's discovery implies that the canonical continued fraction expansion of the [[square root#As periodic continued fractions|square root]] of every non-square integer is periodic and that, if the period is of length ''p'' > 1, it contains a [[palindrome|palindromic]] string of length ''p'' - 1.
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| In 1813 [[Carl Friedrich Gauss|Gauss]] used a very clever trick with the complex-valued [[hypergeometric function]] to derive a versatile continued fraction expression that has since been named in his honor.<ref>1813 [[Carl Friedrich Gauss]], ''Werke'', Vol. 3, pp. 134-138.</ref> That formula can be used to express many elementary functions (and even some more advanced functions, like the [[Bessel function]]s) as rapidly convergent continued fractions valid almost everywhere in the complex plane.
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| == Notation ==
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| The long continued fraction expression displayed in the introduction is probably the most intuitive form for the reader. Unfortunately, it takes up a lot of space in a book (and it's not easy for the typesetter, either). So mathematicians have devised several alternative notations. One convenient way to express a generalized continued fraction looks like this:
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| :<math>
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| x = b_0+
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| \frac{a_1}{b_1+}\,
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| \frac{a_2}{b_2+}\,
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| \frac{a_3}{b_3+}\cdots
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| </math>
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| [[Alfred Pringsheim|Pringsheim]] wrote a generalized continued fraction this way:
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| :<math>
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| x = b_0 + \frac{a_1 \mid}{\mid b_1} + \frac{a_2 \mid}{\mid b_2} + \frac{a_3 \mid}{\mid b_3}+\cdots\,
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| </math>.
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| [[Carl Friedrich Gauss]] evoked the more familiar [[infinite product]] Π when he devised this notation:
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| :<math>
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| x = b_0 + \underset{i=1}{\overset{\infty}{\mathrm K}} \frac{a_i}{b_i}.\,
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| </math>
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| Here the "K" stands for ''Kettenbruch'', the German word for "continued fraction". This is probably the most compact and convenient way to express continued fractions; however, it is not widely used by English typesetters.
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| == Some elementary considerations ==
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| Here are some elementary results that are of fundamental importance in the further development of the analytic theory of continued fractions.
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| === Partial numerators and denominators ===
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| If one of the partial numerators ''a''<sub>''n''+1</sub> is zero, the infinite continued fraction
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| :<math>
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| b_0 + \underset{i=1}{\overset{\infty}{\mathrm K}} \frac{a_i}{b_i}\,
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| </math>
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| is really just a finite continued fraction with ''n'' fractional terms, and therefore a [[rational function]] of the first ''n'' ''a''<sub>''i''</sub>'s and the first (''n'' + 1) ''b''<sub>''i''</sub>'s. Such an object is of little interest from the point of view adopted in mathematical analysis, so it is usually assumed that none of the ''a''<sub>''i''</sub> = 0. There is no need to place this restriction on the partial denominators ''b''<sub>''i''</sub>.
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| === The determinant formula ===
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| When the ''n''th convergent of a continued fraction
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| :<math>
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| x_n = b_0 + \underset{i=1}{\overset{n}{\mathrm K}} \frac{a_i}{b_i}\,
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| </math>
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| is expressed as a simple fraction ''x''<sub>''n''</sub> = ''A''<sub>''n''</sub>/''B''<sub>''n''</sub> we can use the ''determinant formula''
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| :<math>
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| A_{n-1}B_n - A_nB_{n-1} = (-1)^na_1a_2\cdots a_n = \Pi_{i=1}^n (-a_i)\,
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| </math>
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| to relate the numerators and denominators of successive convergents ''x''<sub>''n''</sub> and ''x''<sub>''n''-1</sub> to one another. Specifically, if neither ''B''<sub>''n''</sub> nor ''B''<sub>''n''-1</sub> is zero we can express the difference between the ''n''-1st and ''n''th (''n'' > 0) convergents like this:
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| :<math>
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| x_{n-1} - x_n = \frac{A_{n-1}}{B_{n-1}} - \frac{A_n}{B_n} =
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| (-1)^n \frac{a_1a_2\cdots a_n}{B_nB_{n-1}} = \frac{\Pi_{i=1}^n (-a_i)}{B_nB_{n-1}}.\,
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| </math>
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| === The equivalence transformation ===
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| If {''c''<sub>''i''</sub>} = {''c''<sub>1</sub>, ''c''<sub>2</sub>, ''c''<sub>3</sub>, ...} is any infinite sequence of non-zero complex numbers we can prove, by [[mathematical induction|induction]], that
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| :<math>
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| b_0 + \cfrac{a_1}{b_1 + \cfrac{a_2}{b_2 + \cfrac{a_3}{b_3 + \cfrac{a_4}{b_4 + \ddots\,}}}} =
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| b_0 + \cfrac{c_1a_1}{c_1b_1 + \cfrac{c_1c_2a_2}{c_2b_2 + \cfrac{c_2c_3a_3}{c_3b_3 + \cfrac{c_3c_4a_4}{c_4b_4 + \ddots\,}}}}
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| </math>
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| where equality is understood as equivalence, which is to say that the successive convergents of the continued fraction on the left are exactly the same as the convergents of the fraction on the right.
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| The equivalence transformation is perfectly general, but two particular cases deserve special mention. First, if none of the ''a''<sub>''i''</sub> are zero a sequence {''c''<sub>''i''</sub>} can be chosen to make each partial numerator a 1:
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| :<math>
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| b_0 + \underset{i=1}{\overset{\infty}{\mathrm K}} \frac{a_i}{b_i} =
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| b_0 + \underset{i=1}{\overset{\infty}{\mathrm K}} \frac{1}{c_i b_i}\,
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| </math>
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| <!--b_0 + \cfrac{a_1}{b_1 + \cfrac{a_2}{b_2 + \cfrac{a_3}{b_3 + \cfrac{a_4}{\ddots\,}}}} =
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| b_0 + \cfrac{1}{c_1b_1 + \cfrac{1}{c_2b_2 + \cfrac{1}{c_3b_3 + \cfrac{1}{c_4b_4 + \ddots\,}}}}-->
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| where ''c''<sub>1</sub> = 1/''a''<sub>1</sub>, ''c''<sub>2</sub> = ''a''<sub>1</sub>/''a''<sub>2</sub>, ''c''<sub>3</sub> = ''a''<sub>2</sub>/(''a''<sub>1</sub>''a''<sub>3</sub>), and in general ''c''<sub>''n''+1</sub> = 1/(''a''<sub>''n''+1</sub>''c''<sub>''n''</sub>).
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| Second, if none of the partial denominators ''b''<sub>''i''</sub> are zero we can use a similar procedure to choose another sequence {''d''<sub>''i''</sub>} to make each partial denominator a 1:
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| :<math>
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| b_0 + \underset{i=1}{\overset{\infty}{\mathrm K}} \frac{a_i}{b_i} =
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| b_0 + \underset{i=1}{\overset{\infty}{\mathrm K}} \frac{d_i a_i}{1}\,
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| </math>
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| <!--b_0 + \cfrac{a_1}{b_1 + \cfrac{a_2}{b_2 + \cfrac{a_3}{b_3 + \cfrac{a_4}{b_4 + \ddots\,}}}} =
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| b_0 + \cfrac{d_1a_1}{1 + \cfrac{d_2a_2}{1 + \cfrac{d_3a_3}{1 + \cfrac{d_4a_4}{1 + \ddots\,}}}}-->
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| where ''d''<sub>1</sub> = 1/''b''<sub>1</sub> and otherwise ''d''<sub>''n''+1</sub> = 1/(''b''<sub>''n''</sub>''b''<sub>''n''+1</sub>).
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| These two special cases of the equivalence transformation are enormously useful when the general [[convergence problem]] is analyzed.
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| === Simple convergence concepts ===
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| It has already been noted that the continued fraction
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| :<math>
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| x = b_0 + \underset{i=1}{\overset{\infty}{\mathrm K}} \frac{a_i}{b_i}\,
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| </math>
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| converges if the sequence of convergents {''x''<sub>''n''</sub>} tends to a finite limit.
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| The notion of [[absolute convergence]] plays a central role in the theory of [[infinite series]]. No corresponding notion exists in the analytic theory of continued fractions – in other words, mathematicians do not speak of an ''absolutely convergent'' continued fraction. Sometimes the notion of absolute convergence does enter the discussion, however, especially in the study of the convergence problem. For instance, a particular continued fraction
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| :<math>
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| x = \underset{i=1}{\overset{\infty}{\mathrm K}} \frac{1}{b_i}\,
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| </math> | |
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| diverges by oscillation if the series ''b''<sub>1</sub> + ''b''<sub>2</sub> + ''b''<sub>3</sub> + ... is absolutely convergent.<ref>1895 [[Helge von Koch]], ''Bull. Soc. Math. de France'', "Sur un théorème de Stieltjes et sur les fractions continues".</ref>
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| Sometimes the partial numerators and partial denominators of a continued fraction are expressed as functions of a complex variable ''z''. For example, a relatively simple function<ref>When ''z'' is taken to be an integer this function is quite famous; it generates the [[golden ratio]] and the closely related sequence of [[silver ratio#Silver means|silver means]].</ref> might be defined as
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| :<math>
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| f(z) = \underset{i=1}{\overset{\infty}{\mathrm K}} \frac{1}{z}.\,
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| </math>
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| For a continued fraction like this one the notion of [[uniform convergence]] arises quite naturally. A continued fraction of one or more complex variables is ''uniformly convergent'' in an [[neighbourhood (mathematics)|open neighborhood]] Ω if the fraction's convergents converge uniformly at every point in Ω. Or, in gory detail: if, for every ''ε'' > 0 an integer ''M'' can be found such that the [[absolute value]] of the difference
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| :<math>
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| f(z) - f_n(z) = \underset{i=1}{\overset{\infty}{\mathrm K}} \frac{a_i(z)}{b_i(z)}
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| - \underset{i=1}{\overset{n}{\mathrm K}} \frac{a_i(z)}{b_i(z)}\,
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| </math>
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| is less than ''ε'' for every point ''z'' in an open neighborhood Ω whenever ''n'' > ''M'', the continued fraction defining ''f''(''z'') is uniformly convergent on Ω. (Here ''f''<sub>''n''</sub>(''z'') denotes the ''n''th convergent of the continued fraction, evaluated at the point ''z'' inside Ω, and ''f''(''z'') is the value of the infinite continued fraction at the point ''z''.)
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| The [[Śleszyński–Pringsheim theorem]] provides a sufficient condition for convergence.
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| === Even and odd convergents ===
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| It is sometimes necessary to separate a continued fraction into its even and odd parts. For example, if the continued fraction diverges by oscillation between two distinct limit points ''p'' and ''q'', then the sequence {''x''<sub>0</sub>, ''x''<sub>2</sub>, ''x''<sub>4</sub>, ...} must converge to one of these, and {''x''<sub>1</sub>, ''x''<sub>3</sub>, ''x''<sub>5</sub>, ...} must converge to the other. In such a situation it may be convenient to express the original continued fraction as two different continued fractions, one of them converging to ''p'', and the other converging to ''q''.
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| The formulas for the even and odd parts of a continued fraction can be written most compactly if the fraction has already been transformed so that all its partial denominators are unity. Specifically, if
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| :<math>
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| x = \underset{i=1}{\overset{\infty}{\mathrm K}} \frac{a_i}{1}\,
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| </math>
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| is a continued fraction, then the even part ''x''<sub>even</sub> and the odd part ''x''<sub>odd</sub> are given by
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| :<math>
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| x_\mathrm{even} = \cfrac{a_1}{1+a_2-\cfrac{a_2a_3} {1+a_3+a_4-\cfrac{a_4a_5} {1+a_5+a_6-\cfrac{a_6a_7} {1+a_7+a_8-\ddots}}}}\,
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| </math>
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| and
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| :<math>
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| x_\mathrm{odd} = a_1 - \cfrac{a_1a_2}{1+a_2+a_3-\cfrac{a_3a_4} {1+a_4+a_5-\cfrac{a_5a_6} {1+a_6+a_7-\cfrac{a_7a_8} {1+a_8+a_9-\ddots}}}}\,
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| </math>
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| respectively. More precisely, if the successive convergents of the continued fraction ''x'' are {''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>, ...}, then the successive convergents of ''x''<sub>even</sub> as written above are {''x''<sub>2</sub>, ''x''<sub>4</sub>, ''x''<sub>6</sub>, ...}, and the successive convergents of ''x''<sub>odd</sub> are {''x''<sub>1</sub>, ''x''<sub>3</sub>, ''x''<sub>5</sub>, ...}.<ref>1929 [[Oskar Perron]], ''Die Lehre von den Kettenbrüchen'' derives even more general extension and contraction formulas for continued fractions.</ref>
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| ===Conditions for irrationality===
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| If <math> a_1,a_2, \text{ . . .}</math> and <math> b_1,b_2, \text{ . . .}</math> are positive integers with <math> a_k </math> ≤ <math> b_k </math> for all sufficiently large <math> k </math>, then
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| :<math>
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| x = b_0 + \underset{i=1}{\overset{\infty}{\mathrm K}} \frac{a_i}{b_i}\,
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| </math>
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| converges to an irrational limit.<ref>{{Cite document
| |
| | last = Angell
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| | first = David
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| | title = Irrationality and Transcendence - Lambert's Irrationality Proofs
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| | publisher = School of Mathematics, University of New South Wales
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| | year = 2007
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| | url = http://web.maths.unsw.edu.au/~angell/5535}}</ref>
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| ===Fundamental recurrence formulas===
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| The partial numerators and denominators of the fraction's successive convergents are related by the ''fundamental recurrence formulas'':
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| :<math>
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| \begin{align}
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| A_{-1}& = 1& B_{-1}& = 0\\
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| A_0& = b_0& B_0& = 1\\
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| A_{n+1}& = b_{n+1} A_n + a_{n+1} A_{n-1}& B_{n+1}& = b_{n+1} B_n + a_{n+1} B_{n-1}\,
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| \end{align}
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| </math>
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| The continued fraction's successive convergents are then given by
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| :<math>x_n=\frac{A_n}{B_n}.\,</math>
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| These recurrence relations are due to [[John Wallis]] (1616-1703) and [[Leonhard Euler]] (1707-1783).<ref name=num_theory>{{cite web|first=Štefan |last=Porubský |title=Basic definitions for continued fractions |work=Interactive Information Portal for Algorithmic Mathematics |publisher=Institute of Computer Science of the Czech Academy of Sciences |location=Prague, Czech Republic, |url=http://www.cs.cas.cz/portal/AlgoMath/NumberTheory/ContinuedFractions/BasicDefinitions.htm |accessdate=9 April 2013}}</ref>
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| As an example, consider the [[continued fraction|regular continued fraction in canonical form]] that represents the [[golden ratio|golden ratio φ]]:
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| :<math>x = 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \ddots\,}}}} </math>
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| Applying the fundamental recurrence formulas we find that the successive numerators ''A''<sub>''n''</sub> are {1, 2, 3, 5, 8, 13, ...} and the successive denominators ''B''<sub>''n''</sub> are {1, 1, 2, 3, 5, 8, ...}, the [[Fibonacci number]]s. Since all the partial numerators in this example are equal to one, the determinant formula assures us that the absolute value of the difference between successive convergents approaches zero quite rapidly.
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| == Linear fractional transformations ==
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| A linear fractional transformation (LFT) is a [[complex function]] of the form
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| :<math>
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| w = f(z) = \frac{a + bz}{c + dz},\,
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| </math>
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| where ''z'' is a complex variable, and ''a'', ''b'', ''c'', ''d'' are arbitrary complex constants. An additional restriction – that ''ad'' ≠ ''bc'' – is customarily imposed, to rule out the cases in which ''w'' = ''f''(''z'') is a constant. The linear fractional transformation, also known as a [[Möbius transformation]], has many fascinating properties. Four of these are of primary importance in developing the analytic theory of continued fractions.
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| *If ''d'' ≠ 0 the LFT has one or two [[fixed point (mathematics)|fixed points]]. This can be seen by considering the equation
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| ::<math>
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| f(z) = z \Rightarrow dz^2 + cz = a + bz\,
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| </math>
| |
| | |
| :which is clearly a [[quadratic equation]] in ''z''. The roots of this equation are the fixed points of ''f''(''z''). If the [[discriminant]] (''c'' − ''b'')<sup>2</sup> + 4''ad'' is zero the LFT fixes a single point; otherwise it has two fixed points.
| |
| | |
| *If ''ad'' ≠ ''bc'' the LFT is an [[bijection|invertible]] [[conformal map]]ping of the [[Riemann sphere|extended complex plane]] onto itself. In other words, this LFT has an inverse function
| |
| | |
| ::<math>
| |
| z = g(w) = \frac{-a + cw}{b - dw}\,
| |
| </math>
| |
| | |
| :such that ''f''(''g''(''z'')) = ''g''(''f''(''z'')) = ''z'' for every point ''z'' in the extended complex plane, and both ''f'' and ''g'' preserve angles and shapes at vanishingly small scales. From the form of ''z'' = ''g''(''w'') we see that ''g'' is also an LFT.
| |
| | |
| *The [[function composition|composition]] of two different LFTs for which ''ad'' ≠ ''bc'' is itself an LFT for which ''ad'' ≠ ''bc''. In other words, the set of all LFTs for which ''ad'' ≠ ''bc'' is closed under composition of functions. The collection of all such LFTs – together with the "group operation" composition of functions – is known as the [[automorphism group]] of the extended complex plane.
| |
| | |
| *If ''b'' = 0 the LFT reduces to
| |
| | |
| ::<math>
| |
| w = f(z) = \frac{a}{c + dz},\,
| |
| </math>
| |
| | |
| :which is a very simple [[meromorphic function]] of ''z'' with one [[pole (complex analysis)|simple pole]] (at −''c''/''d'') and a [[residue theorem|residue]] equal to ''a''/''d''. (See also [[Laurent series]].)
| |
| | |
| === The continued fraction as a composition of LFTs ===
| |
| | |
| Consider a sequence of simple linear fractional transformations
| |
| | |
| :<math> | |
| \tau_0(z) = b_0 + z,\quad \tau_1(z) = \frac{a_1}{b_1 + z},\quad
| |
| \tau_2(z) = \frac{a_2}{b_2 + z},\quad \tau_3(z) = \frac{a_3}{b_3 + z},\quad\cdots\,
| |
| </math>
| |
| | |
| Here we use the Greek letter ''τ'' (tau) to represent each simple LFT, and we adopt the conventional circle notation for composition of functions. We also introduce a new symbol '''''Τ'''''<sub>'''''n'''''</sub> to represent the composition of ''n''+1 little ''τ''s – that is,
| |
| | |
| :<math>
| |
| \boldsymbol{\Tau}_{\boldsymbol{1}}(z) = \tau_0\circ\tau_1(z) = \tau_0(\tau_1(z)),\quad
| |
| \boldsymbol{\Tau}_{\boldsymbol{2}}(z) = \tau_0\circ\tau_1\circ\tau_2(z) = \tau_0(\tau_1(\tau_2(z))),\,
| |
| </math>
| |
| | |
| and so forth. By direct substitution from the first set of expressions into the second we see that
| |
| | |
| :<math>
| |
| \begin{align}
| |
| \boldsymbol{\Tau}_{\boldsymbol{1}}(z)& = \tau_0\circ\tau_1(z)& =&\quad b_0 + \cfrac{a_1}{b_1 + z}\\
| |
| \boldsymbol{\Tau}_{\boldsymbol{2}}(z)& = \tau_0\circ\tau_1\circ\tau_2(z)& =&\quad b_0 + \cfrac{a_1}{b_1 + \cfrac{a_2}{b_2 + z}}\,
| |
| \end{align}
| |
| </math>
| |
| | |
| and, in general,
| |
| | |
| :<math>
| |
| \boldsymbol{\Tau}_{\boldsymbol{n}}(z) = \tau_0\circ\tau_1\circ\tau_2\circ\cdots\circ\tau_n(z) =
| |
| b_0 + \underset{i=1}{\overset{n}{\mathrm K}} \frac{a_i}{b_i}\,
| |
| </math>
| |
| | |
| where the last partial denominator in the finite continued fraction ''K'' is understood to be ''b''<sub>''n''</sub> + ''z''. And, since ''b''<sub>''n''</sub> + 0 = ''b''<sub>''n''</sub>, the image of the point ''z'' = 0 under the iterated LFT '''''Τ'''''<sub>'''''n'''''</sub> is indeed the value of the finite continued fraction with ''n'' partial numerators:
| |
| | |
| :<math>
| |
| \boldsymbol{\Tau}_{\boldsymbol{n}}(0) = \boldsymbol{\Tau}_{\boldsymbol{n+1}}(\infty) =
| |
| b_0 + \underset{i=1}{\overset{n}{\mathrm K}} \frac{a_i}{b_i}.\,
| |
| </math>
| |
| | |
| === A geometric interpretation ===
| |
| | |
| Defining a finite continued fraction as the image of a point under the iterated LFT '''Τ'''<sub>'''n'''</sub>(''z'') leads to an intuitively appealing geometric interpretation of infinite continued fractions.
| |
| | |
| The relationship
| |
| | |
| :<math>
| |
| x_n = b_0 + \underset{i=1}{\overset{n}{\mathrm K}} \frac{a_i}{b_i} = \frac{A_n}{B_n} = \boldsymbol{\Tau}_{\boldsymbol{n}}(0) = \boldsymbol{\Tau}_{\boldsymbol{n+1}}(\infty)\,
| |
| </math>
| |
| | |
| can be understood by rewriting the LFTs '''Τ<sub>''n''</sub>'''(''z'') and '''Τ<sub>''n''+1</sub>'''(''z'') in terms of the [[fundamental recurrence formulas]]:
| |
| | |
| :<math>
| |
| \begin{align}
| |
| \boldsymbol{\Tau}_{\boldsymbol{n}}(z)& = \frac{(b_n+z)A_{n-1} + a_nA_{n-2}}{(b_n+z)B_{n-1} + a_nB_{n-2}}& \boldsymbol{\Tau}_{\boldsymbol{n}}(z)& = \frac{zA_{n-1} + A_n}{zB_{n-1} + B_n};\\
| |
| \boldsymbol{\Tau}_{\boldsymbol{n+1}}(z)& = \frac{(b_{n+1}+z)A_n + a_{n+1}A_{n-1}}{(b_{n+1}+z)B_n + a_{n+1}B_{n-1}}& \boldsymbol{\Tau}_{\boldsymbol{n+1}}(z)& = \frac{zA_n + A_{n+1}} {zB_n + B_{n+1}}.\,
| |
| \end{align}
| |
| </math>
| |
| | |
| In the first of these equations the ratio tends toward ''A''<sub>''n''</sub>/''B''<sub>''n''</sub> as ''z'' tends toward zero. In the second, the ratio tends toward ''A''<sub>''n''</sub>/''B''<sub>''n''</sub> as ''z'' tends to infinity. This leads us to our first geometric interpretation. If the continued fraction converges, the successive convergents ''A''<sub>''n''</sub>/''B''<sub>''n''</sub> are eventually [[cauchy sequence|arbitrarily close together]]. Since the linear fractional transformation '''Τ<sub>''n''</sub>'''(''z'') is a [[continuous function|continuous mapping]], there must be a neighborhood of ''z'' = 0 that is mapped into an arbitrarily small neighborhood of '''Τ<sub>''n''</sub>'''(0) = ''A''<sub>''n''</sub>/''B''<sub>''n''</sub>. Similarly, there must be a neighborhood of the point at infinity which is mapped into an arbitrarily small neighborhood of '''Τ<sub>''n''</sub>'''(∞) = ''A''<sub>''n''-1</sub>/''B''<sub>''n''-1</sub>. So if the continued fraction converges the transformation '''Τ<sub>''n''</sub>'''(''z'') maps both very small ''z'' and very large ''z'' into an arbitrarily small neighborhood of ''x'', the value of the continued fraction, as ''n'' gets larger and larger.
| |
| | |
| What about intermediate values of ''z''? Well, since the successive convergents are getting closer together we must have
| |
| | |
| :<math>
| |
| \frac{A_{n-1}}{B_{n-1}} \approx \frac{A_n}{B_n} \quad\Rightarrow\quad
| |
| \frac{A_{n-1}}{A_n} \approx \frac{B_{n-1}}{B_n} = k\,
| |
| </math>
| |
| | |
| where ''k'' is a constant, introduced for convenience. But then, by substituting in the expression for '''Τ<sub>''n''</sub>'''(''z'') we obtain | |
| | |
| :<math> | |
| \boldsymbol{\Tau}_{\boldsymbol{n}}(z) = \frac{zA_{n-1} + A_n}{zB_{n-1} + B_n}
| |
| = \frac{A_n}{B_n} \left(\frac{z\frac{A_{n-1}}{A_n} + 1}{z\frac{B_{n-1}}{B_n} + 1}\right)
| |
| \approx \frac{A_n}{B_n} \left(\frac{zk + 1}{zk + 1}\right) = \frac{A_n}{B_n}\,
| |
| </math>
| |
| | |
| so that even the intermediate values of ''z'' (except when ''z'' ≈ −''k''<sup>−1</sup>) are mapped into an arbitrarily small neighborhood of ''x'', the value of the continued fraction, as ''n'' gets larger and larger. Intuitively, it is almost as if the convergent continued fraction maps the entire extended complex plane into a single point.<ref>This intuitive interpretation is not rigorous because a continued fraction is not a mapping – it is the ''limit'' of a sequence of mappings. This construction of an infinite continued fraction is roughly analogous to the construction of an irrational number as the limit of a [[Cauchy sequence]] of rational numbers.</ref>
| |
| | |
| Notice that the sequence {'''Τ<sub>''n''</sub>'''} lies within the [[automorphism group]] of the extended complex plane, since each '''Τ<sub>''n''</sub>''' is a linear fractional transformation for which ''ab'' ≠ ''cd''. And every member of that automorphism group maps the extended complex plane into itself – not one of the '''Τ<sub>''n''</sub>'''s can possibly map the plane into a single point. Yet in the limit the sequence {'''Τ<sub>''n''</sub>'''} defines an infinite continued fraction which (if i
| |
| t converges) represents a single point in the complex plane.
| |
| | |
| How is this possible? Think of it this way. When an infinite continued fraction converges, the corresponding sequence {'''Τ<sub>''n''</sub>'''} of LFTs "focuses" the plane in the direction of ''x'', the value of the continued fraction. At each stage of the process a larger and larger region of the plane is mapped into a neighborhood of ''x'', and the smaller and smaller region of the plane that's left over is stretched out ever more thinly to cover everything outside that neighborhood.<ref>Because of analogies like this one, the theory of [[conformal map]]ping is sometimes described as "rubber sheet geometry".</ref>
| |
| | |
| What about divergent continued fractions? Can those also be interpreted geometrically? In a word, yes. We distinguish three cases.
| |
| #The two sequences {'''Τ<sub>2''n''-1</sub>'''} and {'''Τ<sub>2''n''</sub>'''} might themselves define two convergent continued fractions that have two different values, ''x''<sub>odd</sub> and ''x''<sub>even</sub>. In this case the continued fraction defined by the sequence {'''Τ<sub>''n''</sub>'''} diverges by oscillation between two distinct limit points. And in fact this idea can be generalized – sequences {'''Τ<sub>''n''</sub>'''} can be constructed that oscillate among three, or four, or indeed any number of limit points. Interesting instances of this case arise when the sequence {'''Τ<sub>''n''</sub>'''} constitutes a [[subgroup]] of finite order within the group of automorphisms over the extended complex plane.
| |
| # The sequence {'''Τ<sub>''n''</sub>'''} may produce an infinite number of zero denominators ''B''<sub>''i''</sub> while also producing a subsequence of finite convergents. These finite convergents may not repeat themselves or fall into a recognizable oscillating pattern. Or they may converge to a finite limit, or even oscillate among multiple finite limits. No matter how the finite convergents behave, the continued fraction defined by the sequence {'''Τ<sub>''n''</sub>'''} diverges by oscillation with the point at infinity in this case.<ref>One approach to the [[convergence problem]] is to construct ''positive definite'' continued fractions, for which the denominators ''B''<sub>''i''</sub> are never zero.</ref>
| |
| #The sequence {'''Τ<sub>''n''</sub>'''} may produce no more than a finite number of zero denominators ''B''<sub>''i''</sub>. while the subsequence of finite convergents dances wildly around the plane in a pattern that never repeats itself and never approaches any finite limit, either.
| |
| | |
| Interesting examples of cases 1 and 3 can be constructed by studying the simple continued fraction
| |
| | |
| :<math>
| |
| x = 1 + \cfrac{z}{1 + \cfrac{z}{1 + \cfrac{z}{1 + \cfrac{z}{1 + \ddots}}}}\,
| |
| </math>
| |
| | |
| where ''z'' is any real number such that ''z'' < −¼.<ref>This periodic fraction of period one is discussed more fully in the article [[convergence problem]].</ref>
| |
| | |
| == Continued fractions and series ==
| |
| {{main|Euler's continued fraction formula}}
| |
| | |
| [[Leonhard Euler|Euler]] proved the following identity:<ref name="Euler"/>
| |
| | |
| :<math>
| |
| a_0 + a_0a_1 + a_0a_1a_2 + \cdots + a_0a_1a_2\cdots a_n =
| |
| \frac{a_0}{1-}
| |
| \frac{a_1}{1+a_1-}
| |
| \frac{a_2}{1+a_2-}\cdots
| |
| \frac{a_{n}}{1+a_n}.\,
| |
| </math>
| |
| | |
| From this many other results can be derived, such as
| |
| | |
| :<math>
| |
| \frac{1}{u_1}+
| |
| \frac{1}{u_2}+
| |
| \frac{1}{u_3}+
| |
| \cdots+
| |
| \frac{1}{u_n} =
| |
| \frac{1}{u_1-}
| |
| \frac{u_1^2}{u_1+u_2-}
| |
| \frac{u_2^2}{u_2+u_3-}\cdots
| |
| \frac{u_{n-1}^2}{u_{n-1}+u_n},\,
| |
| </math>
| |
| | |
| and
| |
| | |
| :<math>
| |
| \frac{1}{a_0} + \frac{x}{a_0a_1} + \frac{x^2}{a_0a_1a_2} + \cdots +
| |
| \frac{x^n}{a_0a_1a_2 \ldots a_n} =
| |
| \frac{1}{a_0-}
| |
| \frac{a_0x}{a_1+x-}
| |
| \frac{a_1x}{a_2+x-}\cdots
| |
| \frac{a_{n-1}x}{a_n-x}.\,
| |
| </math>
| |
| | |
| Euler's formula connecting continued fractions and series is the motivation for the [[fundamental inequalities]], and also the basis of elementary approaches to the [[convergence problem]].
| |
| | |
| ==Examples==
| |
| ===Transcendental functions and numbers===
| |
| | |
| Here are two continued fractions that can be built via Euler's identity.
| |
| | |
| <math>
| |
| e^x = \frac{x^0}{0!} + \frac{x^1}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots
| |
| = 1+\cfrac{x} {1-\cfrac{1x} {2+x-\cfrac{2x} {3+x-\cfrac{3x} {4+x-\ddots}}}}
| |
| </math>
| |
| | |
| <math>
| |
| \log(1+x) = \frac{x^1}{1} - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots
| |
| =\cfrac{x} {1-0x+\cfrac{1^2x} {2-1x+\cfrac{2^2x} {3-2x+\cfrac{3^2x} {4-3x+\ddots}}}}
| |
| </math>
| |
| | |
| Here are additional generalized continued fractions:
| |
| | |
| <math>
| |
| \tan^{-1}\cfrac{x}{y}=\cfrac{xy} {1y^2+\cfrac{(1xy)^2} {3y^2-1x^2+\cfrac{(3xy)^2} {5y^2-3x^2+\cfrac{(5xy)^2} {7y^2-5x^2+\ddots}}}}
| |
| =\cfrac{x} {1y+\cfrac{(1x)^2} {3y+\cfrac{(2x)^2} {5y+\cfrac{(3x)^2} {7y+\ddots}}}}
| |
| </math>
| |
| | |
| <math>
| |
| e^{x/y} = 1+\cfrac{2x} {2y-x+\cfrac{x^2} {6y+\cfrac{x^2} {10y+\cfrac{x^2} {14y+\cfrac{x^2} {18y+\cfrac{x^2} {22y+\ddots}}}}}};
| |
| e^2 = 7+\cfrac{2} {5+\cfrac{1} {7+\cfrac{1} {9+\cfrac{1} {11+\ddots}}}}
| |
| </math>
| |
| | |
| <math>
| |
| \log \left( 1+\frac{x}{y} \right) = \cfrac{x} {y+\cfrac{1x} {2+\cfrac{1x} {3y+\cfrac{2x} {2+\cfrac{2x} {5y+\cfrac{3x} {2+\ddots}}}}}}
| |
| = \cfrac{2x} {2y+x-\cfrac{(1x)^2} {3(2y+x)-\cfrac{(2x)^2} {5(2y+x)-\cfrac{(3x)^2} {7(2y+x)-\ddots}}}}
| |
| </math>
| |
| | |
| This last is based on an algorithm derived by Alekseĭ Nikolaevich Khovanskiĭ in the 1970's.<ref>[http://math.stackexchange.com/questions/75074/an-alternative-way-to-calculate-logx An alternative way to calculate log(x)]</ref>
| |
| | |
| Example: the [[natural logarithm of 2]] (= [0;1,2,3,1,5,2/3,7,1/2,9,2/5,...,2k-1,2/k,...] ≈ 0.693147...):<ref>[http://www.kurims.kyoto-u.ac.jp/EMIS/journals/EM/expmath/volumes/13/13.3/BorweinCrandallFee.pdf On the Ramanujan AGM Fraction, I: The Real-Parameter Case. Experimental Mathematics, Vol. 13 (2004), No. 3, pages 278,280.]</ref>
| |
| | |
| <math>
| |
| \log 2 = \log (1+1) = \cfrac{1} {1+\cfrac{1} {2+\cfrac{1} {3+\cfrac{2} {2+\cfrac{2}
| |
| {5+\cfrac{3} {2+\ddots}}}}}}
| |
| = \cfrac{2} {3-\cfrac{1^2} {9-\cfrac{2^2} {15-\cfrac{3^2} {21-\ddots}}}}
| |
| </math>
| |
| | |
| ===={{pi}}====
| |
| Here are three of [[pi|{{pi}}'s]] best-known generalized continued fractions, the first and third of which are derived from their respective [[Inverse trigonometric functions#Continued fractions for arctangent|arctangent]] formulas above by setting ''x''=''y''=1 and multiplying by four. The [[Leibniz formula for π|Leibniz formula for {{pi}}]]:
| |
| | |
| <math>
| |
| \pi = \cfrac{4} {1+\cfrac{1^2} {2+\cfrac{3^2} {2+\cfrac{5^2} {2+\ddots}}}}
| |
| = \sum_{n=0}^\infty \frac{4(-1)^n}{2n+1}
| |
| = \frac{4}{1} - \frac{4}{3} + \frac{4}{5} - \frac{4}{7} +- \cdots
| |
| </math>
| |
| | |
| converges too slowly, requiring roughly 3 x 10<sup>''n''</sup> terms to achieve ''n''-decimal precision. The series derived by [[Nilakantha Somayaji]]:
| |
| | |
| <math> | |
| \pi = 3 + \cfrac{1^2} {6+\cfrac{3^2} {6+\cfrac{5^2} {6+\ddots}}}
| |
| = 3 - \sum_{n=1}^\infty \frac{(-1)^n} {n (n+1) (2n+1)}
| |
| = 3 + \frac{1}{1\cdot 2\cdot 3} - \frac{1}{2\cdot 3\cdot 5} + \frac{1}{3\cdot 4\cdot 7} -+ \cdots
| |
| </math> | |
| | |
| is a much more obvious expression but still converges quite slowly, requiring nearly 50 terms for five decimals and nearly 120 for six. Both converge ''sublinearly'' to {{pi}}. On the other hand:
| |
| | |
| <math>
| |
| \pi = \cfrac{4} {1+\cfrac{1^2} {3+\cfrac{2^2} {5+\cfrac{3^2} {7+\ddots}}}}
| |
| = 4 - 1 + \frac{1}{6} - \frac{1}{34} + \frac {16}{3145} - \frac{4}{4551} + \frac{1}{6601} - \frac{1}{38341} +- \cdots
| |
| </math>
| |
| | |
| converges ''linearly'' to {{pi}}, adding at least three decimals digits of precision per four terms, a pace slightly faster than the [[Approximations of π#Arcsine|arcsine formula for {{pi}}]]:
| |
| <math>
| |
| \pi = 6 \sin^{-1} \left( \frac{1}{2} \right)
| |
| = \sum_{n=0}^\infty \frac {3 \cdot \binom {2n} n} {16^n (2n+1)}
| |
| = \frac {3} {16^0 \cdot 1} + \frac {6} {16^1 \cdot 3} + \frac {18} {16^2 \cdot 5} + \frac {60} {16^3 \cdot 7} + \cdots\!
| |
| </math>
| |
| | |
| which adds at least three decimal digits per five terms.
| |
| <ref>{{Cite book
| |
| | first=Petr | last=Beckmann
| |
| | year=1971
| |
| | title=A History of Pi
| |
| | publisher=St. Martin's Press, Inc.
| |
| | pages=131–133, 140–143
| |
| | isbn=0-88029-418-3}}.
| |
| Note: this continued fraction's [[rate of convergence]] μ tends to 3 – √8 ≈ 0.1715729, hence 1/μ tends to 3 + √8 ≈ 5.828427, whose [[common logarithm]] is 0.7655... ≈ 13/17 > 3/4. The same 1/μ = 3 + √8 (the [[silver ratio]] squared) also is observed in the ''unfolded'' general continued fractions of both the [[natural logarithm of 2]] and the [[nth root]] of 2 (which works for ''any'' integer ''n'' > 1) if calculated using 2 = 1 + 1. For the ''folded'' general continued fractions of both expressions, the rate convergence μ = (3–√8)<sup>2</sup> = 17–√288 ≈ 0.02943725, hence 1/μ = (3+√8)<sup>2</sup> = 17+√288 ≈ 33.97056, whose common logarithm is 1.531... ≈ 26/17 > 3/2, thus adding at least three digits per ''two'' terms. This is because the ''folded'' GCF ''folds'' each pair of fractions from the ''unfolded'' GCF into one fraction, thus doubling the convergence pace. The Manny Sardina reference further explains "folded" continued fractions.</ref>
| |
| | |
| Note: combining the last continued fraction with the best-known [[Machin-like formula]] provides an even more rapidly-converging expression:
| |
| | |
| <math>
| |
| \pi = 16 \tan^{-1} \cfrac{1}{5} - 4 \tan^{-1} \cfrac{1}{239}
| |
| = \cfrac{16} {5+\cfrac{1^2} {15+\cfrac{2^2} {25+\cfrac{3^2} {35+\ddots}}}}
| |
| - \cfrac{4} {239+\cfrac{1^2} {717+\cfrac{2^2} {1195+\cfrac{3^2} {1673+\ddots}}}}.
| |
| </math>
| |
| | |
| ===Roots of positive numbers===
| |
| | |
| The [[nth root]] of any positive number ''z''<sup>''m''</sup> can be expressed by restating ''z'' = ''x''<sup>''n''</sup> + ''y'', resulting in
| |
| | |
| <math>
| |
| \sqrt[n]{z^m} = \sqrt[n]{(x^n+y)^m} = x^m+\cfrac{my} {nx^{n-m}+\cfrac{(n-m)y} {2x^m+\cfrac{(n+m)y} {3nx^{n-m}+\cfrac{(2n-m)y} {2x^m+\cfrac{(2n+m)y} {5nx^{n-m}+\cfrac{(3n-m)y} {2x^m+\ddots}}}}}}
| |
| </math>
| |
| | |
| which can be simplified, by folding each pair of fractions into one fraction, to
| |
| | |
| <math>
| |
| \sqrt[n]{z^m} = x^m+\cfrac{2x^m \cdot my} {n(2z - y)-my-\cfrac{(1^2n^2-m^2)y^2} {3n(2z - y)-\cfrac{(2^2n^2-m^2)y^2} {5n(2z - y)-\cfrac{(3^2n^2-m^2)y^2} {7n(2z - y)-\cfrac{(4^2n^2-m^2)y^2} {9n(2z - y)-\ddots}}}}}.
| |
| </math>
| |
| | |
| The [[square root]] of ''z'' is a special case of this nth root algorithm (''m''=1, ''n''=2):
| |
| | |
| <math>
| |
| \sqrt{z} = \sqrt{x^2+y} = x+\cfrac{y} {2x+\cfrac{y} {2x+\cfrac{3y} {6x+\cfrac{3y} {2x+\ddots}}}}
| |
| = x+\cfrac{2x \cdot y} {2(2z - y)-y-\cfrac{1\cdot 3y^2} {6(2z - y)-\cfrac{3\cdot 5y^2} {10(2z - y)-\ddots}}}
| |
| </math>
| |
| | |
| which can be simplified by noting that 5/10 = 3/6 = 1/2:
| |
| | |
| <math>
| |
| \sqrt{z} = \sqrt{x^2+y} = x+\cfrac{y} {2x+\cfrac{y} {2x+\cfrac{y} {2x+\cfrac{y} {2x+\ddots}}}}
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| = x+\cfrac{2x \cdot y} {2(2z - y)-y-\cfrac{y^2} {2(2z - y)-\cfrac{y^2} {2(2z - y)-\ddots}}}.
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| </math>
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| The square root can also be expressed by a [[periodic continued fraction]], but the above form converges more quickly with the proper ''x'' and ''y''.
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| ====Example 1====
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| The [[cube root of two]] (2<sup>1/3</sup> or <sup>3</sup>√2 ≈ 1.259921...):
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| (A) "Standard notation" of ''x'' = 1, ''y'' = 1, and 2''z - y'' = 3:
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| <math>
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| \sqrt[3]2 = 1+\cfrac{1} {3+\cfrac{2} {2+\cfrac{4} {9+\cfrac{5} {2+\cfrac{7} {15+\cfrac{8} {2+\cfrac{10} {21+\cfrac{11} {2+\ddots}}}}}}}} = 1+\cfrac{2 \cdot 1} {9-1-\cfrac{2 \cdot 4} {27-\cfrac{5 \cdot 7} {45-\cfrac{8 \cdot 10} {63-\cfrac{11 \cdot 13} {81-\ddots}}}}}.
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| </math>
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| (B) Rapid convergence with ''x'' = 5, ''y'' = 3 and 2''z - y'' = 253:
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| <math>
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| \sqrt[3]2 = \cfrac{5}{4}+\cfrac{0.5} {50+\cfrac{2} {5+\cfrac{4} {150+\cfrac{5} {5+\cfrac{7} {250+\cfrac{8} {5+\cfrac{10} {350+\cfrac{11} {5+\ddots}}}}}}}} = \cfrac{5}{4}+\cfrac{2.5 \cdot 1} {253-1-\cfrac{2 \cdot 4} {759-\cfrac{5 \cdot 7} {1265-\cfrac{8 \cdot 10} {1771-\ddots}}}}.
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| </math>
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| ====Example 2====
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| [[Pogson's ratio]] (100<sup>1/5</sup> or <sup>5</sup>√100 ≈ 2.511886...), with ''x'' = 5, ''y'' = 75 and 2''z - y'' = 6325:
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| <math>
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| \sqrt[5]{100} = \cfrac{5}{2}+\cfrac{3} {250+\cfrac{12} {5+\cfrac{18} {750+\cfrac{27} {5+\cfrac{33} {1250+\cfrac{42} {5+\ddots}}}}}} = \cfrac{5}{2}+\cfrac{5\cdot 3} {1265-3-\cfrac{12 \cdot 18} {3795-\cfrac{27 \cdot 33} {6325-\cfrac{42 \cdot 48} {8855-\ddots}}}}.
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| </math>
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| ====Example 3====
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| The [[twelfth root of two]] (2<sup>1/12</sup> or <sup>12</sup>√2 ≈ 1.059463...), using "standard notation":
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| <math>
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| \sqrt[12]2 = 1+\cfrac{1} {12+\cfrac{11} {2+\cfrac{13} {36+\cfrac{23} {2+\cfrac{25} {60+\cfrac{35} {2+\cfrac{37} {84+\cfrac{47} {2+\ddots}}}}}}}} = 1+\cfrac{2 \cdot 1} {36-1 - \cfrac{11 \cdot 13} {108-\cfrac{23 \cdot 25} {180-\cfrac{35 \cdot 37} {252-\cfrac{47 \cdot 49} {324-\ddots}}}}}.
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| </math>
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| ====Example 4====
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| [[Equal temperament]]'s [[perfect fifth]] (2<sup>7/12</sup> or <sup>12</sup>√2<sup>7</sup> ≈ 1.498307...), with ''m''=7:
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| (A) "Standard notation":
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| <math>
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| \sqrt[12]{2^7} = 1+\cfrac{7} {12+\cfrac{5} {2+\cfrac{19} {36+\cfrac{17} {2+\cfrac{31} {60+\cfrac{29} {2+\cfrac{43} {84+\cfrac{41} {2+\ddots}}}}}}}} = 1+\cfrac{2 \cdot 7} {36-7 - \cfrac{5 \cdot 19} {108-\cfrac{17 \cdot 31} {180-\cfrac{29 \cdot 43} {252-\cfrac{41 \cdot 55} {324-\ddots}}}}}.
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| </math>
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| (B) Rapid convergence with ''x'' = 3, ''y'' = –7153, and 2''z - y'' = 2<sup>19</sup>+3<sup>12</sup>:
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| <math>\sqrt[12]{2^7} = \cfrac{1}{2} \sqrt[12]{3^{12}-7153} = \cfrac{3}{2} - \cfrac{0.5 \cdot 7153}{4\cdot 3^{12} - \cfrac{11\cdot 7153}{6 - \cfrac{13\cdot 7153}{12\cdot 3^{12}
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| - \cfrac{23\cdot 7153}{6 - \cfrac{25\cdot 7153}{20\cdot 3^{12} - \cfrac{35\cdot 7153}{6 - \cfrac{37\cdot 7153}{28\cdot 3^{12} - \cfrac{47\cdot 7153}{6 - \ddots}}}}}}}} </math>
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| <math>\sqrt[12]{2^7} = \cfrac{3}{2} - \cfrac{3\cdot 7153}{12(2^{19}+3^{12}) + 7153 - \cfrac{11\cdot 13\cdot 7153^2}{36(2^{19}+3^{12})
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| - \cfrac{23\cdot 25\cdot 7153^2}{60(2^{19}+3^{12}) - \cfrac{35\cdot 37\cdot 7153^2}{84(2^{19}+3^{12}) - \ddots}}}}. </math>
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| More details on this technique can be found in ''[http://myreckonings.com/Dead_Reckoning/Online/Materials/General%20Method%20for%20Extracting%20Roots.pdf General Method for Extracting Roots using (Folded) Continued Fractions]''.
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| ==Higher dimensions==
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| Another meaning for '''''generalized continued fraction''''' is a generalization to higher dimensions. For example, there is a close relationship between the simple continued fraction in canonical form for the irrational real number α, and the way [[lattice point]]s in two dimensions lie to either side of the line ''y'' = α''x''. Generalizing this idea, one might ask about something related to lattice points in three or more dimensions. One reason to study this area is to quantify the [[mathematical coincidence]] idea; for example, for [[monomial]]s in several real numbers, take the [[logarithmic form]] and consider how small it can be. Another reason is to find a possible solution to [[Hermite's problem]].
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| There have been numerous attempts to construct a generalized theory. Notable efforts in this direction were made by [[Felix Klein]] (the [[Klein polyhedron]]), [[Georges Poitou]] and [[George Szekeres]].
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| ==See also==
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| * [[Gauss's continued fraction]]
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| * [[Padé table]]
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| * [[Solving quadratic equations with continued fractions]]
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| ==Notes==
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| {{reflist|2}}
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| ==References==
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| * {{citation | last1=Jones | first1=William B. | last2=Thron | first2=W.J. | title=Continued fractions. Analytic theory and applications | series=Encyclopedia of Mathematics and its Applications | volume=11 | location=Reading, MA | publisher=Addison-Wesley | year=1980 | isbn=0-201-13510-8 | zbl=0445.30003 }} (Covers both analytic theory and history).
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| * Lisa Lorentzen and Haakon Waadeland, ''Continued Fractions with Applications'', North Holland, 1992. ISBN 978-0-444-89265-2. (Covers primarily analytic theory and some arithmetic theory).
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| * [[Oskar Perron]], ''Die Lehre von den Kettenbrüchen'' Band I, II, B.G. Teubner, 1954.
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| * George Szekeres, ''Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 13'', "Multidimensional Continued Fractions", pp. 113–140, 1970.
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| * H.S. Wall, ''Analytic Theory of Continued Fractions'', Chelsea, 1973. ISBN 0-8284-0207-8. (This reprint of the D. Van Nostrand edition of 1948 covers both history and analytic theory.)
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| * {{Citation | last1=Press | first1=WH | last2=Teukolsky | first2=SA | last3=Vetterling | first3=WT | last4=Flannery | first4=BP | year=2007 | title=Numerical Recipes: The Art of Scientific Computing | edition=3rd | publisher=Cambridge University Press | publication-place=New York | isbn=978-0-521-88068-8 | chapter=Section 5.2. Evaluation of Continued Fractions | chapter-url=http://apps.nrbook.com/empanel/index.html?pg=206}}
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| * Manny Sardina, ''[http://myreckonings.com/Dead_Reckoning/Online/Materials/General%20Method%20for%20Extracting%20Roots.pdf General Method for Extracting Roots using (Folded) Continued Fractions]'', Surrey (UK), 2007.
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| ==External links==
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| * The [http://assets.cambridge.org/052181/8052/sample/0521818052ws.pdf first twenty pages] of Steven R. Finch, ''Mathematical Constants'', Cambridge University Press, 2003, ISBN 0-521-81805-2, contains generalized continued fractions for √2 and the golden mean.
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| *{{SloanesRef |sequencenumber=A133593|name=Exact continued fraction for Pi}}
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| {{DEFAULTSORT:Generalized Continued Fraction}}
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| [[Category:Continued fractions]]
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| [[he:שבר משולב]]
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| [[sl:posplošeni verižni ulomek]]
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