Continued fraction

Template:Short description Template:Thumb}</math> |caption=An infinite continued fraction is defined by the sequences <math>\{a_i\},\{b_i\}</math>, for <math>i=0,1,2,\ldots</math>, with <math>a_0=0</math>. }}

A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or not, the continued fraction is finite or infinite.

Different fields of mathematics have different terminology and notation for continued fraction. In number theory the standard unqualified use of the term continued fraction refers to the special case where all numerators are 1, and is treated in the article simple continued fraction. The present article treats the case where numerators and denominators are sequences <math>\{a_i\},\{b_i\}</math> of constants or functions. From the perspective of number theory, these are called generalized continued fraction. From the perspective of complex analysis or numerical analysis, however, they are just standard, and in the present article they will simply be called "continued fraction".

FormulationEdit

A continued fraction is an expression of the form

<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>

where the Template:Math (Template:Math) are the partial numerators, the Template:Math are the partial denominators, and the leading term Template:Math is called the integer part of the continued fraction.

The successive convergents of the continued fraction are formed by applying the fundamental recurrence formulas:

<math>\begin{align}

x_0 &= \frac{A_0}{B_0} = b_0, \\ x_1 &= \frac{A_1}{B_1} = \frac{b_1b_0+a_1}{b_1}, \\ x_2 &= \frac{A_2}{B_2} = \frac{b_2(b_1b_0+a_1) + a_2b_0}{b_2b_1 + a_2},\ \dots \end{align}</math> where Template:Math is the numerator and Template:Math is the denominator, called continuants,Template:Sfn Template:Sfn of the Template:Mathth convergent. They are given by the three-term recurrence relation Template:Sfn

<math>\begin{align}

A_n &= b_n A_{n-1} + a_n A_{n-2}, \\ B_n &= b_n B_{n-1} + a_n B_{n-2} \qquad \text{for } n \ge 1 \end{align}</math> with initial values

<math>\begin{align}

A_{-1} &= 1,& A_0&=b_0,\\ B_{-1}&=0, & B_0&=1. \end{align}</math>

If the sequence of convergents Template:Math 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 Template:Math.

HistoryEdit

The story of continued fractions begins with the Euclidean algorithm,<ref>Template:Harvtxt - The Euclidean algorithm generates a continued fraction as a by-product.</ref> a procedure for finding the greatest common divisor of two natural numbers Template:Math and Template:Math. 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 Template:Harvtxt 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.Template:Sfn Cataldi represented a continued fraction as

with the dots indicating where the next fraction goes, and each Template:Math representing a modern plus sign.

Late in the seventeenth century John Wallis introduced the term "continued fraction" into mathematical literature.Template:Sfn 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.Template:Sfn 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 π is irrational#Lambert's proof|proof that Template:Pi is irrational]], by using the following continued fraction for Template:Math:Template:Sfn

<math>\tan(x) = \cfrac{x}{1 + \cfrac{-x^2}{3 + \cfrac{-x^2}{5 + \cfrac{-x^2}{7 + {}\ddots}}}}</math>

Continued fractions can also be applied to problems in number theory, and are especially useful in the study of Diophantine equations. In the late eighteenth century 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> Lagrange's discovery implies that the canonical continued fraction expansion of the square root of every non-square integer is periodic and that, if the period is of length Template:Math, it contains a palindromic string of length Template:Math.

In 1813 Gauss derived from complex-valued hypergeometric functions what is now called Gauss's continued fractions.Template:Sfn They can be used to express many elementary functions and some more advanced functions (such as the Bessel functions), as continued fractions that are rapidly convergent almost everywhere in the complex plane.

NotationEdit

The long continued fraction expression displayed in the introduction is easy for an unfamiliar reader to interpret. However, it takes up a lot of space and can be difficult to typeset. So mathematicians have devised several alternative notations. One convenient way to express a generalized continued fraction sets each nested fraction on the same line, indicating the nesting by dangling plus signs in the denominators:

<math>

x = b_0+ \frac{a_1}{b_1+}\, \frac{a_2}{b_2+}\, \frac{a_3}{b_3+ \cdots} </math>

Sometimes the plus signs are typeset to vertically align with the denominators but not under the fraction bars:

<math>

x = b_0 + \frac{a_1}{b_1}{{}\atop+} \frac{a_2}{b_2}{{}\atop+} \frac{a_3}{b_3}{{}\atop\!{}+\cdots} </math>

Pringsheim wrote a generalized continued fraction this way:

<math>

x = b_0 + {{}\atop{\big|\!}}\! \frac{a_1}{\,b_1\,} \!Template:\!\big + {{}\atop{\big|\!}}\! \frac{a_2}{\,b_2\,} \!Template:\!\big + {{}\atop{\big|\!}}\! \frac{a_3}{\,b_3\,} \!Template:\!\big + \cdots</math>

Carl Friedrich Gauss evoked the more familiar infinite product Template:Math when he devised this notation:

<math>

x = b_0 + \underset{i=1}\overset{\infty}\operatorname{K} \frac{a_i}{b_i}.\, </math>

Here the "Template:Math" 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.

Some elementary considerations

Here are some elementary results that are of fundamental importance in the further development of the analytic theory of continued fractions.

Partial numerators and denominators

If one of the partial numerators Template:Math is zero, the infinite continued fraction

<math>

b_0 + \underset{i=1}\overset{\infty}\operatorname{K} \frac{a_i}{b_i}\, </math>

is really just a finite continued fraction with Template:Mvar fractional terms, and therefore a rational function of Template:Math to Template:Mvar and Template:Math to Template:Math. Such an object is of little interest from the point of view adopted in mathematical analysis, so it is usually assumed that all Template:Math. There is no need to place this restriction on the partial denominators Template:Mvar.

The determinant formula

When the Template:Mathth convergent of a continued fraction

<math>

x_n = b_0 + \underset{i=1}\overset{n}\operatorname{K} \frac{a_i}{b_i}\, </math>

is expressed as a simple fraction Template:Math we can use the determinant formula

Template:NumBlk

to relate the numerators and denominators of successive convergents Template:Math and Template:Math to one another. The proof for this can be easily seen by induction.

Template:Collapse top Base case

The case Template:Math results from a very simple computation.

Inductive step

Assume that (Template:EquationNote) holds for Template:Math. Then we need to see the same relation holding true for Template:Math. Substituting the value of Template:Math and Template:Math in (Template:EquationNote) we obtain:

<math>

\begin{align}

&=b_n A_{n-1} B_{n-1} + a_n A_{n-1} B_{n-2} - b_n A_{n-1} B_{n-1} - a_n A_{n-2} B_{n-1} \\
&=a_n(A_{n-1}B_{n-2} - A_{n-2} B_{n-1})

\end{align} </math>

which is true because of our induction hypothesis.

<math>

A_{n-1}B_n - A_nB_{n-1} = \left(-1\right)^na_1a_2\cdots a_n = \prod_{i=1}^n (-a_i)\, </math>

Specifically, if neither Template:Math nor Template:Math is zero (Template:Math) we can express the difference between the Template:Mathth and Template:Mathth convergents like this:

<math>

x_{n-1} - x_n = \frac{A_{n-1}}{B_{n-1}} - \frac{A_n}{B_n} = \left(-1\right)^n \frac{a_1a_2\cdots a_n}{B_nB_{n-1}} = \frac{\prod_{i=1}^n (-a_i)}{B_nB_{n-1}}.\, </math> Template:Collapse bottom

The equivalence transformationEdit

If Template:Math is any infinite sequence of non-zero complex numbers we can prove, by induction, that

<math>

b_0 + \cfrac{a_1}{b_1 + \cfrac{a_2}{b_2 + \cfrac{a_3}{b_3 + \cfrac{a_4}{b_4 + \ddots\,}}}} = 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\,}}}} </math>

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.

The equivalence transformation is perfectly general, but two particular cases deserve special mention. First, if none of the Template:Mvar are zero, a sequence Template:Math can be chosen to make each partial numerator a 1:

<math>

b_0 + \underset{i=1}\overset{\infty}\operatorname{K} \frac{a_i}{b_i} = b_0 + \underset{i=1}\overset{\infty}\operatorname{K} \frac{1}{c_i b_i}\, </math>

where Template:Math, Template:Math, Template:Math, and in general Template:Math.

Second, if none of the partial denominators Template:Mvar are zero we can use a similar procedure to choose another sequence Template:Math to make each partial denominator a 1:

<math>

b_0 + \underset{i=1}\overset{\infty}\operatorname{K} \frac{a_i}{b_i} = b_0 + \underset{i=1}\overset{\infty}\operatorname{K} \frac{d_i a_i}{1}\, </math>

where Template:Math and otherwise Template:Math.

These two special cases of the equivalence transformation are enormously useful when the general convergence problem is analyzed.

Notions of convergenceEdit

As mentioned in the introduction, the continued fraction

<math>

x = b_0 + \underset{i=1}\overset{\infty}\operatorname{K} \frac{a_i}{b_i}\, </math>

converges if the sequence of convergents Template:Math} tends to a finite limit. This notion of convergence is very natural, but it is sometimes too restrictive. It is therefore useful to introduce the notion of general convergence of a continued fraction. Roughly speaking, this consists in replacing the <math>\operatorname{K}_{i = n}^\infty \tfrac{a_i}{b_i}</math> part of the fraction by Template:Math, instead of by 0, to compute the convergents. The convergents thus obtained are called modified convergents. We say that the continued fraction converges generally if there exists a sequence <math>\{w_n^*\}</math> such that the sequence of modified convergents converges for all <math>\{w_n\}</math> sufficiently distinct from <math>\{w_n^*\}</math>. The sequence <math>\{w_n^*\}</math> is then called an exceptional sequence for the continued fraction. See Chapter 2 of Template:Harvtxt for a rigorous definition.

There also exists a notion of absolute convergence for continued fractions, which is based on the notion of absolute convergence of a series: a continued fraction is said to be absolutely convergent when the series

<math> f = \sum_n \left( f_n - f_{n-1}\right),</math>

where <math>f_n = \operatorname{K}_{i = 1}^n \tfrac{a_i}{b_i}</math> are the convergents of the continued fraction, converges absolutely.Template:Sfn The Śleszyński–Pringsheim theorem provides a sufficient condition for absolute convergence.

Finally, a continued fraction of one or more complex variables is uniformly convergent in an open neighborhood Template:Math when its convergents converge uniformly on Template:Math; that is, when for every Template:Math there exists Template:Math such that for all Template:Math, for all <math>z \in \Omega</math>,

<math>

|f(z) - f_n(z)| < \varepsilon. </math>

Even and odd convergentsEdit

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 Template:Math and Template:Math, then the sequence Template:Math must converge to one of these, and Template:Math 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 Template:Math, and the other converging to Template:Math.

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

<math>

x = \underset{i=1}\overset{\infty}\operatorname{K} \frac{a_i}{1}\, </math>

is a continued fraction, then the even part Template:Math and the odd part Template:Math are given by

<math>

x_\text{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}}}}\, </math>

and

<math>

x_\text{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}}}}\, </math>

respectively. More precisely, if the successive convergents of the continued fraction Template:Math are Template:Math, then the successive convergents of Template:Math as written above are Template:Math, and the successive convergents of Template:Math are Template:Math.<ref>Oskar Perron derives even more general extension and contraction formulas for continued fractions. See Template:Harvtxt, Template:Harvtxt.</ref>

Conditions for irrationalityEdit

If Template:Math and Template:Math are positive integers with Template:Math for all sufficiently large Template:Math, then

<math>

x = b_0 + \underset{i=1}\overset{\infty}\operatorname{K} \frac{a_i}{b_i}\, </math>

converges to an irrational limit.Template:Sfn

Fundamental recurrence formulasEdit

The partial numerators and denominators of the fraction's successive convergents are related by the fundamental recurrence formulas:

<math>

\begin{align} A_{-1}& = 1& B_{-1}& = 0\\ A_0& = b_0& B_0& = 1\\ 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}\, \end{align} </math>

The continued fraction's successive convergents are then given by

These recurrence relations are due to John Wallis (1616–1703) and Leonhard Euler (1707–1783).Template:Sfn These recurrence relations are simply a different notation for the relations obtained by Pietro Antonio Cataldi (1548-1626).

As an example, consider the regular continued fraction in canonical form that represents the [[golden ratio|golden ratio Template:Mvar]]:

<math>\varphi = 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \ddots\,}}}} </math>

Applying the fundamental recurrence formulas we find that the successive numerators Template:Math are Template:Math and the successive denominators Template:Math are Template:Math, the Fibonacci numbers. 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.

Linear fractional transformationsEdit

A linear fractional transformation (LFT) is a complex function of the form

<math>

w = f(z) = \frac{az + b}{cz + d},\, </math>

where Template:Mvar is a complex variable, and Template:Math are arbitrary complex constants such that Template:Math. An additional restriction that Template:Math is customarily imposed, to rule out the cases in which Template:Math 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.

If Template:Math the LFT has one or two fixed points. This can be seen by considering the equation

<math>

f(z) = z \Rightarrow az + b = cz^2 + dz \Rightarrow cz^2 + (d-a)z - b = 0 , </math>

which is clearly a quadratic equation in Template:Mvar. The roots of this equation are the fixed points of Template:Math. If the discriminant Template:Math is zero the LFT fixes a single point; otherwise it has two fixed points.

If Template:Math the LFT is an invertible conformal mapping of the extended complex plane onto itself. In other words, this LFT has an inverse function

<math>

z = g(w) = \frac{\phantom{+}dw - b}{-cw + a}\, </math>

such that Template:Math for every point Template:Mvar in the extended complex plane, and both Template:Mvar and Template:Mvar preserve angles and shapes at vanishingly small scales. From the form of Template:Math we see that Template:Mvar is also an LFT.

The composition of two different LFTs for which Template:Math is itself an LFT for which Template:Math. In other words, the set of all LFTs for which Template:Math 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 Template:Math the LFT reduces to

<math>

w = f(z) = \frac{b}{cz + d},\, </math>

which is a very simple meromorphic function of Template:Mvar with one simple pole (at Template:Math) and a residue equal to Template:Math. (See also Laurent series.)

The continued fraction as a composition of LFTsEdit

Consider a sequence of simple linear fractional transformations

<math>\begin{align}

\tau_0(z) &= b_0 + z, \\[4px] \tau_1(z) &= \frac{a_1}{b_1 + z}, \\[4px] \tau_2(z) &= \frac{a_2}{b_2 + z},\\[4px] \tau_3(z) &= \frac{a_3}{b_3 + z},\\&\;\vdots \end{align}</math>

Here we use Template:Mvar to represent each simple LFT, and we adopt the conventional circle notation for composition of functions. We also introduce a new symbol Template:Math to represent the composition of Template:Math transformations Template:Mvar; that is,

<math>\begin{align}

\boldsymbol{\Tau}_\boldsymbol{1}(z) &= \tau_0\circ\tau_1(z) = \tau_0\big(\tau_1(z)\big),\\ \boldsymbol{\Tau}_\boldsymbol{2}(z) &= \tau_0\circ\tau_1\circ\tau_2(z) = \tau_0\Big(\tau_1\big(\tau_2(z)\big)\Big),\, \end{align}</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}\\[4px] \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}\operatorname{K} \frac{a_i}{b_i}\, </math>

where the last partial denominator in the finite continued fraction Template:Math is understood to be Template:Math. And, since Template:Math, the image of the point Template:Math under the iterated LFT Template:Math is indeed the value of the finite continued fraction with Template:Mvar partial numerators:

<math>

\boldsymbol{\Tau}_\boldsymbol{n}(0) = \boldsymbol{\Tau}_\boldsymbol{n+1}(\infty) = b_0 + \underset{i=1}\overset{n}\operatorname{K} \frac{a_i}{b_i}.\, </math>

A geometric interpretationEdit

Defining a finite continued fraction as the image of a point under the iterated linear fractional transformation Template:Math leads to an intuitively appealing geometric interpretation of infinite continued fractions.

The relationship

<math>

x_n = b_0 + \underset{i=1}\overset{n}\operatorname{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 Template:Math and Template:Math 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};\\[6px] \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 Template:Math as Template:Mvar tends toward zero. In the second, the ratio tends toward Template:Math as Template:Mvar tends to infinity. This leads us to our first geometric interpretation. If the continued fraction converges, the successive convergents Template:Math are eventually arbitrarily close together. Since the linear fractional transformation Template:Math is a continuous mapping, there must be a neighborhood of Template:Math that is mapped into an arbitrarily small neighborhood of Template:Math. Similarly, there must be a neighborhood of the point at infinity which is mapped into an arbitrarily small neighborhood of Template:Math. So if the continued fraction converges the transformation Template:Math maps both very small Template:Mvar and very large Template:Mvar into an arbitrarily small neighborhood of Template:Mvar, the value of the continued fraction, as Template:Mvar gets larger and larger.

For intermediate values of Template:Mvar, 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 Template:Mvar is a constant, introduced for convenience. But then, by substituting in the expression for Template:Math 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 Template:Mvar (except when Template:Math) are mapped into an arbitrarily small neighborhood of Template:Mvar, the value of the continued fraction, as Template:Mvar 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 an infinite 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 Template:Math lies within the automorphism group of the extended complex plane, since each Template:Math is a linear fractional transformation for which Template:Math. And every member of that automorphism group maps the extended complex plane into itself: not one of the Template:Math can possibly map the plane into a single point. Yet in the limit the sequence Template:Math defines an infinite continued fraction which (if it converges) represents a single point in the complex plane.

When an infinite continued fraction converges, the corresponding sequence Template:Math of LFTs "focuses" the plane in the direction of Template:Mvar, 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 Template:Mvar, 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 mapping is sometimes described as "rubber sheet geometry".</ref>

For divergent continued fractions, we can distinguish three cases:

The two sequences Template:Math and Template:Math might themselves define two convergent continued fractions that have two different values, Template:Math and Template:Math. In this case the continued fraction defined by the sequence Template:Math diverges by oscillation between two distinct limit points. And in fact this idea can be generalized: sequences Template:Math 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 Template:Math constitutes a subgroup of finite order within the group of automorphisms over the extended complex plane.
The sequence Template:Math may produce an infinite number of zero denominators Template:Mvar 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 Template:Math 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 Template:Mvar are never zero.</ref>
The sequence Template:Math may produce no more than a finite number of zero denominators Template:Mvar. 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 Template:Mvar is any real number such that Template:Math.<ref>This periodic fraction of period one is discussed more fully in the article convergence problem.</ref>

Euler's continued fraction formulaEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

Euler proved the following identity:Template:Sfn

<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 Template:Clarify, and also the basis of elementary approaches to the convergence problem.

ExamplesEdit

Transcendental functions and numbersEdit

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>

\arctan\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^\frac{x}{y} = 1+\cfrac{2x} {2y-x+\cfrac{x^2} {6y+\cfrac{x^2} {10y+\cfrac{x^2} {14y+\cfrac{x^2} {18y+\ddots}}}}} \quad\Rightarrow\quad 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 Aleksei Nikolaevich Khovansky in the 1970s.<ref>An alternative way to calculate log(x)</ref>

Example: the natural logarithm of 2 (= Template:Math ≈ 0.693147...):Template:Sfn

<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>

Template:PiEdit

Here are three of [[pi|Template:Pi's]] best-known generalized continued fractions, the first and third of which are derived from their respective arctangent formulas above by setting Template:Math and multiplying by 4. The [[Leibniz formula for π|Leibniz formula for Template: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 Template:Math terms to achieve Template:Math correct decimal places. 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 Template: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 Template:Pi, adding at least three digits of precision per four terms, a pace slightly faster than the [[Approximations of π#Arcsine|arcsine formula for Template: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.Template:Sfn

Note: this continued fraction's rate of convergence Template:Math tends to Template:Math, hence Template:Math tends to Template:Math, whose common logarithm is Template:Math. The same Template:Math (the silver ratio squared) also is observed in the unfolded general continued fractions of both the natural logarithm of 2 and the [[nth root|Template:Mathth root]] of 2 (which works for any integer Template:Math) if calculated using Template:Math. For the folded general continued fractions of both expressions, the rate convergence Template:Math, hence Template:Math, whose common logarithm is Template:Math, 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.
Note: Using the continued fraction for Template:Math cited above with the best-known Machin-like formula provides an even more rapidly, although still linearly, converging expression:

<math>

\pi = 16 \tan^{-1} \cfrac{1}{5}\, -\, 4 \tan^{-1} \cfrac{1}{239} = \cfrac{16} {u+\cfrac{1^2} {3u+\cfrac{2^2} {5u+\cfrac{3^2} {7u+\ddots}}}} \, -\, \cfrac{4} {v+\cfrac{1^2} {3v+\cfrac{2^2} {5v+\cfrac{3^2} {7v+\ddots}}}}. </math>

with Template:Math and Template:Math.

Roots of positive numbersEdit

The [[nth root|Template:Mathth root]] of any positive number Template:Math can be expressed by restating Template:Math, resulting in

<math>

\sqrt[n]{z^m} = \sqrt[n]{\left(x^n+y\right)^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(2x^n + y)-my-\cfrac{(1^2n^2-m^2)y^2} {3n(2x^n + y)-\cfrac{(2^2n^2-m^2)y^2} {5n(2x^n + y)-\cfrac{(3^2n^2-m^2)y^2} {7n(2x^n + y)-\cfrac{(4^2n^2-m^2)y^2} {9n(2x^n + y)-\ddots}}}}}. </math>

The square root of Template:Math is a special case with Template:Math and Template:Math:

<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(2x^2 + y)-y-\cfrac{1\cdot 3y^2} {6(2x^2 + y)-\cfrac{3\cdot 5y^2} {10(2x^2 + y)-\ddots}}} </math>

which can be simplified by noting that Template:Math:

<math>

\sqrt{z} = \sqrt{x^2+y} = x+\cfrac{y} {2x+\cfrac{y} {2x+\cfrac{y} {2x+\cfrac{y} {2x+\ddots}}}} = x+\cfrac{2x \cdot y} {2(2x^2 + y)-y-\cfrac{y^2} {2(2x^2 + y)-\cfrac{y^2} {2(2x^2 + y)-\ddots}}}. </math>

The square root can also be expressed by a periodic continued fraction, but the above form converges more quickly with the proper Template:Math and Template:Math.

Example 1Edit

The cube root of two (2^1/3 or Template:Radic ≈ 1.259921...) can be calculated in two ways:

Firstly, "standard notation" of Template:Math, Template:Math, and Template:Math:

<math>

\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}}}}}. </math>

Secondly, a rapid convergence with Template:Math, Template:Math and Template:Math:

<math>

\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}}}}. </math>

Example 2Edit

Pogson's ratio (100^1/5 or Template:Radic ≈ 2.511886...), with Template:Math, Template:Math and Template:Math:

<math>

\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}}}}. </math>

Example 3Edit

The twelfth root of two (2^1/12 or Template:Radic ≈ 1.059463...), using "standard notation":

<math>

\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}}}}}. </math>

Example 4Edit

Equal temperament's perfect fifth (2^7/12 or Template:Radic ≈ 1.498307...), with Template:Math:

With "standard notation":

<math>

\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}}}}}. </math>

A rapid convergence with Template:Math, Template:Math, and Template:Math:

<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}

- \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>

<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})

- \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>

More details on this technique can be found in General Method for Extracting Roots using (Folded) Continued Fractions.

Higher dimensionsEdit

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 Template:Math, and the way lattice points in two dimensions lie to either side of the line Template:Math. 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 monomials 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.

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.