HOME

TheInfoList



OR:

In
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
, a branch of
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a generalized continued fraction is a generalization of regular
continued fraction In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer ...
s in canonical form, in which the partial numerators and partial denominators can assume arbitrary complex values. A generalized continued fraction is an expression of the form :x = b_0 + \cfrac where the () are the partial numerators, the are the partial denominators, and the leading term is called the ''integer'' part of the continued fraction. The successive convergents of the continued fraction are formed by applying the fundamental recurrence formulas: :\begin x_0 &= \frac = b_0, \\ pxx_1 &= \frac = \frac, \\ pxx_2 &= \frac = \frac,\ \dots \end where is the ''numerator'' and is the ''denominator'', called
continuant In phonetics, a continuant is a speech sound produced without a complete closure in the oral cavity, namely fricatives, approximants, vowels, and trills. While vowels are included in continuants, the term is often reserved for consonant sound ...
s, of the th convergent. They are given by the recursion :\begin A_n &= b_n A_ + a_n A_, \\ B_n &= b_n B_ + a_n B_ \qquad \text n \ge 1 \end with initial values :\begin A_ &= 1,& A_0&=b_0,\\ B_&=0, & B_0&=1. \end If the sequence of convergents approaches a
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
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 .


History

The story of continued fractions begins with the
Euclidean algorithm In mathematics, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's ''Topics in Algebra'' and Serge Lang's ''Algebra'', use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is an e ...
, a procedure for finding the
greatest common divisor In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is ...
of two natural numbers and . 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 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 Pietro Antonio Cataldi (15 April 1548, Bologna – 11 February 1626, Bologna) was an Italian mathematician. A citizen of Bologna, he taught mathematics and astronomy and also worked on military problems. His work included the development of conti ...
introduced the first formal notation for the generalized continued fraction. Cataldi represented a continued fraction as : \,\&\, \frac \,\&\, \frac \,\&\, \frac with the dots indicating where the next fraction goes, and each representing a modern plus sign. Late in the seventeenth century
John Wallis John Wallis (; la, Wallisius; ) was an English clergyman and mathematician who is given partial credit for the development of infinitesimal calculus. Between 1643 and 1689 he served as chief cryptographer for Parliament and, later, the royal ...
introduced the term "continued fraction" into mathematical literature. New techniques for mathematical analysis ( Newton's and Leibniz's
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
) had recently come onto the scene, and a generation of Wallis' contemporaries put the new phrase to use. In 1748
Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ...
published a theorem showing that a particular kind of continued fraction is equivalent to a certain very general
infinite series In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
. Euler's continued fraction formula is still the basis of many modern proofs of convergence of continued fractions. In 1761,
Johann Heinrich Lambert Johann Heinrich Lambert (, ''Jean-Henri Lambert'' in French; 26 or 28 August 1728 – 25 September 1777) was a polymath from the Republic of Mulhouse, generally referred to as either Swiss or French, who made important contributions to the subject ...
gave the first proof that is irrational, by using the following continued fraction for : :\tan(x) = \cfrac Continued fractions can also be applied to problems in
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...
, and are especially useful in the study of
Diophantine equation In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a ...
s. In the late eighteenth century
Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaPell's equation Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form x^2 - ny^2 = 1, where ''n'' is a given positive nonsquare integer, and integer solutions are sought for ''x'' and ''y''. In Cartesian coordinates, ...
, thus answering a question that had fascinated mathematicians for more than a thousand years. Amazingly, Lagrange's discovery implies that the canonical continued fraction expansion of the
square root In mathematics, a square root of a number is a number such that ; in other words, a number whose '' square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . ...
of every non-square integer is periodic and that, if the period is of length , it contains a palindromic string of length . In 1813 Gauss derived from complex-valued hypergeometric functions what is now called Gauss's continued fractions. They can be used to express many elementary functions and some more advanced functions (such as the
Bessel function Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrar ...
s), as continued fractions that are rapidly convergent almost everywhere in the complex plane.


Notation

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 is 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: : x = b_0+ \frac\, \frac\, \frac\cdots Pringsheim wrote a generalized continued fraction this way: : x = b_0 + \frac + \frac + \frac+\cdots.\,
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
evoked the more familiar
infinite product In mathematics, for a sequence of complex numbers ''a''1, ''a''2, ''a''3, ... the infinite product : \prod_^ a_n = a_1 a_2 a_3 \cdots is defined to be the limit of the partial products ''a''1''a''2...''a'n'' as ''n'' increases without bound. ...
when he devised this notation: : x = b_0 + \underset\overset\operatorname \frac.\, Here the "" 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 is zero, the infinite continued fraction : b_0 + \underset\overset\operatorname \frac\, is really just a finite continued fraction with fractional terms, and therefore a
rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...
of to and to . Such an object is of little interest from the point of view adopted in mathematical analysis, so it is usually assumed that all . There is no need to place this restriction on the partial denominators .


The determinant formula

When the th convergent of a continued fraction : x_n = b_0 + \underset\overset\operatorname \frac\, is expressed as a simple fraction we can use the ''determinant formula'' to relate the numerators and denominators of successive convergents and to one another. The proof for this can be easily seen by induction. Base case :The case results from a very simple computation. Inductive step :Assume that () holds for . Then we need to see the same relation holding true for . Substituting the value of and in () we obtain: :: \begin &=b_n A_ B_ + a_n A_ B_ - b_n A_ B_ - a_n A_ B_ \\ &=a_n(A_B_ - A_ B_) \end :which is true because of our induction hypothesis. :: A_B_n - A_nB_ = \left(-1\right)^na_1a_2\cdots a_n = \prod_^n (-a_i)\, :Specifically, if neither nor is zero () we can express the difference between the th and th convergents like this: :: x_ - x_n = \frac - \frac = \left(-1\right)^n \frac = \frac.\,


The equivalence transformation

If is any infinite sequence of non-zero complex numbers we can prove, by induction, that : b_0 + \cfrac = b_0 + \cfrac 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 are zero a sequence can be chosen to make each partial numerator a 1: : b_0 + \underset\overset\operatorname \frac = b_0 + \underset\overset\operatorname \frac\, where , , , and in general . Second, if none of the partial denominators are zero we can use a similar procedure to choose another sequence to make each partial denominator a 1: : b_0 + \underset\overset\operatorname \frac = b_0 + \underset\overset\operatorname \frac\, where and otherwise . These two special cases of the equivalence transformation are enormously useful when the general convergence problem is analyzed.


Notions of convergence

As mentioned in the introduction, the continued fraction : x = b_0 + \underset\overset\operatorname \frac\, converges if the sequence of convergents 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 \operatorname_^\infty \tfrac part of the fraction by , 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 \ such that the sequence of modified convergents converges for all \ sufficiently distinct from \. The sequence \ is then called an ''exceptional sequence'' for the continued fraction. See Chapter 2 of for a rigorous definition. There also exists a notion of
absolute convergence In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series \textstyle\sum_^\infty a_n is s ...
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 : f = \sum_n \left( f_n - f_\right), where f_n = \operatorname_^n \tfrac are the convergents of the continued fraction, converges absolutely. 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 when its convergents converge uniformly on ; that is, when for every there exists such that for all , for all z \in \Omega, : , f(z) - f_n(z), < \varepsilon.


Even and odd convergents

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 and , then the sequence must converge to one of these, and 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 , and the other converging to . 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 : x = \underset\overset\operatorname \frac\, is a continued fraction, then the even part and the odd part are given by : x_\text = \cfrac\, and : x_\text = a_1 - \cfrac\, respectively. More precisely, if the successive convergents of the continued fraction are , then the successive convergents of as written above are , and the successive convergents of are .


Conditions for irrationality

If and are positive integers with for all sufficiently large , then : x = b_0 + \underset\overset\operatorname \frac\, converges to an irrational limit.


Fundamental recurrence formulas

The partial numerators and denominators of the fraction's successive convergents are related by the ''fundamental recurrence formulas'': : \begin A_& = 1& B_& = 0\\ A_0& = b_0& B_0& = 1\\ A_& = b_ A_n + a_ A_& B_& = b_ B_n + a_ B_\, \end The continued fraction's successive convergents are then given by :x_n=\frac.\, These recurrence relations are due to
John Wallis John Wallis (; la, Wallisius; ) was an English clergyman and mathematician who is given partial credit for the development of infinitesimal calculus. Between 1643 and 1689 he served as chief cryptographer for Parliament and, later, the royal ...
(1616–1703) and
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries ...
(1707–1783). As an example, consider the regular continued fraction in canonical form that represents the golden ratio : :x = 1 + \cfrac Applying the fundamental recurrence formulas we find that the successive numerators are and the successive denominators are , the
Fibonacci number In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...
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.


Linear fractional transformations

A linear fractional transformation (LFT) is a complex function of the form : w = f(z) = \frac,\, where is a complex variable, and are arbitrary complex constants such that . An additional restriction that is customarily imposed, to rule out the cases in which is a constant. The linear fractional transformation, also known as a
Möbius transformation In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form f(z) = \frac of one complex variable ''z''; here the coefficients ''a'', ''b'', ''c'', ''d'' are complex numbers satisfying ''ad' ...
, has many fascinating properties. Four of these are of primary importance in developing the analytic theory of continued fractions. *If the LFT has one or two fixed points. This can be seen by considering the equation :: f(z) = z \Rightarrow dz^2 + cz = a + bz\, :which is clearly a
quadratic equation In algebra, a quadratic equation () is any equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where represents an unknown value, and , , and represent known numbers, where . (If and then the equation is linear, not qu ...
in . The roots of this equation are the fixed points of . If the
discriminant In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the orig ...
is zero the LFT fixes a single point; otherwise it has two fixed points. *If the LFT is an invertible
conformal map In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths. More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-preserving) at a point u_0\in ...
ping of the
extended complex plane In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers ...
onto itself. In other words, this LFT has an inverse function :: z = g(w) = \frac\, :such that for every point in the extended complex plane, and both and preserve angles and shapes at vanishingly small scales. From the form of we see that is also an LFT. *The composition of two different LFTs for which is itself an LFT for which . In other words, the set of all LFTs for which 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 In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
of the extended complex plane. *If the LFT reduces to :: w = f(z) = \frac,\, :which is a very simple
meromorphic function In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are poles of the function. The ...
of with one
simple pole In complex analysis (a branch of mathematics), a pole is a certain type of singularity of a complex-valued function of a complex variable. In some sense, it is the simplest type of singularity. Technically, a point is a pole of a function if ...
(at ) and a residue equal to . (See also
Laurent series In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion c ...
.)


The continued fraction as a composition of LFTs

Consider a sequence of simple linear fractional transformations :\begin \tau_0(z) &= b_0 + z, \\ px\tau_1(z) &= \frac, \\ px\tau_2(z) &= \frac,\\ px\tau_3(z) &= \frac,\\&\;\vdots \end Here we use to represent each simple LFT, and we adopt the conventional circle notation for composition of functions. We also introduce a new symbol to represent the composition of transformations ; that is, :\begin \boldsymbol_\boldsymbol(z) &= \tau_0\circ\tau_1(z) = \tau_0\big(\tau_1(z)\big),\\ \boldsymbol_\boldsymbol(z) &= \tau_0\circ\tau_1\circ\tau_2(z) = \tau_0\Big(\tau_1\big(\tau_2(z)\big)\Big),\, \end and so forth. By direct substitution from the first set of expressions into the second we see that : \begin \boldsymbol_\boldsymbol(z)& = \tau_0\circ\tau_1(z)& =&\quad b_0 + \cfrac\\ px\boldsymbol_\boldsymbol(z)& = \tau_0\circ\tau_1\circ\tau_2(z)& =&\quad b_0 + \cfrac\, \end and, in general, : \boldsymbol_\boldsymbol(z) = \tau_0\circ\tau_1\circ\tau_2\circ\cdots\circ\tau_n(z) = b_0 + \underset\overset\operatorname \frac\, where the last partial denominator in the finite continued fraction is understood to be . And, since , the image of the point under the iterated LFT is indeed the value of the finite continued fraction with partial numerators: : \boldsymbol_\boldsymbol(0) = \boldsymbol_\boldsymbol(\infty) = b_0 + \underset\overset\operatorname \frac.\,


A geometric interpretation

Defining a finite continued fraction as the image of a point under the iterated linear functional transformation leads to an intuitively appealing geometric interpretation of infinite continued fractions. The relationship : x_n = b_0 + \underset\overset\operatorname \frac = \frac = \boldsymbol_(0) = \boldsymbol_(\infty)\, can be understood by rewriting and in terms of the fundamental recurrence formulas: : \begin \boldsymbol_(z)& = \frac& \boldsymbol_(z)& = \frac;\\ px\boldsymbol_(z)& = \frac& \boldsymbol_(z)& = \frac .\, \end In the first of these equations the ratio tends toward as tends toward zero. In the second, the ratio tends toward as tends to infinity. This leads us to our first geometric interpretation. If the continued fraction converges, the successive convergents are eventually arbitrarily close together. Since the linear fractional transformation is a
continuous mapping In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in valu ...
, there must be a neighborhood of that is mapped into an arbitrarily small neighborhood of . Similarly, there must be a neighborhood of the point at infinity which is mapped into an arbitrarily small neighborhood of . So if the continued fraction converges the transformation maps both very small and very large into an arbitrarily small neighborhood of , the value of the continued fraction, as gets larger and larger. For intermediate values of , since the successive convergents are getting closer together we must have : \frac \approx \frac \quad\Rightarrow\quad \frac \approx \frac = k\, where is a constant, introduced for convenience. But then, by substituting in the expression for we obtain : \boldsymbol_(z) = \frac = \frac \left(\frac\right) \approx \frac \left(\frac\right) = \frac\, so that even the intermediate values of (except when ) are mapped into an arbitrarily small neighborhood of , the value of the continued fraction, as gets larger and larger. Intuitively, it is almost as if the convergent continued fraction maps the entire extended complex plane into a single point. Notice that the sequence lies within the
automorphism group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
of the extended complex plane, since each is a linear fractional transformation for which . And every member of that automorphism group maps the extended complex plane into itself: not one of the can possibly map the plane into a single point. Yet in the limit the sequence 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 of LFTs "focuses" the plane in the direction of , 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 , 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. For divergent continued fractions, we can distinguish three cases: #The two sequences and might themselves define two convergent continued fractions that have two different values, and . In this case the continued fraction defined by the sequence diverges by oscillation between two distinct limit points. And in fact this idea can be generalized: sequences 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 constitutes a
subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
of finite order within the group of automorphisms over the extended complex plane. # The sequence may produce an infinite number of zero denominators 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 diverges by oscillation with the point at infinity in this case. #The sequence may produce no more than a finite number of zero denominators . 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 : x = 1 + \cfrac\, where is any real number such that .


Euler's continued fraction formula

Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ...
proved the following identity: : a_0 + a_0a_1 + a_0a_1a_2 + \cdots + a_0a_1a_2\cdots a_n = \frac \frac \frac\cdots \frac.\, From this many other results can be derived, such as : \frac+ \frac+ \frac+ \cdots+ \frac = \frac \frac \frac\cdots \frac,\, and : \frac + \frac + \frac + \cdots + \frac = \frac \frac \frac\cdots \frac.\, Euler's formula connecting continued fractions and series is the motivation for the , 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 In mathematics, Euler's identity (also known as Euler's equation) is the equality e^ + 1 = 0 where : is Euler's number, the base of natural logarithms, : is the imaginary unit, which by definition satisfies , and : is pi, the ratio of the circ ...
. : e^x = \frac + \frac + \frac + \frac + \frac + \cdots = 1+\cfrac : \log(1+x) = \frac - \frac + \frac - \frac + \cdots =\cfrac Here are additional generalized continued fractions: : \arctan\cfrac=\cfrac =\cfrac : e^\frac = 1+\cfrac \quad\Rightarrow\quad e^2 = 7+\cfrac : \log \left( 1+\frac \right) = \cfrac = \cfrac This last is based on an algorithm derived by Aleksei Nikolaevich Khovansky in the 1970s.An alternative way to calculate log(x)
/ref> Example: the
natural logarithm of 2 The decimal value of the natural logarithm of 2 is approximately :\ln 2 \approx 0.693\,147\,180\,559\,945\,309\,417\,232\,121\,458. The logarithm of 2 in other bases is obtained with the formula :\log_b 2 = \frac. The common logarithm in particul ...
(= ≈ 0.693147...): : \log 2 = \log (1+1) = \cfrac = \cfrac


Here are three of 's best-known generalized continued fractions, the first and third of which are derived from their respective
arctangent In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). Spe ...
formulas above by setting and multiplying by 4. The Leibniz formula for : : \pi = \cfrac = \sum_^\infty \frac = \frac - \frac + \frac - \frac +- \cdots converges too slowly, requiring roughly terms to achieve correct decimal places. The series derived by
Nilakantha Somayaji Keļallur Nilakantha Somayaji (14 June 1444 – 1544), also referred to as Keļallur Comatiri, was a major mathematician and astronomer of the Kerala school of astronomy and mathematics. One of his most influential works was the comprehens ...
: : \pi = 3 + \cfrac = 3 - \sum_^\infty \frac = 3 + \frac - \frac + \frac -+ \cdots 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 . On the other hand: : \pi = \cfrac = 4 - 1 + \frac - \frac + \frac - \frac + \frac - \frac +- \cdots converges ''linearly'' to , adding at least three digits of precision per four terms, a pace slightly faster than the arcsine formula for : : \pi = 6 \sin^ \left( \frac \right) = \sum_^\infty \frac = \frac + \frac + \frac + \frac + \cdots\! which adds at least three decimal digits per five terms. *Note: this continued fraction's
rate of convergence In numerical analysis, the order of convergence and the rate of convergence of a convergent sequence are quantities that represent how quickly the sequence approaches its limit. A sequence (x_n) that converges to x^* is said to have ''order of c ...
tends to , hence tends to , whose
common logarithm In mathematics, the common logarithm is the logarithm with base 10. It is also known as the decadic logarithm and as the decimal logarithm, named after its base, or Briggsian logarithm, after Henry Briggs, an English mathematician who pioneered ...
is . The same (the silver ratio squared) also is observed in the ''unfolded'' general continued fractions of both the
natural logarithm of 2 The decimal value of the natural logarithm of 2 is approximately :\ln 2 \approx 0.693\,147\,180\,559\,945\,309\,417\,232\,121\,458. The logarithm of 2 in other bases is obtained with the formula :\log_b 2 = \frac. The common logarithm in particul ...
and the th root of 2 (which works for any integer ) if calculated using . For the ''folded'' general continued fractions of both expressions, the rate convergence , hence , whose common logarithm is , 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 cited above with the best-known Machin-like formula provides an even more rapidly, although still linearly, converging expression: :: \pi = 16 \tan^ \cfrac\, -\, 4 \tan^ \cfrac = \cfrac \, -\, \cfrac . with and .


Roots of positive numbers

The th root of any positive number can be expressed by restating , resulting in : \sqrt = \sqrt = x^m+\cfrac which can be simplified, by folding each pair of fractions into one fraction, to : \sqrt = x^m+\cfrac . The
square root In mathematics, a square root of a number is a number such that ; in other words, a number whose '' square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . ...
of is a special case with and : : \sqrt = \sqrt = x+\cfrac = x+\cfrac which can be simplified by noting that : : \sqrt = \sqrt = x+\cfrac = x+\cfrac . The square root can also be expressed by a periodic continued fraction, but the above form converges more quickly with the proper and .


Example 1

The
cube root of two In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only ...
(21/3 or ≈ 1.259921...) can be calculated in two ways: Firstly, "standard notation" of , , and : : \sqrt = 1+\cfrac = 1+\cfrac . Secondly, a rapid convergence with , and : : \sqrt = \cfrac+\cfrac = \cfrac+\cfrac .


Example 2

Pogson's ratio (1001/5 or ≈ 2.511886...), with , and : : \sqrt = \cfrac+\cfrac = \cfrac+\cfrac .


Example 3

The
twelfth root of two The twelfth root of two or \sqrt 2/math> (or equivalently 2^) is an algebraic irrational number, approximately equal to 1.0594631. It is most important in Western music theory, where it represents the frequency ratio ( musical interval) of a se ...
(21/12 or ≈ 1.059463...), using "standard notation": : \sqrt 2 = 1+\cfrac = 1+\cfrac .


Example 4

Equal temperament An equal temperament is a musical temperament or tuning system, which approximates just intervals by dividing an octave (or other interval) into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, ...
's
perfect fifth In music theory, a perfect fifth is the musical interval corresponding to a pair of pitches with a frequency ratio of 3:2, or very nearly so. In classical music from Western culture, a fifth is the interval from the first to the last of five ...
(27/12 or ≈ 1.498307...), with : With "standard notation": : \sqrt 2= 1+\cfrac = 1+\cfrac . A rapid convergence with , , and : :\sqrt 2= \cfrac \sqrt 2= \cfrac - \cfrac :\sqrt 2 = \cfrac - \cfrac. More details on this technique can be found in
General Method for Extracting Roots using (Folded) Continued Fractions
'.


Higher dimensions

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 , and the way lattice points in two dimensions lie to either side of the line . 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 In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered: # A monomial, also called power product, is a product of powers of variables with nonnegative integer expon ...
s in several real numbers, take the
logarithmic form In contexts including complex manifolds and algebraic geometry, a logarithmic differential form is a meromorphic differential form with poles of a certain kind. The concept was introduced by Deligne. Let ''X'' be a complex manifold, ''D'' ⊂ ''X' ...
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 Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and grou ...
(the Klein polyhedron), Georges Poitou and
George Szekeres George Szekeres AM FAA (; 29 May 1911 – 28 August 2005) was a Hungarian–Australian mathematician. Early years Szekeres was born in Budapest, Hungary, as Szekeres György and received his degree in chemistry at the Technical University of ...
.


See also

* Gauss's continued fraction *
Padé table In complex analysis, a Padé table is an array, possibly of infinite extent, of the rational Padé approximants :''R'm'', ''n'' to a given complex formal power series. Certain sequences of approximants lying within a Padé table can often b ...
* Solving quadratic equations with continued fractions * Convergence problem *
Infinite compositions of analytic functions In mathematics, infinite compositions of analytic functions (ICAF) offer alternative formulations of analytic continued fractions, series, products and other infinite expansions, and the theory evolving from such compositions may shed light on the ...


Notes


References

* * * * * * * * * * * * * (Covers both analytic theory and history.) * (Covers primarily analytic theory and some arithmetic theory.) * * * * * * * * (This reprint of the D. Van Nostrand edition of 1948 covers both history and analytic theory.) *


External links

* Th
first twenty pages
of Steven R. Finch, ''Mathematical Constants'',
Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambridge University Pr ...
, 2003, , contains generalized continued fractions for and the golden mean. * {{DEFAULTSORT:Generalized Continued Fraction Continued fractions he:שבר משולב