Generalized continued fraction

In complex analysis, a branch of mathematics, a generalized continued fraction is a generalization of regular continued fractions in canonical form, in which the partial numerators and partial denominators can assume arbitrary complex values.

where the an (n > 0) are the partial numerators, the bn are the partial denominators, and the leading term b0 is called the integer part of the continued fraction.

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

where An is the numerator and Bn is the denominator, called continuants,[1][2] of the nth convergent. They are given by the recursion[3]

The story of continued fractions begins with the Euclidean algorithm,[4] 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 repeatedly.

Nearly two thousand years passed before Rafael Bombelli[5] devised a 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[6] for the generalized continued fraction. Cataldi represented a continued fraction as

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

Late in the seventeenth century John Wallis[7] introduced the term "continued fraction" into mathematical literature. 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.[8] 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, by using the following continued fraction for tan x:[9]

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.[10] Amazingly, 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 p > 1, it contains a palindromic string of length p − 1.

In 1813 Gauss derived from complex-valued hypergeometric functions what is now called Gauss's continued fractions.[11] 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.

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:

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

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.

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

If one of the partial numerators an + 1 is zero, the infinite continued fraction

is really just a finite continued fraction with n fractional terms, and therefore a rational function of a1 to an and b0 to bn + 1. Such an object is of little interest from the point of view adopted in mathematical analysis, so it is usually assumed that all ai ≠ 0. There is no need to place this restriction on the partial denominators bi.

is expressed as a simple fraction xn = An/Bn we can use the determinant formula

to relate the numerators and denominators of successive convergents xn and xn − 1 to one another. The proof for this can be easily seen by induction.

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.

where c1 = 1/a1, c2 = a1/a2, c3 = a2/a1a3, and in general cn + 1 = 1/an + 1cn.

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

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

diverges by oscillation if the series b1 + b2 + b3 + ... is absolutely convergent.[12]

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[13] might be defined as

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 open neighborhood Ω if the fraction's convergents converge uniformly at every point in Ω. Or, more precisely: if, for every ε > 0 an integer M can be found such that the absolute value of the difference

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 fn(z) denotes the nth 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.)

The Śleszyński–Pringsheim theorem provides a sufficient condition for convergence.

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

is a continued fraction, then the even part xeven and the odd part xodd are given by

If a1, a2,... and b1, b2,... are positive integers with akbk for all sufficiently large k, then

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

These recurrence relations are due to John Wallis (1616–1703) and Leonhard Euler (1707–1783).[16]

As an example, consider the regular continued fraction in canonical form that represents the golden ratio φ:

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

where z is a complex variable, and a, b, c, d are arbitrary complex constants such that c + dz ≠ 0. An additional restriction that adbc 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.

Here we use τ to represent each simple LFT, and we adopt the conventional circle notation for composition of functions. We also introduce a new symbol Τn to represent the composition of n + 1 transformations τi; that is,

and so forth. By direct substitution from the first set of expressions into the second we see that

where the last partial denominator in the finite continued fraction K is understood to be bn + z. And, since bn + 0 = bn, the image of the point z = 0 under the iterated LFT Τn is indeed the value of the finite continued fraction with n partial numerators:

Defining a finite continued fraction as the image of a point under the iterated linear functional transformation Τn(z) leads to an intuitively appealing geometric interpretation of infinite continued fractions.

can be understood by rewriting Τn(z) and Τn + 1(z) in terms of the fundamental recurrence formulas:

In the first of these equations the ratio tends toward An/Bn as z tends toward zero. In the second, the ratio tends toward An/Bn as z tends to infinity. This leads us to our first geometric interpretation. If the continued fraction converges, the successive convergents An/Bn are eventually arbitrarily close together. Since the linear fractional transformation Τn(z) is a continuous mapping, there must be a neighborhood of z = 0 that is mapped into an arbitrarily small neighborhood of Τn(0) = An/Bn. Similarly, there must be a neighborhood of the point at infinity which is mapped into an arbitrarily small neighborhood of Τn(∞) = An − 1/Bn − 1. So if the continued fraction converges the transformation Τn(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.

For intermediate values of z, since the successive convergents are getting closer together we must have

where k is a constant, introduced for convenience. But then, by substituting in the expression for Τn(z) we obtain

so that even the intermediate values of z (except when z ≈ −k−1) 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.[17]

Interesting examples of cases 1 and 3 can be constructed by studying the simple continued fraction

Euler's formula connecting continued fractions and series is the motivation for the fundamental inequalities[link or clarification needed], and also the basis of elementary approaches to the convergence problem.

Here are two continued fractions that can be built via Euler's identity.

This last is based on an algorithm derived by Aleksei Nikolaevich Khovansky in the 1970s.[21]

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...):[22]

Here are three of π's best-known generalized continued fractions, the first and third of which are derived from their respective arctangent formulas above by setting x = y = 1 and multiplying by 4. The Leibniz formula for π:

converges too slowly, requiring roughly 3 × 10n terms to achieve n correct decimal places. The series derived by Nilakantha Somayaji:

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 π. On the other hand:

converges linearly to π, adding at least three digits of precision per four terms, a pace slightly faster than the arcsine formula for π:

The nth root of any positive number zm can be expressed by restating z = xn + y, resulting in

which can be simplified, by folding each pair of fractions into one fraction, to

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.

The cube root of two (21/3 or 32 ≈ 1.259921...) can be calculated in two ways:

Pogson's ratio (1001/5 or 5100 ≈ 2.511886...), with x = 5, y = 75 and 2zy = 6325:

The twelfth root of two (21/12 or 122 ≈ 1.059463...), using "standard notation":

Equal temperament's perfect fifth (27/12 or 1227 ≈ 1.498307...), with m = 7:

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 points in two dimensions lie to either side of the line y = αx. 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.