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.
A generalized continued fraction is an expression of the form
where the an 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, of the nth convergent. They are given by the recursion
with initial values
If the sequence of convergents 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, or it may produce an infinite number of zero denominators Bn.

History

The story of continued fractions begins with the Euclidean algorithm, 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 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. 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 introduced the term "continued fraction" into mathematical literature. New techniques for mathematical analysis 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. 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 is irrational, by using the following continued fraction for :
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. 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. They can be used to express many elementary functions and some more advanced functions, 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. So mathematicians have devised several alternative notations. One convenient way to express a generalized continued fraction looks like this:
Pringsheim wrote a generalized continued fraction this way:
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.

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 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 the first n ai's and the first bi'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 ai = 0. There is no need to place this restriction on the partial denominators bi.

The determinant formula

When the nth convergent of a continued fraction
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.
Base Case
It is trivially true.
Inductive Step
Assume the holds for.
Then we need to see the same relation holding true for.
Substituting the value of and in we obtain:
which is true because of our induction hypothesis.
Specifically, if neither Bn nor Bn-1 is zero we can express the difference between the n-1st and nth convergents like this:

The equivalence transformation

If = is any infinite sequence of non-zero complex numbers we can prove, by induction, that
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 ai are zero a sequence can be chosen to make each partial numerator a 1:
where c1 = 1/a1, c2 = a1/a2, c3 = a2/, and in general cn+1 = 1/.
Second, if none of the partial denominators bi are zero we can use a similar procedure to choose another sequence to make each partial denominator a 1:
where d1 = 1/b1 and otherwise dn+1 = 1/.
These two special cases of the equivalence transformation are enormously useful when the general convergence problem is analyzed.

Simple convergence concepts

It has already been noted that the continued fraction
converges if the sequence of convergents tends to a finite limit.
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.
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 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 is uniformly convergent on Ω. denotes the nth convergent of the continued fraction, evaluated at the point z inside Ω, and f
The Śleszyński–Pringsheim theorem provides a sufficient condition for convergence.

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 p and q, 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 p, and the other converging to q.
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
and
respectively. More precisely, if the successive convergents of the continued fraction x are, then the successive convergents of xeven as written above are, and the successive convergents of xodd are.

Conditions for irrationality

If and are positive integers with ≤ for all sufficiently large, then
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:
The continued fraction's successive convergents are then given by
These recurrence relations are due to John Wallis and Leonhard Euler.
As an example, consider the regular continued fraction in canonical form that represents the golden ratio φ:
Applying the fundamental recurrence formulas we find that the successive numerators An are and the successive denominators Bn are, 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 transformations

A linear fractional transformation is a complex function of the form
where z is a complex variable, and a, b, c, d are arbitrary complex constants such that. An additional restriction - that ad ≠ bc - is customarily imposed, to rule out the cases in which w = f 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.
Consider a sequence of simple linear fractional transformations
Here we use the Greek letter 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 little τs - that is,
and so forth. By direct substitution from the first set of expressions into the second we see that
and, in general,
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:

A geometric interpretation

Defining a finite continued fraction as the image of a point under the iterated linear functional transformation Τn leads to an intuitively appealing geometric interpretation of infinite continued fractions.
The relationship
can be understood by rewriting Τn and Τn+1 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 is a continuous mapping, there must be a neighborhood of z = 0 that is mapped into an arbitrarily small neighborhood of Τn = 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 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
where k is a constant, introduced for convenience. But then, by substituting in the expression for Τn we obtain
so that even the intermediate values of z 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.
Notice that the sequence lies within the automorphism group of the extended complex plane, since each Τn 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 Τns can possibly map the plane into a single point. Yet in the limit the sequence defines an infinite continued fraction which 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 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.
What about divergent continued fractions? Can those also be interpreted geometrically? In a word, yes. We distinguish three cases.
  1. The two sequences and might themselves define two convergent continued fractions that have two different values, xodd and xeven. 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 of finite order within the group of automorphisms over the extended complex plane.
  2. The sequence may produce an infinite number of zero denominators Bi 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.
  3. The sequence may produce no more than a finite number of zero denominators Bi. 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
where z is any real number such that z < −¼.

Euler's continued fraction formula

proved the following identity:
From this many other results can be derived, such as
and
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.
Here are additional generalized continued fractions:
This last is based on an algorithm derived by Alekseĭ Nikolaevich Khovanskiĭ in the 1970s.
Example: the natural logarithm of 2 :

Here are three of 's best-known generalized continued fractions, the first and third of which are derived from their respective formulas above by setting x=y=1 and multiplying by four. The Leibniz formula for :
converges too slowly, requiring roughly 3 x 10n terms to achieve n-decimal precision. 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. On the other hand:
converges linearly to, adding at least three decimals digits of precision per four terms, a pace slightly faster than the arcsine formula for :
which adds at least three decimal digits per five terms.
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:
where

Roots of positive numbers

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 of z is a special case of this nth root algorithm :
which can be simplified by noting that 5/10 = 3/6 = 1/2:
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.

Example 1

The cube root of two :
"Standard notation" of x = 1, y = 1, and 2z - y = 3:
Rapid convergence with x = 5, y = 3 and 2z - y = 253:

Example 2

, with x = 5, y = 75 and 2z - y = 6325:

Example 3

The twelfth root of two, using "standard notation":

Example 4

's perfect fifth, with m=7:
"Standard notation":
Rapid convergence with x = 3, y = -7153, and 2z - y = 219+312:
More details on this technique can be found in .

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 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, Georges Poitou and George Szekeres.