Generalized continued fraction

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search

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 (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]

with initial values

If the sequence of convergents {xn} 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 Bn.

History[edit]

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 Bombelli (1579) 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.[5] 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.[6] 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.[7] 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:[8]

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.[9] 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.[10] 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.

Notation[edit]

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:

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[edit]

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[edit]

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.

The determinant formula[edit]

When the nth convergent of a continued fraction

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

 

 

 

 

(1)

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

The case n = 1 results from a very simple computation.

Inductive step

Assume that (1) holds for n − 1. Then we need to see the same relation holding true for n. Substituting the value of An and Bn in (1) we obtain:
which is true because of our induction hypothesis.
Specifically, if neither Bn nor Bn − 1 is zero (n > 0) we can express the difference between the (n − 1)th and nth convergents like this:

The equivalence transformation[edit]

If {ci} = {c1, c2, c3, ...} 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 {ci} can be chosen to make each partial numerator a 1:

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

Second, if none of the partial denominators bi are zero we can use a similar procedure to choose another sequence {di} to make each partial denominator a 1:

where d1 = 1/b1 and otherwise dn + 1 = 1/bnbn + 1.

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

Notions of convergence[edit]

As mentioned in the introduction, the continued fraction

converges if the sequence of convergents {xn} 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 part of the fraction by wn, 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 Lorentzen & Waadeland (1992) 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

where are the convergents of the continued fraction, converges absolutely.[11] 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 Ω when its convergents converge uniformly on Ω; that is, when for every ε > 0 there exists M such that for all n > M, for all ,

Even and odd convergents[edit]

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 {x0, x2, x4, ...} must converge to one of these, and {x1, x3, x5, ...} 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 {x1, x2, x3, ...}, then the successive convergents of xeven as written above are {x2, x4, x6, ...}, and the successive convergents of xodd are {x1, x3, x5, ...}.[12]

Conditions for irrationality[edit]

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

converges to an irrational limit.[13]

Fundamental recurrence formulas[edit]

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 (1616–1703) and Leonhard Euler (1707–1783).[14]

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 {1, 2, 3, 5, 8, 13, ...} and the successive denominators Bn are {1, 1, 2, 3, 5, 8, ...}, 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[edit]

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.

  • If d ≠ 0 the LFT has one or two fixed points. This can be seen by considering the equation
which is clearly a quadratic equation in z. The roots of this equation are the fixed points of f(z). If the discriminant (cb)2 + 4ad is zero the LFT fixes a single point; otherwise it has two fixed points.
such that f(g(z)) = g(f(z)) = z for every point z in the extended complex plane, and both f and g preserve angles and shapes at vanishingly small scales. From the form of z = g(w) we see that g is also an LFT.
  • The composition of two different LFTs for which adbc is itself an LFT for which adbc. In other words, the set of all LFTs for which adbc is closed under composition of functions. The collection of all such LFTs, together with the "group operation" composition of functions, is known as the automorphism group of the extended complex plane.
  • If b = 0 the LFT reduces to
which is a very simple meromorphic function of z with one simple pole (at c/d) and a residue equal to a/d. (See also Laurent series.)

The continued fraction as a composition of LFTs[edit]

Consider a sequence of simple linear fractional transformations

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

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[edit]

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.

The relationship

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.[15]

Notice that the sequence {Τn} lies within the automorphism group of the extended complex plane, since each Τn is a linear fractional transformation for which abcd. And every member of that automorphism group maps the extended complex plane into itself: not one of the Τn can possibly map the plane into a single point. Yet in the limit the sequence {Τn} 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 {Τn} 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.[16]

For divergent continued fractions, we can distinguish three cases:

  1. The two sequences {Τ2n − 1} and {Τ2n} 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 {Τn} diverges by oscillation between two distinct limit points. And in fact this idea can be generalized: sequences {Τn} 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 {Τn} constitutes a subgroup of finite order within the group of automorphisms over the extended complex plane.
  2. The sequence {Τn} 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 {Τn} diverges by oscillation with the point at infinity in this case.[17]
  3. The sequence {Τn} 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 < −1/4.[18]

Euler's continued fraction formula[edit]

Euler proved the following identity:[7]

From this many other results can be derived, such as

and

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.

Examples[edit]

Transcendental functions and numbers[edit]

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 Aleksei Nikolaevich Khovansky in the 1970s.[19]

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...):[20]

π[edit]

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:

which adds at least three decimal digits per five terms.[21]

  • Note: this continued fraction's rate of convergence μ tends to 3 − 8 ≈ 0.1715729, hence 1/μ tends to 3 + 8 ≈ 5.828427, whose common logarithm is 0.7655... ≈ 13/17 > 3/4. The same 1/μ = 3 + 8 (the silver ratio squared) also is observed in the unfolded general continued fractions of both the natural logarithm of 2 and the nth root of 2 (which works for any integer n > 1) if calculated using 2 = 1 + 1. For the folded general continued fractions of both expressions, the rate convergence μ = (3 − 8)2 = 17 − 288 ≈ 0.02943725, hence 1/μ = (3 + 8)2 = 17 + 288 ≈ 33.97056, whose common logarithm is 1.531... ≈ 26/17 > 3/2, thus adding at least three digits per two terms. This is because the folded GCF folds each pair of fractions from the unfolded GCF into one fraction, thus doubling the convergence pace. The Manny Sardina reference further explains "folded" continued fractions.
  • Note: Using the continued fraction for arctan x/y cited above with the best-known Machin-like formula provides an even more rapidly, although still linearly, converging expression:

with u = 5 and v = 239.

Roots of positive numbers[edit]

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 with m = 1 and n = 2:

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[edit]

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

Firstly, "standard notation" of x = 1, y = 1, and 2zy = 3:

Secondly, a rapid convergence with x = 5, y = 3 and 2zy = 253:

Example 2[edit]

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

Example 3[edit]

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

Example 4[edit]

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

With "standard notation":

A rapid convergence with x = 3, y = −7153, and 2zy = 219 + 312:

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

Higher dimensions[edit]

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.

See also[edit]

Notes[edit]

  1. ^ Cusick & Flahive 1989.
  2. ^ Chrystal 1999.
  3. ^ Jones & Thron 1980, p. 20.
  4. ^ Euclid (2008) - The Euclidean algorithm generates a continued fraction as a by-product.
  5. ^ Cataldi 1613.
  6. ^ Wallis 1699.
  7. ^ a b Euler 1748, Chapter 18.
  8. ^ Havil 2012, pp. 104–105.
  9. ^ Brahmagupta (598–670) was the first mathematician to make a systematic study of Pell's equation.
  10. ^ Gauss 1813.
  11. ^ Lorentzen & Waadeland 1992.
  12. ^ Oskar Perron derives even more general extension and contraction formulas for continued fractions. See Perron (1977a), Perron (1977b).
  13. ^ Angell 2021.
  14. ^ Porubský 2008.
  15. ^ 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.
  16. ^ Because of analogies like this one, the theory of conformal mapping is sometimes described as "rubber sheet geometry".
  17. ^ One approach to the convergence problem is to construct positive definite continued fractions, for which the denominators Bi are never zero.
  18. ^ This periodic fraction of period one is discussed more fully in the article convergence problem.
  19. ^ An alternative way to calculate log(x)
  20. ^ Borwein, Crandall & Fee 2004, p. 278, 280.
  21. ^ Beckmann 1971.

References[edit]

  • Angell, David (2010). "A family of continued fractions" (PDF). Journal of Number Theory. Elsevier. 130 (4): 904–911. doi:10.1016/j.jnt.2009.12.003.
  • Angell, David (2021). Irrationality and Transcendence in Number Theory. Chapman and Hall/CRC. ISBN 9780367628376.
  • Chrystal, George (1999). Algebra, an Elementary Text-book for the Higher Classes of Secondary Schools and for Colleges: Pt. 1. American Mathematical Society. p. 500. ISBN 0-8218-1649-7.
  • Lorentzen, Lisa; Waadeland, Haakon (1992). Continued Fractions with Applications. Reading, MA: North Holland. ISBN 978-0-444-89265-2. (Covers primarily analytic theory and some arithmetic theory.)
  • Perron, Oskar (1977a) [1954]. Die Lehre von den Kettenbrüchen. Vol. Band I: Elementare Kettenbrüche (3 ed.). Vieweg + Teubner Verlag. ISBN 9783519020219.
  • Perron, Oskar (1977b) [1954]. Die Lehre von den Kettenbrüchen. Vol. Band II: Analytisch-funktionentheoretische Kettenbrüche (3 ed.). Vieweg + Teubner Verlag. ISBN 9783519020226.
  • Porubský, Štefan (2008). "Basic definitions for continued fractions". Interactive Information Portal for Algorithmic Mathematics. Prague, Czech Republic: Institute of Computer Science of the Czech Academy of Sciences. Retrieved 2 May 2022.
  • Szekeres, George (1970). "Multidimensional continued fractions". Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 13: 113–140.
  • Wall, Hubert Stanley (1967). Analytic Theory of Continued Fractions (Reprint ed.). Chelsea Pub Co. ISBN 0-8284-0207-8. (This reprint of the D. Van Nostrand edition of 1948 covers both history and analytic theory.)
  • Wallis, John (1699). Opera mathematica [Mathematical Works].

External links[edit]